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Deck 6: The Trigonometric Functions
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Question
Simplify the expression. 
A)
B)
C)
D)
E)
A)
B)
C)
D)
E)
Question
Approximate, to the nearest 0.01 radian, all angles 
in the interval [ 0, 2 
) that satisfy the equation. 
= 0.8412
A)
B)
C)
D)
E)
in the interval [ 0, 2 ) that satisfy the equation. = 0.8412A)
B)
C)
D)
E)
Question
Verify the identity by transforming the left-hand side into the right-hand side. 
A)False
B)True
A)False
B)True
Question
Express 
= 2 in terms of degrees, minutes, and seconds, to the nearest second.
A)
B)
C)
D)
E)
= 2 in terms of degrees, minutes, and seconds, to the nearest second.A)
B)
C)
D)
E)
Question
Find the quadrant containing 
if the given conditions are true. 
and 
A)IV
B)II
C)I
D)III
if the given conditions are true. and A)IV
B)II
C)I
D)III
Question
Let P ( t ) be the point on the unit circle U that corresponds to t. If P ( t ) has the coordinates 
, find 
, 
, 
, 
.
A)
B)
C)
D)
E)
, find , , , .A)
B)
C)
D)
E)
Question
If a circular arc of the length s = 11 cm subtends the central angle 
= 2 on a circle, find the radius of the circle.
A)6.5 cm
B)5.5 cm
C)7 cm
D)22 cm
E)6 cm
= 2 on a circle, find the radius of the circle.A)6.5 cm
B)5.5 cm
C)7 cm
D)22 cm
E)6 cm
Question
Two stars that are very close may appear to be one. The ability of a telescope to separate their images is called its resolution. The smaller the resolution, the better a telescope's ability to separate images in the sky. In a refracting telescope, resolution 
( see the figure ) can be improved by using a lens with a larger diameter D. The relationship between 
in degrees and D in meters is given by 
, where 
is the wavelength of light in meters. The largest refracting telescope in the world is at the University of Chicago. At a wavelength of 
meter, its resolution is 
. Approximate the diameter of the lens to the nearest hundredth of a meter. 
A)0.02 meters
B)1.17 meters
C)0.97 meters
D)0.77 meters
E)1.02 meters
( see the figure ) can be improved by using a lens with a larger diameter D. The relationship between in degrees and D in meters is given by , where is the wavelength of light in meters. The largest refracting telescope in the world is at the University of Chicago. At a wavelength of meter, its resolution is . Approximate the diameter of the lens to the nearest hundredth of a meter. A)0.02 meters
B)1.17 meters
C)0.97 meters
D)0.77 meters
E)1.02 meters
Question
Scientists sometimes use the formula 
to simulate temperature variations during the day, with time t in hours, temperature f ( t ) in 
, and t=0 corresponding to midnight. Assume that f ( t ) is decreasing at midnight. Determine values of a, b, c, and d that fit the information: the high temperature of 32 
occurs at 2 P.M., and the average temperature of 23 
occurs 6 hours later.
A)
B)
C)
D)
E)
to simulate temperature variations during the day, with time t in hours, temperature f ( t ) in , and t=0 corresponding to midnight. Assume that f ( t ) is decreasing at midnight. Determine values of a, b, c, and d that fit the information: the high temperature of 32 occurs at 2 P.M., and the average temperature of 23 occurs 6 hours later.A)
B)
C)
D)
E)
Question
Find the period of the equation. 
A)
B)
C)
D)
E)
A)
B)
C)
D)
E)
Question
Find the amplitude, the period, and the phase shift. 
A)
B)
C)
D)
E)
A)
B)
C)
D)
E)
Question
Express the angle as a decimal, to the nearest ten-thousandth of a degree. 
A)
B)
C)
D)
E)
A)
B)
C)
D)
E)
Question
Points on the terminal sides of angles play an important part in the design of arms for robots. Suppose a robot has a straight arm 18 inches long that can rotate about the origin in a coordinate plane. If the robot's hand is located at (18, 0) and then rotates through an angle of 60 o , what is the new location of the hand?
A)
B)
C)
D)
E)
A)
B)
C)
D)
E)
Question
Approximate to four decimal places. 
A)0.2683
B)0.2713
C)0.2837
D)0.5425
E)0.2327
A)0.2683
B)0.2713
C)0.2837
D)0.5425
E)0.2327
Question
Approximate the acute angle 
to the nearest 
. 
A)
B)
C)
D)
E)
to the nearest . A)
B)
C)
D)
E)
Question
The popular biorhythm theory uses the graphs of three simple sine functions to make predictions about an individual's physical, emotional, and intellectual potential for a particular day. The graphs are given by 
with t in days and t = 0 corresponding to birth. Find the value of b for the physical cycle, which has a period of 19 days.
A)
B)
C)
D)
E)
with t in days and t = 0 corresponding to birth. Find the value of b for the physical cycle, which has a period of 19 days.A)
B)
C)
D)
E)
Question
Refer to the graph of y = cos x to find the separate values of x in the interval 
that satisfy the equation. 
A)
B)
C)
D)
E)
that satisfy the equation. A)
B)
C)
D)
E)
Question
The amount of sunshine illuminating a wall of a building can greatly affect the energy efficiency of the building. The solar radiation striking a vertical wall that faces east is given by the formula 
where R 0 is the maximum solar radiation possible, 
is the angle that the sun makes with the horizontal, and 
is the direction of the sun in the sky, with 
= 90 o when the sun is in the east and 
= 0 o when the sun is in the south. What percentage of R 0 is striking the wall when 
is equal to 30 o and the sun is in the southeast?
Round the answer to the nearest integer.
A)71 %
B)61 %
C)56 %
D)76 %
E)29 %
where R 0 is the maximum solar radiation possible, is the angle that the sun makes with the horizontal, and is the direction of the sun in the sky, with = 90 o when the sun is in the east and = 0 o when the sun is in the south. What percentage of R 0 is striking the wall when is equal to 30 o and the sun is in the southeast?Round the answer to the nearest integer.
A)71 %
B)61 %
C)56 %
D)76 %
E)29 %
Question
As 
for some real number 
. Use a graph to predict 
. 
A)
B)
C)
D)
E)
for some real number . Use a graph to predict . A)
B)
C)
D)
E)
Question
On a certain spring day with 12 hours of daylight, the light intensity I takes on its largest value of 390 calories/cm 2 at midday. If t = 0 corresponds to sunrise and t is measured in hours, find a formula I = a sin bt that fits this information.
A)
B)
C)
D)
E)
A)
B)
C)
D)
E)
Question
Simplify the expression. 
A)
B)
C)
D)
E)
A)
B)
C)
D)
E)
Question
Approximate to four decimal places. 
A)0.2837
B)0.2713
C)0.2683
D)0.2327
E)0.5425
A)0.2837
B)0.2713
C)0.2683
D)0.2327
E)0.5425
Question
Express 
= 4.5 in terms of degrees, minutes, and seconds, to the nearest second.
A)
B)
C)
D)
E)
= 4.5 in terms of degrees, minutes, and seconds, to the nearest second.A)
B)
C)
D)
E)
Question
Approximate the acute angle 
to the nearest 
. 
A)
B)
C)
D)
E)
to the nearest . A)
B)
C)
D)
E)
Question
Let P ( t ) be the point on the unit circle U that corresponds to t. If P ( t ) has the coordinates 
, find 
, 
, 
, 
.
A)
B)
C)
D)
E)
, find , , , .A)
B)
C)
D)
E)
Question
An airplane flying at a speed of 360 mi/hr flies from a point A in the direction 
for 15 minutes and then flies in the direction 
for 15 minutes. Approximate, to the nearest mile, the distance from the airplane to A.
A)104 mi
B)127 mi
C)1 mi
D)254 mi
E)180 mi
for 15 minutes and then flies in the direction for 15 minutes. Approximate, to the nearest mile, the distance from the airplane to A.A)104 mi
B)127 mi
C)1 mi
D)254 mi
E)180 mi
Question
Points on the terminal sides of angles play an important part in the design of arms for robots. Suppose a robot has a straight arm 20 inches long that can rotate about the origin in a coordinate plane. If the robot's hand is located at (20, 0) and then rotates through an angle of 60 o , what is the new location of the hand?
A)
B)
C)
D)
E)
A)
B)
C)
D)
E)
Question
Verify the identity by transforming the left-hand side into the right-hand side. 
A)False
B)True
A)False
B)True
Question
Radio stations often have more than one broadcasting tower because federal guidelines do not usually permit a radio station to broadcast its signal in all directions with equal power. Since radio waves can travel over long distances, it is important to control their directional patterns so that radio stations do not interfere with one another. Suppose that a radio station has two broadcasting towers located along a north-south line, as shown in the figure. If the radio station is broadcasting at a wavelength 
and the distance between the two radio towers is equal to 
, then the intensity I of the signal in the direction 
is given by 
where I o is the maximum intensity. 
Approximate I in terms of I o for 
.
A)0.055I o
B)I o
C)0.033I o
D)0.066I o
E)0.044I o
and the distance between the two radio towers is equal to , then the intensity I of the signal in the direction is given by where I o is the maximum intensity. Approximate I in terms of I o for .A)0.055I o
B)I o
C)0.033I o
D)0.066I o
E)0.044I o
Question
Refer to the graph of y = cos x to find the separate values of x in the interval 
that satisfy the equation. 
A)
B)
C)
D)
E)
that satisfy the equation. A)
B)
C)
D)
E)
Question
A ship leaves port at 4:00 P.M. and sails in the direction 
at a rate of 19 mi/hr. Another ship leaves port at 4:30 P.M. and sails in the direction 
at a rate of 16 mi/hr. Approximately how far apart are the ships at 7:00 P.M.?
A)70 mi
B)17 mi
C)75 mi
D)25 mi
E)140 mi
at a rate of 19 mi/hr. Another ship leaves port at 4:30 P.M. and sails in the direction at a rate of 16 mi/hr. Approximately how far apart are the ships at 7:00 P.M.?A)70 mi
B)17 mi
C)75 mi
D)25 mi
E)140 mi
Question
If a circular arc of the length s = 13.5 cm subtends the central angle 
= 3 on a circle, find the radius of the circle.
A)4.5 cm
B)5.5 cm
C)6 cm
D)40.5 cm
E)5 cm
= 3 on a circle, find the radius of the circle.A)4.5 cm
B)5.5 cm
C)6 cm
D)40.5 cm
E)5 cm
Question
The amount of sunshine illuminating a wall of a building can greatly affect the energy efficiency of the building. The solar radiation striking a vertical wall that faces east is given by the formula 
where R 0 is the maximum solar radiation possible, 
is the angle that the sun makes with the horizontal, and 
is the direction of the sun in the sky, with 
= 90 o when the sun is in the east and 
= 0 o when the sun is in the south. What percentage of R 0 is striking the wall when 
is equal to 60 o and the sun is in the southeast?
Round the answer to the nearest integer.
A)35 %
B)30 %
C)20 %
D)45 %
E)59 %
where R 0 is the maximum solar radiation possible, is the angle that the sun makes with the horizontal, and is the direction of the sun in the sky, with = 90 o when the sun is in the east and = 0 o when the sun is in the south. What percentage of R 0 is striking the wall when is equal to 60 o and the sun is in the southeast?Round the answer to the nearest integer.
A)35 %
B)30 %
C)20 %
D)45 %
E)59 %
Question
Express the angle as a decimal, to the nearest ten-thousandth of a degree. 
A)
B)
C)
D)
E)
A)
B)
C)
D)
E)
Question
Two stars that are very close may appear to be one. The ability of a telescope to separate their images is called its resolution. The smaller the resolution, the better a telescope's ability to separate images in the sky. In a refracting telescope, resolution 
( see the figure ) can be improved by using a lens with a larger diameter D. The relationship between 
in degrees and D in meters is given by 
, where 
is the wavelength of light in meters. The largest refracting telescope in the world is at the University of Chicago. At a wavelength of 
meter, its resolution is 
. Approximate the diameter of the lens to the nearest hundredth of a meter. 
A)1.17 meters
B)0.77 meters
C)0.02 meters
D)0.97 meters
E)1.02 meters
( see the figure ) can be improved by using a lens with a larger diameter D. The relationship between in degrees and D in meters is given by , where is the wavelength of light in meters. The largest refracting telescope in the world is at the University of Chicago. At a wavelength of meter, its resolution is . Approximate the diameter of the lens to the nearest hundredth of a meter. A)1.17 meters
B)0.77 meters
C)0.02 meters
D)0.97 meters
E)1.02 meters
Question
Approximate, to the nearest 0.01 radian, all angles 
in the interval [ 0, 2 
) that satisfy the equation. 
= 0.2518
A)
B)
C)
D)
E)
in the interval [ 0, 2 ) that satisfy the equation. = 0.2518A)
B)
C)
D)
E)
Question
A CB antenna is located on the top of a garage that is 17 feet tall. From a point on level ground that is 120 feet from a point directly below the antenna, the antenna subtends an angle of 
, as shown in the figure. Approximate the length of the antenna. Give the answer to one decimal place. 
A)345.6 ft
B)26.8 ft
C)24.2 ft
D)36.8 ft
E)95.7 ft
, as shown in the figure. Approximate the length of the antenna. Give the answer to one decimal place. A)345.6 ft
B)26.8 ft
C)24.2 ft
D)36.8 ft
E)95.7 ft
Question
Find the quadrant containing 
if the given conditions are true. 
and 
A)I
B)IV
C)II
D)III
if the given conditions are true. and A)I
B)IV
C)II
D)III
Question
Use the graph of a trigonometric function to aid in sketching the graph of the equation without plotting points. 
A)
B)
C)
D)
E)
A)
B)
C)
D)
E)
Question
As 
for some real number 
. Use a graph to predict 
. 
A)
B)
C)
D)
E)
for some real number . Use a graph to predict . A)
B)
C)
D)
E)
Question
Let P ( t ) be the point on the unit circle U that corresponds to t. If P ( t ) has the coordinates 
, find 
, 
, 
, 
.
A)
B)
C)
D)
E)
, find , , , .A)
B)
C)
D)
E)
Question
Two stars that are very close may appear to be one. The ability of a telescope to separate their images is called its resolution. The smaller the resolution, the better a telescope's ability to separate images in the sky. In a refracting telescope, resolution 
( see the figure ) can be improved by using a lens with a larger diameter D. The relationship between 
in degrees and D in meters is given by 
, where 
is the wavelength of light in meters. The largest refracting telescope in the world is at the University of Chicago. At a wavelength of 
meter, its resolution is 
. Approximate the diameter of the lens to the nearest hundredth of a meter. 
A)0.02 meters
B)0.77 meters
C)1.17 meters
D)0.97 meters
E)1.02 meters
( see the figure ) can be improved by using a lens with a larger diameter D. The relationship between in degrees and D in meters is given by , where is the wavelength of light in meters. The largest refracting telescope in the world is at the University of Chicago. At a wavelength of meter, its resolution is . Approximate the diameter of the lens to the nearest hundredth of a meter. A)0.02 meters
B)0.77 meters
C)1.17 meters
D)0.97 meters
E)1.02 meters
Question
If a circular arc of the length s = 16 cm subtends the central angle 
= 2 on a circle, find the radius of the circle.
A)8.5 cm
B)32 cm
C)8 cm
D)9 cm
E)9.5 cm
= 2 on a circle, find the radius of the circle.A)8.5 cm
B)32 cm
C)8 cm
D)9 cm
E)9.5 cm
Question
On a certain spring day with 12 hours of daylight, the light intensity I takes on its largest value of 480 calories/cm 2 at midday. If t = 0 corresponds to sunrise and t is measured in hours, find a formula I = a sin bt that fits this information.
A)
B)
C)
D)
E)
A)
B)
C)
D)
E)
Question
Scientists sometimes use the formula 
to simulate temperature variations during the day, with time t in hours, temperature f ( t ) in 
, and t=0 corresponding to midnight. Assume that f ( t ) is decreasing at midnight. Determine values of a, b, c, and d that fit the information: the high temperature of 29 
occurs at 2 P.M., and the average temperature of 21 
occurs 6 hours later.
A)
B)
C)
D)
E)
to simulate temperature variations during the day, with time t in hours, temperature f ( t ) in , and t=0 corresponding to midnight. Assume that f ( t ) is decreasing at midnight. Determine values of a, b, c, and d that fit the information: the high temperature of 29 occurs at 2 P.M., and the average temperature of 21 occurs 6 hours later.A)
B)
C)
D)
E)
Question
A ship leaves port at 4:00 P.M. and sails in the direction 
at a rate of 24 mi/hr. Another ship leaves port at 4:30 P.M. and sails in the direction 
at a rate of 22 mi/hr. Approximately how far apart are the ships at 7:00 P.M.?
A)17 mi
B)182 mi
C)33 mi
D)98 mi
E)91 mi
at a rate of 24 mi/hr. Another ship leaves port at 4:30 P.M. and sails in the direction at a rate of 22 mi/hr. Approximately how far apart are the ships at 7:00 P.M.?A)17 mi
B)182 mi
C)33 mi
D)98 mi
E)91 mi
Question
As 
for some real number 
. Use a graph to predict 
. 
A)
B)
C)
D)
E)
for some real number . Use a graph to predict . A)
B)
C)
D)
E)
Question
Find the period of the equation. 
A)
B)
C)
D)
E)
A)
B)
C)
D)
E)
Question
A CB antenna is located on the top of a garage that is 17 feet tall. From a point on level ground that is 120 feet from a point directly below the antenna, the antenna subtends an angle of 
, as shown in the figure. Approximate the length of the antenna. Give the answer to one decimal place. 
A)26.8 ft
B)345.6 ft
C)95.7 ft
D)36.8 ft
E)24.2 ft
, as shown in the figure. Approximate the length of the antenna. Give the answer to one decimal place. A)26.8 ft
B)345.6 ft
C)95.7 ft
D)36.8 ft
E)24.2 ft
Question
Express 
= 2.5 in terms of degrees, minutes, and seconds, to the nearest second.
A)
B)
C)
D)
E)
= 2.5 in terms of degrees, minutes, and seconds, to the nearest second.A)
B)
C)
D)
E)
Question
Simplify the expression. 
A)
B)
C)
D)
E)
A)
B)
C)
D)
E)
Question
The popular biorhythm theory uses the graphs of three simple sine functions to make predictions about an individual's physical, emotional, and intellectual potential for a particular day. The graphs are given by 
with t in days and t = 0 corresponding to birth. Find the value of b for the physical cycle, which has a period of 29 days.
A)
B)
C)
D)
E)
with t in days and t = 0 corresponding to birth. Find the value of b for the physical cycle, which has a period of 29 days.A)
B)
C)
D)
E)
Question
Radio stations often have more than one broadcasting tower because federal guidelines do not usually permit a radio station to broadcast its signal in all directions with equal power. Since radio waves can travel over long distances, it is important to control their directional patterns so that radio stations do not interfere with one another. Suppose that a radio station has two broadcasting towers located along a north-south line, as shown in the figure. If the radio station is broadcasting at a wavelength 
and the distance between the two radio towers is equal to 
, then the intensity I of the signal in the direction 
is given by 
where I o is the maximum intensity. 
Approximate I in terms of I o for 
.
A)0.055I o
B)0.044I o
C)0.066I o
D)I o
E)0.033I o
and the distance between the two radio towers is equal to , then the intensity I of the signal in the direction is given by where I o is the maximum intensity. Approximate I in terms of I o for .A)0.055I o
B)0.044I o
C)0.066I o
D)I o
E)0.033I o
Question
Refer to the graph of y = cos x to find the separate values of x in the interval 
that satisfy the equation. 
A)
B)
C)
D)
E)
that satisfy the equation. A)
B)
C)
D)
E)
Question
Use the graph of a trigonometric function to aid in sketching the graph of the equation without plotting points. 
A)
B)
C)
D)
E)
A)
B)
C)
D)
E)
Question
Find the amplitude, the period, and the phase shift. 
A)
B)
C)
D)
E)
A)
B)
C)
D)
E)
Question
Express the angle as a decimal, to the nearest ten-thousandth of a degree. 
A)
B)
C)
D)
E)
A)
B)
C)
D)
E)
Question
An airplane flying at a speed of 360 mi/hr flies from a point A in the direction 
for 45 minutes and then flies in the direction 
for 45 minutes. Approximate, to the nearest mile, the distance from the airplane to A.
A)764 mi
B)540 mi
C)180 mi
D)1 mi
E)382 mi
for 45 minutes and then flies in the direction for 45 minutes. Approximate, to the nearest mile, the distance from the airplane to A.A)764 mi
B)540 mi
C)180 mi
D)1 mi
E)382 mi
Question
Verify the identity by transforming the left-hand side into the right-hand side. 
A)False
B)True
A)False
B)True
Question
Find the quadrant containing 
if the given conditions are true. 
and 
A)IV
B)III
C)II
D)I
if the given conditions are true. and A)IV
B)III
C)II
D)I
Question
A ship leaves port at 1:00 P.M. and sails in the direction 
at a rate of 20 mi/hr. Another ship leaves port at 1:30 P.M. and sails in the direction 
at a rate of 18 mi/hr. Approximately how far apart are the ships at 4:00 P.M.?
A)75 mi
B)150 mi
C)27 mi
D)81 mi
E)15 mi
at a rate of 20 mi/hr. Another ship leaves port at 1:30 P.M. and sails in the direction at a rate of 18 mi/hr. Approximately how far apart are the ships at 4:00 P.M.?A)75 mi
B)150 mi
C)27 mi
D)81 mi
E)15 mi
Question
Express the angle as a decimal, to the nearest ten-thousandth of a degree. 
A)
B)
C)
D)
E)
A)
B)
C)
D)
E)
Question
Points on the terminal sides of angles play an important part in the design of arms for robots. Suppose a robot has a straight arm 20 inches long that can rotate about the origin in a coordinate plane. If the robot's hand is located at (20, 0) and then rotates through an angle of 60 o , what is the new location of the hand?
A)
B)
C)
D)
E)
A)
B)
C)
D)
E)
Question
The amount of sunshine illuminating a wall of a building can greatly affect the energy efficiency of the building. The solar radiation striking a vertical wall that faces east is given by the formula 
where R 0 is the maximum solar radiation possible, 
is the angle that the sun makes with the horizontal, and 
is the direction of the sun in the sky, with 
= 90 o when the sun is in the east and 
= 0 o when the sun is in the south. What percentage of R 0 is striking the wall when 
is equal to 60 o and the sun is in the southeast?
Round the answer to the nearest integer.
A)40 %
B)59 %
C)20 %
D)25 %
E)35 %
where R 0 is the maximum solar radiation possible, is the angle that the sun makes with the horizontal, and is the direction of the sun in the sky, with = 90 o when the sun is in the east and = 0 o when the sun is in the south. What percentage of R 0 is striking the wall when is equal to 60 o and the sun is in the southeast?Round the answer to the nearest integer.
A)40 %
B)59 %
C)20 %
D)25 %
E)35 %
Question
Scientists sometimes use the formula 
to simulate temperature variations during the day, with time t in hours, temperature f ( t ) in 
, and t=0 corresponding to midnight. Assume that f ( t ) is decreasing at midnight. Determine values of a, b, c, and d that fit the information: the high temperature of 34 
occurs at 2 P.M., and the average temperature of 19 
occurs 6 hours later.
A)
B)
C)
D)
E)
to simulate temperature variations during the day, with time t in hours, temperature f ( t ) in , and t=0 corresponding to midnight. Assume that f ( t ) is decreasing at midnight. Determine values of a, b, c, and d that fit the information: the high temperature of 34 occurs at 2 P.M., and the average temperature of 19 occurs 6 hours later.A)
B)
C)
D)
E)
Question
Approximate the acute angle 
to the nearest 
. 
A)
B)
C)
D)
E)
to the nearest . A)
B)
C)
D)
E)
Question
Find the amplitude, the period, and the phase shift. 
A)
B)
C)
D)
E)
A)
B)
C)
D)
E)
Question
On a certain spring day with 12 hours of daylight, the light intensity I takes on its largest value of 400 calories/cm 2 at midday. If t = 0 corresponds to sunrise and t is measured in hours, find a formula I = a sin bt that fits this information.
A)
B)
C)
D)
E)
A)
B)
C)
D)
E)
Question
A CB antenna is located on the top of a garage that is 17 feet tall. From a point on level ground that is 120 feet from a point directly below the antenna, the antenna subtends an angle of 
, as shown in the figure. Approximate the length of the antenna. Give the answer to one decimal place. 
A)345.6 ft
B)36.8 ft
C)95.7 ft
D)24.2 ft
E)26.8 ft
, as shown in the figure. Approximate the length of the antenna. Give the answer to one decimal place. A)345.6 ft
B)36.8 ft
C)95.7 ft
D)24.2 ft
E)26.8 ft
Question
A forester, 180 feet from the base of a redwood tree, observes that the angle between the ground and the top of the tree is 
. Estimate the height of the tree.
A)246.8 feet
B)257.1 feet
C)8,133.0 feet
D)265.7 feet
E)0.0 feet
. Estimate the height of the tree.A)246.8 feet
B)257.1 feet
C)8,133.0 feet
D)265.7 feet
E)0.0 feet
Question
Find the period of the equation. 
A)
B)
C)
D)
E)
A)
B)
C)
D)
E)
Question
Approximate, to the nearest 0.01 radian, all angles 
in the interval [ 0, 2 
) that satisfy the equation. 
= 0.0135
A)
B)
C)
D)
E)
in the interval [ 0, 2 ) that satisfy the equation. = 0.0135A)
B)
C)
D)
E)
Question
Find the exact values of the six trigonometric functions of the angle, whenever possible. 
A)
B)
C)
D)
E)
A)
B)
C)
D)
E)
Question
Use fundamental identities to find the exact values of the trigonometric functions for the given conditions. 
A)
B)
C)
D)
E)
A)
B)
C)
D)
E)
Question
Approximate to four decimal places. 
A)0.6569
B)0.6723
C)0.6059
D)0.6693
E)1.3386
A)0.6569
B)0.6723
C)0.6059
D)0.6693
E)1.3386
Question
The popular biorhythm theory uses the graphs of three simple sine functions to make predictions about an individual's physical, emotional, and intellectual potential for a particular day. The graphs are given by 
with t in days and t = 0 corresponding to birth. Find the value of b for the physical cycle, which has a period of 23 days.
A)
B)
C)
D)
E)
with t in days and t = 0 corresponding to birth. Find the value of b for the physical cycle, which has a period of 23 days.A)
B)
C)
D)
E)
Question
An airplane flying at a speed of 300 mi/hr flies from a point A in the direction 
for 15 minutes and then flies in the direction 
for 45 minutes. Approximate, to the nearest mile, the distance from the airplane to A.
A)474 mi
B)237 mi
C)213 mi
D)134 mi
E)300 mi
for 15 minutes and then flies in the direction for 45 minutes. Approximate, to the nearest mile, the distance from the airplane to A.A)474 mi
B)237 mi
C)213 mi
D)134 mi
E)300 mi
Question
Radio stations often have more than one broadcasting tower because federal guidelines do not usually permit a radio station to broadcast its signal in all directions with equal power. Since radio waves can travel over long distances, it is important to control their directional patterns so that radio stations do not interfere with one another. Suppose that a radio station has two broadcasting towers located along a north-south line, as shown in the figure. If the radio station is broadcasting at a wavelength 
and the distance between the two radio towers is equal to 
, then the intensity I of the signal in the direction 
is given by 
where I o is the maximum intensity. 
Approximate I in terms of I o for 
.
A)I o
B)0.614I o
C)0.625I o
D)0.603I o
E)0.592I o
and the distance between the two radio towers is equal to , then the intensity I of the signal in the direction is given by where I o is the maximum intensity. Approximate I in terms of I o for .A)I o
B)0.614I o
C)0.625I o
D)0.603I o
E)0.592I o
Question
The phases of the moon can be described using the phase angle 
, determined by the sun, the moon, and the Earth, as shown in the figure. Because the moon orbits Earth, 
changes during the course of a month. The area of the region A of the moon, which appears illuminated to an observer on Earth, is given by 
, where R = 1,080 mi is the radius of the moon. Approximate A for the following position of the moon: 
A)A = 943,920 mi 2
B)A = 3,434,635 mi 2
C)A = 1,151,699 mi 2
D)A = 3,533,300 mi 2
E)A = 3,775,680 mi 2
, determined by the sun, the moon, and the Earth, as shown in the figure. Because the moon orbits Earth, changes during the course of a month. The area of the region A of the moon, which appears illuminated to an observer on Earth, is given by , where R = 1,080 mi is the radius of the moon. Approximate A for the following position of the moon: A)A = 943,920 mi 2
B)A = 3,434,635 mi 2
C)A = 1,151,699 mi 2
D)A = 3,533,300 mi 2
E)A = 3,775,680 mi 2
Question
Use the graph of a trigonometric function to aid in sketching the graph of the equation without plotting points. 
A)
B)
C)
D)
E)
A)
B)
C)
D)
E)
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Deck 6: The Trigonometric Functions
1
Simplify the expression.
A)
B)
C)
D)
E)
A)
B)
C)
D)
E)
2
Approximate, to the nearest 0.01 radian, all angles in the interval [ 0, 2 ) that satisfy the equation. = 0.8412
A)
B)
C)
D)
E)
in the interval [ 0, 2 ) that satisfy the equation. = 0.8412A)
B)
C)
D)
E)
3
Verify the identity by transforming the left-hand side into the right-hand side.
A)False
B)True
A)False
B)True
False
4
Express = 2 in terms of degrees, minutes, and seconds, to the nearest second.
A)
B)
C)
D)
E)
= 2 in terms of degrees, minutes, and seconds, to the nearest second.A)
B)
C)
D)
E)
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5
Find the quadrant containing if the given conditions are true. and
A)IV
B)II
C)I
D)III
if the given conditions are true. and A)IV
B)II
C)I
D)III
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6
Let P ( t ) be the point on the unit circle U that corresponds to t. If P ( t ) has the coordinates , find , , , .
A)
B)
C)
D)
E)
, find , , , .A)
B)
C)
D)
E)
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7
If a circular arc of the length s = 11 cm subtends the central angle = 2 on a circle, find the radius of the circle.
A)6.5 cm
B)5.5 cm
C)7 cm
D)22 cm
E)6 cm
= 2 on a circle, find the radius of the circle.A)6.5 cm
B)5.5 cm
C)7 cm
D)22 cm
E)6 cm
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8
Two stars that are very close may appear to be one. The ability of a telescope to separate their images is called its resolution. The smaller the resolution, the better a telescope's ability to separate images in the sky. In a refracting telescope, resolution ( see the figure ) can be improved by using a lens with a larger diameter D. The relationship between in degrees and D in meters is given by , where is the wavelength of light in meters. The largest refracting telescope in the world is at the University of Chicago. At a wavelength of meter, its resolution is . Approximate the diameter of the lens to the nearest hundredth of a meter.
A)0.02 meters
B)1.17 meters
C)0.97 meters
D)0.77 meters
E)1.02 meters
( see the figure ) can be improved by using a lens with a larger diameter D. The relationship between in degrees and D in meters is given by , where is the wavelength of light in meters. The largest refracting telescope in the world is at the University of Chicago. At a wavelength of meter, its resolution is . Approximate the diameter of the lens to the nearest hundredth of a meter. A)0.02 meters
B)1.17 meters
C)0.97 meters
D)0.77 meters
E)1.02 meters
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9
Scientists sometimes use the formula to simulate temperature variations during the day, with time t in hours, temperature f ( t ) in , and t=0 corresponding to midnight. Assume that f ( t ) is decreasing at midnight. Determine values of a, b, c, and d that fit the information: the high temperature of 32 occurs at 2 P.M., and the average temperature of 23 occurs 6 hours later.
A)
B)
C)
D)
E)
to simulate temperature variations during the day, with time t in hours, temperature f ( t ) in , and t=0 corresponding to midnight. Assume that f ( t ) is decreasing at midnight. Determine values of a, b, c, and d that fit the information: the high temperature of 32 occurs at 2 P.M., and the average temperature of 23 occurs 6 hours later.A)
B)
C)
D)
E)
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10
Find the period of the equation.
A)
B)
C)
D)
E)
A)
B)
C)
D)
E)
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11
Find the amplitude, the period, and the phase shift.
A)
B)
C)
D)
E)
A)
B)
C)
D)
E)
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12
Express the angle as a decimal, to the nearest ten-thousandth of a degree.
A)
B)
C)
D)
E)
A)
B)
C)
D)
E)
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13
Points on the terminal sides of angles play an important part in the design of arms for robots. Suppose a robot has a straight arm 18 inches long that can rotate about the origin in a coordinate plane. If the robot's hand is located at (18, 0) and then rotates through an angle of 60 o , what is the new location of the hand?
A)
B)
C)
D)
E)
A)
B)
C)
D)
E)
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14
Approximate to four decimal places.
A)0.2683
B)0.2713
C)0.2837
D)0.5425
E)0.2327
A)0.2683
B)0.2713
C)0.2837
D)0.5425
E)0.2327
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15
Approximate the acute angle to the nearest .
A)
B)
C)
D)
E)
to the nearest . A)
B)
C)
D)
E)
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16
The popular biorhythm theory uses the graphs of three simple sine functions to make predictions about an individual's physical, emotional, and intellectual potential for a particular day. The graphs are given by with t in days and t = 0 corresponding to birth. Find the value of b for the physical cycle, which has a period of 19 days.
A)
B)
C)
D)
E)
with t in days and t = 0 corresponding to birth. Find the value of b for the physical cycle, which has a period of 19 days.A)
B)
C)
D)
E)
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17
Refer to the graph of y = cos x to find the separate values of x in the interval that satisfy the equation.
A)
B)
C)
D)
E)
that satisfy the equation. A)
B)
C)
D)
E)
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18
The amount of sunshine illuminating a wall of a building can greatly affect the energy efficiency of the building. The solar radiation striking a vertical wall that faces east is given by the formula where R 0 is the maximum solar radiation possible, is the angle that the sun makes with the horizontal, and is the direction of the sun in the sky, with = 90 o when the sun is in the east and = 0 o when the sun is in the south. What percentage of R 0 is striking the wall when is equal to 30 o and the sun is in the southeast?
Round the answer to the nearest integer.
A)71 %
B)61 %
C)56 %
D)76 %
E)29 %
where R 0 is the maximum solar radiation possible, is the angle that the sun makes with the horizontal, and is the direction of the sun in the sky, with = 90 o when the sun is in the east and = 0 o when the sun is in the south. What percentage of R 0 is striking the wall when is equal to 30 o and the sun is in the southeast?Round the answer to the nearest integer.
A)71 %
B)61 %
C)56 %
D)76 %
E)29 %
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19
As for some real number . Use a graph to predict .
A)
B)
C)
D)
E)
for some real number . Use a graph to predict . A)
B)
C)
D)
E)
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20
On a certain spring day with 12 hours of daylight, the light intensity I takes on its largest value of 390 calories/cm 2 at midday. If t = 0 corresponds to sunrise and t is measured in hours, find a formula I = a sin bt that fits this information.
A)
B)
C)
D)
E)
A)
B)
C)
D)
E)
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21
Simplify the expression.
A)
B)
C)
D)
E)
A)
B)
C)
D)
E)
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22
Approximate to four decimal places.
A)0.2837
B)0.2713
C)0.2683
D)0.2327
E)0.5425
A)0.2837
B)0.2713
C)0.2683
D)0.2327
E)0.5425
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23
Express = 4.5 in terms of degrees, minutes, and seconds, to the nearest second.
A)
B)
C)
D)
E)
= 4.5 in terms of degrees, minutes, and seconds, to the nearest second.A)
B)
C)
D)
E)
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24
Approximate the acute angle to the nearest .
A)
B)
C)
D)
E)
to the nearest . A)
B)
C)
D)
E)
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25
Let P ( t ) be the point on the unit circle U that corresponds to t. If P ( t ) has the coordinates , find , , , .
A)
B)
C)
D)
E)
, find , , , .A)
B)
C)
D)
E)
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26
An airplane flying at a speed of 360 mi/hr flies from a point A in the direction for 15 minutes and then flies in the direction for 15 minutes. Approximate, to the nearest mile, the distance from the airplane to A.
A)104 mi
B)127 mi
C)1 mi
D)254 mi
E)180 mi
for 15 minutes and then flies in the direction for 15 minutes. Approximate, to the nearest mile, the distance from the airplane to A.A)104 mi
B)127 mi
C)1 mi
D)254 mi
E)180 mi
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27
Points on the terminal sides of angles play an important part in the design of arms for robots. Suppose a robot has a straight arm 20 inches long that can rotate about the origin in a coordinate plane. If the robot's hand is located at (20, 0) and then rotates through an angle of 60 o , what is the new location of the hand?
A)
B)
C)
D)
E)
A)
B)
C)
D)
E)
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28
Verify the identity by transforming the left-hand side into the right-hand side.
A)False
B)True
A)False
B)True
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29
Radio stations often have more than one broadcasting tower because federal guidelines do not usually permit a radio station to broadcast its signal in all directions with equal power. Since radio waves can travel over long distances, it is important to control their directional patterns so that radio stations do not interfere with one another. Suppose that a radio station has two broadcasting towers located along a north-south line, as shown in the figure. If the radio station is broadcasting at a wavelength and the distance between the two radio towers is equal to , then the intensity I of the signal in the direction is given by where I o is the maximum intensity. Approximate I in terms of I o for .
A)0.055I o
B)I o
C)0.033I o
D)0.066I o
E)0.044I o
and the distance between the two radio towers is equal to , then the intensity I of the signal in the direction is given by where I o is the maximum intensity. Approximate I in terms of I o for .A)0.055I o
B)I o
C)0.033I o
D)0.066I o
E)0.044I o
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30
Refer to the graph of y = cos x to find the separate values of x in the interval that satisfy the equation.
A)
B)
C)
D)
E)
that satisfy the equation. A)
B)
C)
D)
E)
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31
A ship leaves port at 4:00 P.M. and sails in the direction at a rate of 19 mi/hr. Another ship leaves port at 4:30 P.M. and sails in the direction at a rate of 16 mi/hr. Approximately how far apart are the ships at 7:00 P.M.?
A)70 mi
B)17 mi
C)75 mi
D)25 mi
E)140 mi
at a rate of 19 mi/hr. Another ship leaves port at 4:30 P.M. and sails in the direction at a rate of 16 mi/hr. Approximately how far apart are the ships at 7:00 P.M.?A)70 mi
B)17 mi
C)75 mi
D)25 mi
E)140 mi
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32
If a circular arc of the length s = 13.5 cm subtends the central angle = 3 on a circle, find the radius of the circle.
A)4.5 cm
B)5.5 cm
C)6 cm
D)40.5 cm
E)5 cm
= 3 on a circle, find the radius of the circle.A)4.5 cm
B)5.5 cm
C)6 cm
D)40.5 cm
E)5 cm
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33
The amount of sunshine illuminating a wall of a building can greatly affect the energy efficiency of the building. The solar radiation striking a vertical wall that faces east is given by the formula where R 0 is the maximum solar radiation possible, is the angle that the sun makes with the horizontal, and is the direction of the sun in the sky, with = 90 o when the sun is in the east and = 0 o when the sun is in the south. What percentage of R 0 is striking the wall when is equal to 60 o and the sun is in the southeast?
Round the answer to the nearest integer.
A)35 %
B)30 %
C)20 %
D)45 %
E)59 %
where R 0 is the maximum solar radiation possible, is the angle that the sun makes with the horizontal, and is the direction of the sun in the sky, with = 90 o when the sun is in the east and = 0 o when the sun is in the south. What percentage of R 0 is striking the wall when is equal to 60 o and the sun is in the southeast?Round the answer to the nearest integer.
A)35 %
B)30 %
C)20 %
D)45 %
E)59 %
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34
Express the angle as a decimal, to the nearest ten-thousandth of a degree.
A)
B)
C)
D)
E)
A)
B)
C)
D)
E)
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35
Two stars that are very close may appear to be one. The ability of a telescope to separate their images is called its resolution. The smaller the resolution, the better a telescope's ability to separate images in the sky. In a refracting telescope, resolution ( see the figure ) can be improved by using a lens with a larger diameter D. The relationship between in degrees and D in meters is given by , where is the wavelength of light in meters. The largest refracting telescope in the world is at the University of Chicago. At a wavelength of meter, its resolution is . Approximate the diameter of the lens to the nearest hundredth of a meter.
A)1.17 meters
B)0.77 meters
C)0.02 meters
D)0.97 meters
E)1.02 meters
( see the figure ) can be improved by using a lens with a larger diameter D. The relationship between in degrees and D in meters is given by , where is the wavelength of light in meters. The largest refracting telescope in the world is at the University of Chicago. At a wavelength of meter, its resolution is . Approximate the diameter of the lens to the nearest hundredth of a meter. A)1.17 meters
B)0.77 meters
C)0.02 meters
D)0.97 meters
E)1.02 meters
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36
Approximate, to the nearest 0.01 radian, all angles in the interval [ 0, 2 ) that satisfy the equation. = 0.2518
A)
B)
C)
D)
E)
in the interval [ 0, 2 ) that satisfy the equation. = 0.2518A)
B)
C)
D)
E)
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37
A CB antenna is located on the top of a garage that is 17 feet tall. From a point on level ground that is 120 feet from a point directly below the antenna, the antenna subtends an angle of , as shown in the figure. Approximate the length of the antenna. Give the answer to one decimal place.
A)345.6 ft
B)26.8 ft
C)24.2 ft
D)36.8 ft
E)95.7 ft
, as shown in the figure. Approximate the length of the antenna. Give the answer to one decimal place. A)345.6 ft
B)26.8 ft
C)24.2 ft
D)36.8 ft
E)95.7 ft
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38
Find the quadrant containing if the given conditions are true. and
A)I
B)IV
C)II
D)III
if the given conditions are true. and A)I
B)IV
C)II
D)III
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39
Use the graph of a trigonometric function to aid in sketching the graph of the equation without plotting points.
A)
B)
C)
D)
E)
A)
B)
C)
D)
E)
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40
As for some real number . Use a graph to predict .
A)
B)
C)
D)
E)
for some real number . Use a graph to predict . A)
B)
C)
D)
E)
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41
Let P ( t ) be the point on the unit circle U that corresponds to t. If P ( t ) has the coordinates , find , , , .
A)
B)
C)
D)
E)
, find , , , .A)
B)
C)
D)
E)
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42
Two stars that are very close may appear to be one. The ability of a telescope to separate their images is called its resolution. The smaller the resolution, the better a telescope's ability to separate images in the sky. In a refracting telescope, resolution ( see the figure ) can be improved by using a lens with a larger diameter D. The relationship between in degrees and D in meters is given by , where is the wavelength of light in meters. The largest refracting telescope in the world is at the University of Chicago. At a wavelength of meter, its resolution is . Approximate the diameter of the lens to the nearest hundredth of a meter.
A)0.02 meters
B)0.77 meters
C)1.17 meters
D)0.97 meters
E)1.02 meters
( see the figure ) can be improved by using a lens with a larger diameter D. The relationship between in degrees and D in meters is given by , where is the wavelength of light in meters. The largest refracting telescope in the world is at the University of Chicago. At a wavelength of meter, its resolution is . Approximate the diameter of the lens to the nearest hundredth of a meter. A)0.02 meters
B)0.77 meters
C)1.17 meters
D)0.97 meters
E)1.02 meters
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43
If a circular arc of the length s = 16 cm subtends the central angle = 2 on a circle, find the radius of the circle.
A)8.5 cm
B)32 cm
C)8 cm
D)9 cm
E)9.5 cm
= 2 on a circle, find the radius of the circle.A)8.5 cm
B)32 cm
C)8 cm
D)9 cm
E)9.5 cm
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44
On a certain spring day with 12 hours of daylight, the light intensity I takes on its largest value of 480 calories/cm 2 at midday. If t = 0 corresponds to sunrise and t is measured in hours, find a formula I = a sin bt that fits this information.
A)
B)
C)
D)
E)
A)
B)
C)
D)
E)
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45
Scientists sometimes use the formula to simulate temperature variations during the day, with time t in hours, temperature f ( t ) in , and t=0 corresponding to midnight. Assume that f ( t ) is decreasing at midnight. Determine values of a, b, c, and d that fit the information: the high temperature of 29 occurs at 2 P.M., and the average temperature of 21 occurs 6 hours later.
A)
B)
C)
D)
E)
to simulate temperature variations during the day, with time t in hours, temperature f ( t ) in , and t=0 corresponding to midnight. Assume that f ( t ) is decreasing at midnight. Determine values of a, b, c, and d that fit the information: the high temperature of 29 occurs at 2 P.M., and the average temperature of 21 occurs 6 hours later.A)
B)
C)
D)
E)
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46
A ship leaves port at 4:00 P.M. and sails in the direction at a rate of 24 mi/hr. Another ship leaves port at 4:30 P.M. and sails in the direction at a rate of 22 mi/hr. Approximately how far apart are the ships at 7:00 P.M.?
A)17 mi
B)182 mi
C)33 mi
D)98 mi
E)91 mi
at a rate of 24 mi/hr. Another ship leaves port at 4:30 P.M. and sails in the direction at a rate of 22 mi/hr. Approximately how far apart are the ships at 7:00 P.M.?A)17 mi
B)182 mi
C)33 mi
D)98 mi
E)91 mi
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47
As for some real number . Use a graph to predict .
A)
B)
C)
D)
E)
for some real number . Use a graph to predict . A)
B)
C)
D)
E)
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48
Find the period of the equation.
A)
B)
C)
D)
E)
A)
B)
C)
D)
E)
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49
A CB antenna is located on the top of a garage that is 17 feet tall. From a point on level ground that is 120 feet from a point directly below the antenna, the antenna subtends an angle of , as shown in the figure. Approximate the length of the antenna. Give the answer to one decimal place.
A)26.8 ft
B)345.6 ft
C)95.7 ft
D)36.8 ft
E)24.2 ft
, as shown in the figure. Approximate the length of the antenna. Give the answer to one decimal place. A)26.8 ft
B)345.6 ft
C)95.7 ft
D)36.8 ft
E)24.2 ft
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50
Express = 2.5 in terms of degrees, minutes, and seconds, to the nearest second.
A)
B)
C)
D)
E)
= 2.5 in terms of degrees, minutes, and seconds, to the nearest second.A)
B)
C)
D)
E)
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51
Simplify the expression.
A)
B)
C)
D)
E)
A)
B)
C)
D)
E)
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52
The popular biorhythm theory uses the graphs of three simple sine functions to make predictions about an individual's physical, emotional, and intellectual potential for a particular day. The graphs are given by with t in days and t = 0 corresponding to birth. Find the value of b for the physical cycle, which has a period of 29 days.
A)
B)
C)
D)
E)
with t in days and t = 0 corresponding to birth. Find the value of b for the physical cycle, which has a period of 29 days.A)
B)
C)
D)
E)
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53
Radio stations often have more than one broadcasting tower because federal guidelines do not usually permit a radio station to broadcast its signal in all directions with equal power. Since radio waves can travel over long distances, it is important to control their directional patterns so that radio stations do not interfere with one another. Suppose that a radio station has two broadcasting towers located along a north-south line, as shown in the figure. If the radio station is broadcasting at a wavelength and the distance between the two radio towers is equal to , then the intensity I of the signal in the direction is given by where I o is the maximum intensity. Approximate I in terms of I o for .
A)0.055I o
B)0.044I o
C)0.066I o
D)I o
E)0.033I o
and the distance between the two radio towers is equal to , then the intensity I of the signal in the direction is given by where I o is the maximum intensity. Approximate I in terms of I o for .A)0.055I o
B)0.044I o
C)0.066I o
D)I o
E)0.033I o
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54
Refer to the graph of y = cos x to find the separate values of x in the interval that satisfy the equation.
A)
B)
C)
D)
E)
that satisfy the equation. A)
B)
C)
D)
E)
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55
Use the graph of a trigonometric function to aid in sketching the graph of the equation without plotting points.
A)
B)
C)
D)
E)
A)
B)
C)
D)
E)
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56
Find the amplitude, the period, and the phase shift.
A)
B)
C)
D)
E)
A)
B)
C)
D)
E)
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57
Express the angle as a decimal, to the nearest ten-thousandth of a degree.
A)
B)
C)
D)
E)
A)
B)
C)
D)
E)
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58
An airplane flying at a speed of 360 mi/hr flies from a point A in the direction for 45 minutes and then flies in the direction for 45 minutes. Approximate, to the nearest mile, the distance from the airplane to A.
A)764 mi
B)540 mi
C)180 mi
D)1 mi
E)382 mi
for 45 minutes and then flies in the direction for 45 minutes. Approximate, to the nearest mile, the distance from the airplane to A.A)764 mi
B)540 mi
C)180 mi
D)1 mi
E)382 mi
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59
Verify the identity by transforming the left-hand side into the right-hand side.
A)False
B)True
A)False
B)True
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60
Find the quadrant containing if the given conditions are true. and
A)IV
B)III
C)II
D)I
if the given conditions are true. and A)IV
B)III
C)II
D)I
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61
A ship leaves port at 1:00 P.M. and sails in the direction at a rate of 20 mi/hr. Another ship leaves port at 1:30 P.M. and sails in the direction at a rate of 18 mi/hr. Approximately how far apart are the ships at 4:00 P.M.?
A)75 mi
B)150 mi
C)27 mi
D)81 mi
E)15 mi
at a rate of 20 mi/hr. Another ship leaves port at 1:30 P.M. and sails in the direction at a rate of 18 mi/hr. Approximately how far apart are the ships at 4:00 P.M.?A)75 mi
B)150 mi
C)27 mi
D)81 mi
E)15 mi
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62
Express the angle as a decimal, to the nearest ten-thousandth of a degree.
A)
B)
C)
D)
E)
A)
B)
C)
D)
E)
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63
Points on the terminal sides of angles play an important part in the design of arms for robots. Suppose a robot has a straight arm 20 inches long that can rotate about the origin in a coordinate plane. If the robot's hand is located at (20, 0) and then rotates through an angle of 60 o , what is the new location of the hand?
A)
B)
C)
D)
E)
A)
B)
C)
D)
E)
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64
The amount of sunshine illuminating a wall of a building can greatly affect the energy efficiency of the building. The solar radiation striking a vertical wall that faces east is given by the formula where R 0 is the maximum solar radiation possible, is the angle that the sun makes with the horizontal, and is the direction of the sun in the sky, with = 90 o when the sun is in the east and = 0 o when the sun is in the south. What percentage of R 0 is striking the wall when is equal to 60 o and the sun is in the southeast?
Round the answer to the nearest integer.
A)40 %
B)59 %
C)20 %
D)25 %
E)35 %
where R 0 is the maximum solar radiation possible, is the angle that the sun makes with the horizontal, and is the direction of the sun in the sky, with = 90 o when the sun is in the east and = 0 o when the sun is in the south. What percentage of R 0 is striking the wall when is equal to 60 o and the sun is in the southeast?Round the answer to the nearest integer.
A)40 %
B)59 %
C)20 %
D)25 %
E)35 %
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65
Scientists sometimes use the formula to simulate temperature variations during the day, with time t in hours, temperature f ( t ) in , and t=0 corresponding to midnight. Assume that f ( t ) is decreasing at midnight. Determine values of a, b, c, and d that fit the information: the high temperature of 34 occurs at 2 P.M., and the average temperature of 19 occurs 6 hours later.
A)
B)
C)
D)
E)
to simulate temperature variations during the day, with time t in hours, temperature f ( t ) in , and t=0 corresponding to midnight. Assume that f ( t ) is decreasing at midnight. Determine values of a, b, c, and d that fit the information: the high temperature of 34 occurs at 2 P.M., and the average temperature of 19 occurs 6 hours later.A)
B)
C)
D)
E)
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66
Approximate the acute angle to the nearest .
A)
B)
C)
D)
E)
to the nearest . A)
B)
C)
D)
E)
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67
Find the amplitude, the period, and the phase shift.
A)
B)
C)
D)
E)
A)
B)
C)
D)
E)
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68
On a certain spring day with 12 hours of daylight, the light intensity I takes on its largest value of 400 calories/cm 2 at midday. If t = 0 corresponds to sunrise and t is measured in hours, find a formula I = a sin bt that fits this information.
A)
B)
C)
D)
E)
A)
B)
C)
D)
E)
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69
A CB antenna is located on the top of a garage that is 17 feet tall. From a point on level ground that is 120 feet from a point directly below the antenna, the antenna subtends an angle of , as shown in the figure. Approximate the length of the antenna. Give the answer to one decimal place.
A)345.6 ft
B)36.8 ft
C)95.7 ft
D)24.2 ft
E)26.8 ft
, as shown in the figure. Approximate the length of the antenna. Give the answer to one decimal place. A)345.6 ft
B)36.8 ft
C)95.7 ft
D)24.2 ft
E)26.8 ft
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70
A forester, 180 feet from the base of a redwood tree, observes that the angle between the ground and the top of the tree is . Estimate the height of the tree.
A)246.8 feet
B)257.1 feet
C)8,133.0 feet
D)265.7 feet
E)0.0 feet
. Estimate the height of the tree.A)246.8 feet
B)257.1 feet
C)8,133.0 feet
D)265.7 feet
E)0.0 feet
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71
Find the period of the equation.
A)
B)
C)
D)
E)
A)
B)
C)
D)
E)
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72
Approximate, to the nearest 0.01 radian, all angles in the interval [ 0, 2 ) that satisfy the equation. = 0.0135
A)
B)
C)
D)
E)
in the interval [ 0, 2 ) that satisfy the equation. = 0.0135A)
B)
C)
D)
E)
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73
Find the exact values of the six trigonometric functions of the angle, whenever possible.
A)
B)
C)
D)
E)
A)
B)
C)
D)
E)
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74
Use fundamental identities to find the exact values of the trigonometric functions for the given conditions.
A)
B)
C)
D)
E)
A)
B)
C)
D)
E)
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75
Approximate to four decimal places.
A)0.6569
B)0.6723
C)0.6059
D)0.6693
E)1.3386
A)0.6569
B)0.6723
C)0.6059
D)0.6693
E)1.3386
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76
The popular biorhythm theory uses the graphs of three simple sine functions to make predictions about an individual's physical, emotional, and intellectual potential for a particular day. The graphs are given by with t in days and t = 0 corresponding to birth. Find the value of b for the physical cycle, which has a period of 23 days.
A)
B)
C)
D)
E)
with t in days and t = 0 corresponding to birth. Find the value of b for the physical cycle, which has a period of 23 days.A)
B)
C)
D)
E)
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77
An airplane flying at a speed of 300 mi/hr flies from a point A in the direction for 15 minutes and then flies in the direction for 45 minutes. Approximate, to the nearest mile, the distance from the airplane to A.
A)474 mi
B)237 mi
C)213 mi
D)134 mi
E)300 mi
for 15 minutes and then flies in the direction for 45 minutes. Approximate, to the nearest mile, the distance from the airplane to A.A)474 mi
B)237 mi
C)213 mi
D)134 mi
E)300 mi
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78
Radio stations often have more than one broadcasting tower because federal guidelines do not usually permit a radio station to broadcast its signal in all directions with equal power. Since radio waves can travel over long distances, it is important to control their directional patterns so that radio stations do not interfere with one another. Suppose that a radio station has two broadcasting towers located along a north-south line, as shown in the figure. If the radio station is broadcasting at a wavelength and the distance between the two radio towers is equal to , then the intensity I of the signal in the direction is given by where I o is the maximum intensity. Approximate I in terms of I o for .
A)I o
B)0.614I o
C)0.625I o
D)0.603I o
E)0.592I o
and the distance between the two radio towers is equal to , then the intensity I of the signal in the direction is given by where I o is the maximum intensity. Approximate I in terms of I o for .A)I o
B)0.614I o
C)0.625I o
D)0.603I o
E)0.592I o
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79
The phases of the moon can be described using the phase angle , determined by the sun, the moon, and the Earth, as shown in the figure. Because the moon orbits Earth, changes during the course of a month. The area of the region A of the moon, which appears illuminated to an observer on Earth, is given by , where R = 1,080 mi is the radius of the moon. Approximate A for the following position of the moon:
A)A = 943,920 mi 2
B)A = 3,434,635 mi 2
C)A = 1,151,699 mi 2
D)A = 3,533,300 mi 2
E)A = 3,775,680 mi 2
, determined by the sun, the moon, and the Earth, as shown in the figure. Because the moon orbits Earth, changes during the course of a month. The area of the region A of the moon, which appears illuminated to an observer on Earth, is given by , where R = 1,080 mi is the radius of the moon. Approximate A for the following position of the moon: A)A = 943,920 mi 2
B)A = 3,434,635 mi 2
C)A = 1,151,699 mi 2
D)A = 3,533,300 mi 2
E)A = 3,775,680 mi 2
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80
Use the graph of a trigonometric function to aid in sketching the graph of the equation without plotting points.
A)
B)
C)
D)
E)
A)
B)
C)
D)
E)
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