Question 1

Solve the equation for solutions in the interval [0, 2 $$[ 0,2 \pi ) .$$
-$$\sin x \cos x = \frac { 1 } { 2 }$$

Accepted Answer

A)  $\{ 0 \}$ 
B)  $\left\{ \frac { \pi } { 4 } , \frac { 5 \pi } { 4 } \right\}$ 
C)  $\left\{ 0 , \frac { \pi } { 4 } , \pi , \frac { 5 \pi } { 4 } \right\}$ 
D)  $\left\{ \frac { \pi } { 12 } , \frac { \pi } { 6 } , \frac { 2 \pi } { 3 } , \frac { 7 \pi } { 12 } , \frac { 7 \pi } { 6 } , \frac { 13 \pi } { 12 } , \frac { 5 \pi } { 3 } \right\}$ 
A)  $\{ 0 \}$ 
B)  $\left\{ \frac { \pi } { 4 } , \frac { 5 \pi } { 4 } \right\}$ 
C)  $\left\{ 0 , \frac { \pi } { 4 } , \pi , \frac { 5 \pi } { 4 } \right\}$ 
D)  $\left\{ \frac { \pi } { 12 } , \frac { \pi } { 6 } , \frac { 2 \pi } { 3 } , \frac { 7 \pi } { 12 } , \frac { 7 \pi } { 6 } , \frac { 13 \pi } { 12 } , \frac { 5 \pi } { 3 } \right\}$

Question 2

Solve the problem.
-A painting 1 meter high and 3 meters from the floor will cut off an angle $\theta$ to an observer, where $\theta = \tan ^ { - 1 } \left( \frac { x } { x ^ { 2 } + 1.6 } \right)$, assuming that the observer is $x$ feet from the wall where the painting is displayed and that the eyes of the observer are $1.6$ meters above the ground (see the figure). Find the value of $\theta$ for $x = 3$. Round to the nearest tenth of a degree.

Accepted Answer

A)  $13.3 ^ { \circ }$ 
B)  $18.3 ^ { \circ }$ 
C)  $15.8 ^ { \circ }$ 
D)  $33.1 ^ { \circ }$ 
A)  $13.3 ^ { \circ }$ 
B)  $18.3 ^ { \circ }$ 
C)  $15.8 ^ { \circ }$ 
D)  $33.1 ^ { \circ }$

Question 3

Give the exact value of the expression.
-$\cos \left( 2 \arcsin \frac { 1 } { 4 } \right)$

Accepted Answer

A)  $\frac { 5 } { 8 }$ 
B)  $\frac { 1 } { 8 }$ 
C)  $\frac { 7 } { 8 }$ 
D)  $\frac { 3 } { 8 }$ 
A)  $\frac { 5 } { 8 }$ 
B)  $\frac { 1 } { 8 }$ 
C)  $\frac { 7 } { 8 }$ 
D)  $\frac { 3 } { 8 }$

Question 4

Use the parallelogram rule to find the magnitude of the resultant force for the two forces shown in the figure. Round to
one decimal place.
-Explain what is wrong with the following solution for the equation $\cos 2 \theta = \sqrt { 3 }$ in the interval $[ 0,2 \pi )$.
$$\cos 2 \theta = \sqrt { 3 }$$
$$\cos \theta = \frac { \sqrt { 3 } } { 2 }$$
$$\theta = \frac { \pi } { 6 } \text { or } \theta = \frac { 11 \pi } { 6 }$$

Accepted Answer

The answer of Use the parallelogram rule to find the...

Question 5

Solve the equation for exact solutions.
-$6 \arcsin x = \pi$

Accepted Answer

A)  $\{ - 5 + 3 \sqrt { 2 } \}$ 
B)  $\left\{ \frac { 1 } { 2 } \right\}$ 
C)  $\left\{ - \frac { 3 } { 4 } , \frac { 3 } { 4 } \right\}$ 
D)  $\{ 2 \}$ 
A)  $\{ - 5 + 3 \sqrt { 2 } \}$ 
B)  $\left\{ \frac { 1 } { 2 } \right\}$ 
C)  $\left\{ - \frac { 3 } { 4 } , \frac { 3 } { 4 } \right\}$ 
D)  $\{ 2 \}$

Question 6

Use a calculator to give the real number value. Round the answer to 7 decimal places.
-$$y = \sin ^ { - 1 } ( 0.2079 )$$

Accepted Answer

A)  $0.2094276$ 
B)  $0.2444444$ 
C)  $0.2619048$ 
D)  $0.1746032$ 
A)  $0.2094276$ 
B)  $0.2444444$ 
C)  $0.2619048$ 
D)  $0.1746032$

Question 7

Give the exact value of the expression.
-$\cos \left( \arcsin \frac { 5 } { 13 } + \arccos \frac { 3 } { 5 } \right)$

Accepted Answer

A)  $\frac { 48 } { 65 }$ 
B)  $\frac { 56 } { 65 }$ 
C)  $\frac { 16 } { 65 }$ 
D)  $\frac { 72 } { 65 }$ 
A)  $\frac { 48 } { 65 }$ 
B)  $\frac { 56 } { 65 }$ 
C)  $\frac { 16 } { 65 }$ 
D)  $\frac { 72 } { 65 }$

Question 8

Solve the problem.
-The formula for the up-and-down motion of a weight on a spring is given by $$\mathrm { S } = \mathrm { a } \sin \left( \frac { \sqrt { \mathrm { k } } } { \mathrm { m } } \right) \mathrm { t }$$
where a is the radius of the circle, $\mathrm { k }$ is the spring constant, $\mathrm { m }$ is the mass, and $\mathrm { t }$ is the time. Solve the equation for t.

Accepted Answer

A)  $t = \frac { m } { \sqrt { k } } \arcsin \left( \frac { S } { a } \right)$ 
B)  $t = \frac { m } { \sqrt { k } } \sin \left( \frac { S } { a } \right)$ 
C)  $t = \frac { \sqrt { k } } { m } \sin \left( \frac { a } { S } \right)$ 
D)  $t = \frac { \sqrt { k } } { m } \arcsin \left( \frac { S } { a } \right)$ 
A)  $t = \frac { m } { \sqrt { k } } \arcsin \left( \frac { S } { a } \right)$ 
B)  $t = \frac { m } { \sqrt { k } } \sin \left( \frac { S } { a } \right)$ 
C)  $t = \frac { \sqrt { k } } { m } \sin \left( \frac { a } { S } \right)$ 
D)  $t = \frac { \sqrt { k } } { m } \arcsin \left( \frac { S } { a } \right)$

Question 9

Use a calculator to give the real number value. Round the answer to 7 decimal places.
-$$y = \cos ^ { - 1 } ( - 0.3907 )$$

Accepted Answer

A)  $2.0079383$ 
B)  $1.9721882$ 
C)  $- 1.1698423$ 
D)  $1.9206367$ 
A)  $2.0079383$ 
B)  $1.9721882$ 
C)  $- 1.1698423$ 
D)  $1.9206367$

Question 10

Use a calculator to find the value. Give answers as real numbers and round to 4 decimal places, if necessary.
-$\sin \left( \cos ^ { - 1 } 0.8324 \right)$

Accepted Answer

A)  $0.0175$ 
B)  $0.5542$ 
C)  $0.6324$ 
D)  $0.1389$ 
A)  $0.0175$ 
B)  $0.5542$ 
C)  $0.6324$ 
D)  $0.1389$

Question 11

Solve the equation for x, where x is restricted to the given interval.
-$$y = 3 \tan 2 x - 1 \text {, for } x \text { in } \left( - \frac { \pi } { 4 } , \frac { \pi } { 4 } \right)$$

Accepted Answer

A)  $x = 2 \arctan ( 3 y + 3 )$ 
B)  $x = \frac { 1 } { 2 } \arctan \frac { y + 1 } { 3 }$ 
C)  $x = \frac { 1 } { 2 } \arctan ( 3 y + 3 )$ 
D)  $x = 2 \arctan \frac { y + 1 } { 3 }$ 
A)  $x = 2 \arctan ( 3 y + 3 )$ 
B)  $x = \frac { 1 } { 2 } \arctan \frac { y + 1 } { 3 }$ 
C)  $x = \frac { 1 } { 2 } \arctan ( 3 y + 3 )$ 
D)  $x = 2 \arctan \frac { y + 1 } { 3 }$

Question 12

Give the degree measure of .
-ϴ = arcsin (1)

Accepted Answer

A) 45° 
B) 180° 
C) 0° 
D) 90° 
A) 45° 
B) 180° 
C) 0° 
D) 90°

Question 13

Solve.
-The output voltage for an $\mathrm { AC }$ generator is approximated by $\mathrm { v } = 156 \cos \left( 120 \pi \mathrm { t } - \frac { \pi } { 3 } \right)$. Find the smallest positive value of $\mathrm { t }$ for which the output is 75 volts. Round values to 4 decimal places.

Accepted Answer

A)  $0.0056$ 
B)  $0.0046$ 
C)  $0.0122$ 
D)  $0.0112$ 
A)  $0.0056$ 
B)  $0.0046$ 
C)  $0.0122$ 
D)  $0.0112$

Question 14

Solve the problem.
-The position of a weight on a spring relative to the point of equilibrium is given by y = 4 cos 6t - 2 sin 6t,
Where the arguments are in radians and t is in seconds. Find the smallest value of t for which the
Weight is at the point of equilibrium (y = 0). Give your answer to the nearest hundredth.

Accepted Answer

A) 0.13 
B) 0.22 
C) 0.08 
D) 0.18 
A) 0.13 
B) 0.22 
C) 0.08 
D) 0.18

Question 15

Solve the equation for exact solutions over the interval [0, 2 $$[ 0,2 \pi )$$
-$\csc ^ { 5 } x - 4 \csc x = 0$

Accepted Answer

A)  $\left\{ \frac { \pi } { 4 } , \frac { 3 \pi } { 4 } , \frac { \pi } { 3 } , \frac { 5 \pi } { 6 } \right\}$ 
B)  $\left\{ \frac { \pi } { 4 } , \frac { 5 \pi } { 4 } , \frac { \pi } { 3 } , \frac { 5 \pi } { 3 } \right\}$ 
C)  $\left\{ \frac { \pi } { 4 } , \frac { 3 \pi } { 4 } , \frac { \pi } { 6 } , \frac { 5 \pi } { 6 } \right\}$ 
D)  $\left\{ \frac { \pi } { 4 } , \frac { 3 \pi } { 4 } , \frac { 5 \pi } { 4 } , \frac { 7 \pi } { 4 } \right\}$ 
A)  $\left\{ \frac { \pi } { 4 } , \frac { 3 \pi } { 4 } , \frac { \pi } { 3 } , \frac { 5 \pi } { 6 } \right\}$ 
B)  $\left\{ \frac { \pi } { 4 } , \frac { 5 \pi } { 4 } , \frac { \pi } { 3 } , \frac { 5 \pi } { 3 } \right\}$ 
C)  $\left\{ \frac { \pi } { 4 } , \frac { 3 \pi } { 4 } , \frac { \pi } { 6 } , \frac { 5 \pi } { 6 } \right\}$ 
D)  $\left\{ \frac { \pi } { 4 } , \frac { 3 \pi } { 4 } , \frac { 5 \pi } { 4 } , \frac { 7 \pi } { 4 } \right\}$

Question 16

Solve the equation for exact solutions.
-$\sin ^ { - 1 } x + \tan ^ { - 1 } x = 0$

Accepted Answer

A)  $\left\{ - \frac { \sqrt { 3 } } { 2 } , \frac { \sqrt { 3 } } { 2 } \right\}$ 
B)  $\{ 1 \}$ 
C)  $\{ 0 \}$ 
D)  $\left\{ - \frac { \sqrt { 3 } } { 4 } , \frac { \sqrt { 3 } } { 4 } \right\}$ 
A)  $\left\{ - \frac { \sqrt { 3 } } { 2 } , \frac { \sqrt { 3 } } { 2 } \right\}$ 
B)  $\{ 1 \}$ 
C)  $\{ 0 \}$ 
D)  $\left\{ - \frac { \sqrt { 3 } } { 4 } , \frac { \sqrt { 3 } } { 4 } \right\}$

Question 17

Find the exact value of the real number y.
-$$y = \csc ^ { - 1 } ( - 1 )$$

Accepted Answer

A)  $2 \pi$ 
B)  $\frac { \pi } { 4 }$ 
C)  $\pi$ 
D)  $- \frac { \pi } { 2 }$ 
A)  $2 \pi$ 
B)  $\frac { \pi } { 4 }$ 
C)  $\pi$ 
D)  $- \frac { \pi } { 2 }$

Question 18

Solve the equation for exact solutions over the interval [0, 2 $$[ 0,2 \pi )$$
-$$\sec ^ { 2 } x - 2 = \tan ^ { 2 } x$$

Accepted Answer

A)  $\left\{ \frac { \pi } { 3 } \right\}$ 
B)  $\left\{ \frac { \pi } { 4 } \right\}$ 
C)  $\varnothing$ 
D)  $\left\{ \frac { \pi } { 6 } \right\}$ 
A)  $\left\{ \frac { \pi } { 3 } \right\}$ 
B)  $\left\{ \frac { \pi } { 4 } \right\}$ 
C)  $\varnothing$ 
D)  $\left\{ \frac { \pi } { 6 } \right\}$

Question 19

Solve the problem.
-The figure shows a stationary spy satellite positioned 12,000 miles above the equator. What percent, to the nearest tenth, of the equator can be seen from the satellite? The diameter of Earth is 7927
Miles at the equator.

Accepted Answer

A)  $42.9 \%$ 
B)  $40.4 \%$ 
C)  $75.6 \%$ 
D)  $42.0 \%$ 
A)  $42.9 \%$ 
B)  $40.4 \%$ 
C)  $75.6 \%$ 
D)  $42.0 \%$

Solve the equation for solutions in the interval [0, 2 $[ 0,2 \pi ) .$ - $\sin x \cos x = \frac { 1 } { 2 }$

Give the exact value of the expression. - $\cos \left( 2 \arcsin \frac { 1 } { 4 } \right)$

Solve the equation for exact solutions. - $6 \arcsin x = \pi$

Use a calculator to give the real number value. Round the answer to 7 decimal places. - $y = \sin ^ { - 1 } ( 0.2079 )$

Give the exact value of the expression. - $\cos \left( \arcsin \frac { 5 } { 13 } + \arccos \frac { 3 } { 5 } \right)$

Use a calculator to give the real number value. Round the answer to 7 decimal places. - $y = \cos ^ { - 1 } ( - 0.3907 )$

Use a calculator to find the value. Give answers as real numbers and round to 4 decimal places, if necessary. - $\sin \left( \cos ^ { - 1 } 0.8324 \right)$

Solve the equation for x, where x is restricted to the given interval. - $y = 3 \tan 2 x - 1 \text {, for } x \text { in } \left( - \frac { \pi } { 4 } , \frac { \pi } { 4 } \right)$

Give the degree measure of . -ϴ = arcsin (1)

Solve. -The output voltage for an $\mathrm { AC }$ generator is approximated by $\mathrm { v } = 156 \cos \left( 120 \pi \mathrm { t } - \frac { \pi } { 3 } \right)$ . Find the smallest positive value of $\mathrm { t }$ for which the output is 75 volts. Round values to 4 decimal places.

Solve the equation for exact solutions over the interval [0, 2 $[ 0,2 \pi )$ - $\csc ^ { 5 } x - 4 \csc x = 0$

Solve the equation for exact solutions. - $\sin ^ { - 1 } x + \tan ^ { - 1 } x = 0$

Find the exact value of the real number y. - $y = \csc ^ { - 1 } ( - 1 )$

Solve the equation for exact solutions over the interval [0, 2 $[ 0,2 \pi )$ - $\sec ^ { 2 } x - 2 = \tan ^ { 2 } x$

Solve the problem. -The figure shows a stationary spy satellite positioned 12,000 miles above the equator. What percent, to the nearest tenth, of the equator can be seen from the satellite? The diameter of Earth is 7927 Miles at the equator.

Trigonometric Functions

Acute Angles and Right Triangles

Radian Measure and the Unit Circle

Graphs of the Circular Functions

Trigonometric Identities

Applications of Trigonometry and Vectors

Complex Numbers, Polar Equations, and Parametric Equations

Filters

Exam 6: Inverse Circular Functions and Trigonometric Equations

Solve the equation for solutions in the interval [0, 2 [0,2π).[ 0,2 \pi ) .[0,2π). - sin⁡xcos⁡x=12\sin x \cos x = \frac { 1 } { 2 }sinxcosx=21​

Give the exact value of the expression. - cos⁡(2arcsin⁡14)\cos \left( 2 \arcsin \frac { 1 } { 4 } \right)cos(2arcsin41​)

Solve the equation for exact solutions. - 6arcsin⁡x=π6 \arcsin x = \pi6arcsinx=π

Use a calculator to give the real number value. Round the answer to 7 decimal places. - y=sin⁡−1(0.2079)y = \sin ^ { - 1 } ( 0.2079 )y=sin−1(0.2079)

Give the exact value of the expression. - cos⁡(arcsin⁡513+arccos⁡35)\cos \left( \arcsin \frac { 5 } { 13 } + \arccos \frac { 3 } { 5 } \right)cos(arcsin135​+arccos53​)

Use a calculator to give the real number value. Round the answer to 7 decimal places. - y=cos⁡−1(−0.3907)y = \cos ^ { - 1 } ( - 0.3907 )y=cos−1(−0.3907)

Use a calculator to find the value. Give answers as real numbers and round to 4 decimal places, if necessary. - sin⁡(cos⁡−10.8324)\sin \left( \cos ^ { - 1 } 0.8324 \right)sin(cos−10.8324)

Solve the equation for x, where x is restricted to the given interval. - y=3tan⁡2x−1, for x in (−π4,π4)y = 3 \tan 2 x - 1 \text {, for } x \text { in } \left( - \frac { \pi } { 4 } , \frac { \pi } { 4 } \right)y=3tan2x−1, for x in (−4π​,4π​)