Exam 6: The Circular Functions and Their Graphs

Solve the problem. -Suppose that a weight on a spring has an initial position of $s ( 0 ) = 6$ inches and a period of $P = 2.5$ seconds. Find a function $s ( t ) = a \cos ( 2 \pi F t )$ that models the displacement of the weight.

(Multiple Choice)

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Question 201

Solve the problem. -The chart represents the amount of fuel consumed by a machine used in manufacturing. The machine is turned on at the beginning of the day, takes a certain amount of time to reach its full power (the point at which it uses the most fuel per hour), runs for a certain number of hours, and is shut off at the end of the work day. The fuel usage per hour of the machine is represented by a periodic function. When does the machine first reach its full power?

(Multiple Choice)

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Question 202

Graph the function. - $y=\frac{2}{3} \csc \left(\frac{2}{5} x-\frac{\pi}{2}\right)$ $Graph the function. - y=\frac{2}{3} \csc \left(\frac{2}{5} x-\frac{\pi}{2}\right)$

(Multiple Choice)

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Question 203

Solve the problem. -A pendulum swinging through a central angle of $108 ^ { \circ }$ completes an arc of length $24 \mathrm {~cm}$ . What is the length of the pendulum? Round to the nearest hundredth.

(Multiple Choice)

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Question 204

Solve the problem. -A weight attached to a spring is pulled down 7 inches below the equilibrium position. Assuming that the frequency of the system is $\frac { 5 } { \pi }$ cycles per second, determine a trigonometric model that gives the position of the weight at time $t$ second.

(Multiple Choice)

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Question 205

Graph the function. - $y=\frac{1}{2} \sin \frac{1}{2} x$ $Graph the function. - y=\frac{1}{2} \sin \frac{1}{2} x$

(Multiple Choice)

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Question 206

Graph the function. - $y=-\frac{1}{2} \cos \left(x+\frac{\pi}{3}\right)$ $Graph the function. - y=-\frac{1}{2} \cos \left(x+\frac{\pi}{3}\right)$

(Multiple Choice)

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Question 207

Provide an appropriate response. -Describe how an angle measure can be converted from radians to degrees.

(Multiple Choice)

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Question 208

Convert the degree measure to radians. Leave answer as a multiple of π. - $36 ^ { \circ }$

(Multiple Choice)

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Question 209

Solve the problem. -Find the measure (in radians) of a central angle of a sector of area 45 square inches in a circle of radius 9 inches. Round to the nearest hundredth.

(Multiple Choice)

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Question 210

Match the function with its graph. -1) $y = \sec x$ 2) $y = \csc x$ 3) $y = - \sec x$ 4) $y = - \csc x$ A. $Match the function with its graph. -1) y = \sec x 2) y = \csc x 3) y = - \sec x 4) y = - \csc x A. B. C. D.$ B. $Match the function with its graph. -1) y = \sec x 2) y = \csc x 3) y = - \sec x 4) y = - \csc x A. B. C. D.$ C. $Match the function with its graph. -1) y = \sec x 2) y = \csc x 3) y = - \sec x 4) y = - \csc x A. B. C. D.$ D. $Match the function with its graph. -1) y = \sec x 2) y = \csc x 3) y = - \sec x 4) y = - \csc x A. B. C. D.$

(Multiple Choice)

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Question 211

Graph the function. - $y=\csc \left(x-\frac{\pi}{4}\right)$ $Graph the function. - y=\csc \left(x-\frac{\pi}{4}\right) \$ \

(Multiple Choice)

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Question 212

Solve the problem. -The position of a weight attached to a spring is $s ( t ) = - 5 \cos 16 \pi t$ inches after $t$ seconds. What is the maximum height that the weight reaches above the equilibrium position and when does it first reach the maximum height?

(Multiple Choice)

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Question 213

Convert the degree measure to radians, correct to four decimal places. Use 3.1416 for π. - $16 ^ { \circ } 10 ^ { \prime }$

(Multiple Choice)

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Question 214

Solve the problem. -A weight attached to a spring is pulled down 4 inches below the equilibrium position. Assuming that the period of the system is $\frac { 1 } { 3 } \mathrm { sec }$ , what is the frequency of the system?

(Multiple Choice)

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Question 215

Solve the problem. -Suppose the tip of the minute hand of a clock is 8 in. from the center of the clock. Determine the distance traveled by the tip of the minute hand in $3 \frac { 1 } { 2 }$ hours. Give an exact answer.

(Multiple Choice)

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Question 216

Solve the problem. -The temperature in Fairbanks is approximated by $T ( x ) = 37 \sin \left[ \frac { 2 \pi } { 365 } ( x - 101 ) \right] + 25$ where $\mathrm { T } ( \mathrm { x } )$ is the temperature on day $\mathrm { x }$ , with $\mathrm { x } = 1$ corresponding to Jan. 1 and $\mathrm { x } = 365$ corresponding to Dec. 31. Estimate the temperature on day 230 .

(Multiple Choice)

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Question 217

Solve the problem. -A generator produces an alternating current according to the equation $\mathrm { I } = 40 \sin 28 \pi \mathrm { t }$ , where $\mathrm { t }$ is time in seconds and $\mathrm { I }$ is the current in amperes. What is the smallest time $t$ such that $\mathrm { I } = 20$ ?

(Multiple Choice)

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Question 218

Give the amplitude or period as requested. -Amplitude of $y = - 5 \sin x$

(Multiple Choice)

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Question 219

Solve the problem. -The position of a weight attached to a spring is $s ( t ) = - 5 \cos ( 6 \pi t )$ inches after $t$ seconds. What is the maximum height that the weight rises above the equilibrium position?

(Multiple Choice)

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Question 220

Showing 201 - 220 of 289

Solve the problem. -Suppose that a weight on a spring has an initial position of $s ( 0 ) = 6$ inches and a period of $P = 2.5$ seconds. Find a function $s ( t ) = a \cos ( 2 \pi F t )$ that models the displacement of the weight.

Graph the function. - $y=\frac{2}{3} \csc \left(\frac{2}{5} x-\frac{\pi}{2}\right)$ $Graph the function. - y=\frac{2}{3} \csc \left(\frac{2}{5} x-\frac{\pi}{2}\right)$

Solve the problem. -A pendulum swinging through a central angle of $108 ^ { \circ }$ completes an arc of length $24 \mathrm {~cm}$ . What is the length of the pendulum? Round to the nearest hundredth.

Solve the problem. -A weight attached to a spring is pulled down 7 inches below the equilibrium position. Assuming that the frequency of the system is $\frac { 5 } { \pi }$ cycles per second, determine a trigonometric model that gives the position of the weight at time $t$ second.

Graph the function. - $y=\frac{1}{2} \sin \frac{1}{2} x$ $Graph the function. - y=\frac{1}{2} \sin \frac{1}{2} x$

Graph the function. - $y=-\frac{1}{2} \cos \left(x+\frac{\pi}{3}\right)$ $Graph the function. - y=-\frac{1}{2} \cos \left(x+\frac{\pi}{3}\right)$

Provide an appropriate response. -Describe how an angle measure can be converted from radians to degrees.

Convert the degree measure to radians. Leave answer as a multiple of π. - $36 ^ { \circ }$

Solve the problem. -Find the measure (in radians) of a central angle of a sector of area 45 square inches in a circle of radius 9 inches. Round to the nearest hundredth.

Graph the function. - $y=\csc \left(x-\frac{\pi}{4}\right)$ $Graph the function. - y=\csc \left(x-\frac{\pi}{4}\right) \$ \

Solve the problem. -The position of a weight attached to a spring is $s ( t ) = - 5 \cos 16 \pi t$ inches after $t$ seconds. What is the maximum height that the weight reaches above the equilibrium position and when does it first reach the maximum height?

Convert the degree measure to radians, correct to four decimal places. Use 3.1416 for π. - $16 ^ { \circ } 10 ^ { \prime }$

Solve the problem. -A weight attached to a spring is pulled down 4 inches below the equilibrium position. Assuming that the period of the system is $\frac { 1 } { 3 } \mathrm { sec }$ , what is the frequency of the system?

Solve the problem. -Suppose the tip of the minute hand of a clock is 8 in. from the center of the clock. Determine the distance traveled by the tip of the minute hand in $3 \frac { 1 } { 2 }$ hours. Give an exact answer.

Solve the problem. -A generator produces an alternating current according to the equation $\mathrm { I } = 40 \sin 28 \pi \mathrm { t }$ , where $\mathrm { t }$ is time in seconds and $\mathrm { I }$ is the current in amperes. What is the smallest time $t$ such that $\mathrm { I } = 20$ ?

Give the amplitude or period as requested. -Amplitude of $y = - 5 \sin x$

Solve the problem. -The position of a weight attached to a spring is $s ( t ) = - 5 \cos ( 6 \pi t )$ inches after $t$ seconds. What is the maximum height that the weight rises above the equilibrium position?

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Exam 6: The Circular Functions and Their Graphs

Solve the problem. -Suppose that a weight on a spring has an initial position of s(0)=6s ( 0 ) = 6s(0)=6 inches and a period of P=2.5P = 2.5P=2.5 seconds. Find a function s(t)=acos⁡(2πFt)s ( t ) = a \cos ( 2 \pi F t )s(t)=acos(2πFt) that models the displacement of the weight.

Graph the function. - y=23csc⁡(25x−π2)y=\frac{2}{3} \csc \left(\frac{2}{5} x-\frac{\pi}{2}\right)y=32​csc(52​x−2π​)

Solve the problem. -A pendulum swinging through a central angle of 108∘108 ^ { \circ }108∘ completes an arc of length 24 cm24 \mathrm {~cm}24 cm . What is the length of the pendulum? Round to the nearest hundredth.

Solve the problem. -A weight attached to a spring is pulled down 7 inches below the equilibrium position. Assuming that the frequency of the system is 5π\frac { 5 } { \pi }π5​ cycles per second, determine a trigonometric model that gives the position of the weight at time ttt second.

Graph the function. - y=12sin⁡12xy=\frac{1}{2} \sin \frac{1}{2} xy=21​sin21​x

Graph the function. - y=−12cos⁡(x+π3)y=-\frac{1}{2} \cos \left(x+\frac{\pi}{3}\right)y=−21​cos(x+3π​)

Provide an appropriate response. -Describe how an angle measure can be converted from radians to degrees.

Convert the degree measure to radians. Leave answer as a multiple of π. - 36∘36 ^ { \circ }36∘

Solve the problem. -Find the measure (in radians) of a central angle of a sector of area 45 square inches in a circle of radius 9 inches. Round to the nearest hundredth.

Match the function with its graph. -1) y=sec⁡xy = \sec xy=secx 2) y=csc⁡xy = \csc xy=cscx 3) y=−sec⁡xy = - \sec xy=−secx 4) y=−csc⁡xy = - \csc xy=−cscx A. B. C. D.

Graph the function. - y=csc⁡(x−π4)y=\csc \left(x-\frac{\pi}{4}\right)y=csc(x−4π​) \

Solve the problem. -The position of a weight attached to a spring is s(t)=−5cos⁡16πts ( t ) = - 5 \cos 16 \pi ts(t)=−5cos16πt inches after ttt seconds. What is the maximum height that the weight reaches above the equilibrium position and when does it first reach the maximum height?

Convert the degree measure to radians, correct to four decimal places. Use 3.1416 for π. - 16∘10′16 ^ { \circ } 10 ^ { \prime }16∘10′

Solve the problem. -A weight attached to a spring is pulled down 4 inches below the equilibrium position. Assuming that the period of the system is 13sec\frac { 1 } { 3 } \mathrm { sec }31​sec , what is the frequency of the system?

Solve the problem. -Suppose the tip of the minute hand of a clock is 8 in. from the center of the clock. Determine the distance traveled by the tip of the minute hand in 3123 \frac { 1 } { 2 }321​ hours. Give an exact answer.

Give the amplitude or period as requested. -Amplitude of y=−5sin⁡xy = - 5 \sin xy=−5sinx

Solve the problem. -The position of a weight attached to a spring is s(t)=−5cos⁡(6πt)s ( t ) = - 5 \cos ( 6 \pi t )s(t)=−5cos(6πt) inches after ttt seconds. What is the maximum height that the weight rises above the equilibrium position?

Equations and Inequalities

Graphs and Functions

Polynomial and Rational Functions

Inverse, Exponential, and Logarithmic Functions

Trigonometric Functions

Trigonometric Identities and Equations

Applications of Trigonometry

Systems and Matrices

Analytic Geometry

Further Topics in Algebra

Review of Basic Concepts

Filters

Solve the problem. -Suppose that a weight on a spring has an initial position of $s ( 0 ) = 6$ inches and a period of $P = 2.5$ seconds. Find a function $s ( t ) = a \cos ( 2 \pi F t )$ that models the displacement of the weight.

Graph the function. - $y=\frac{2}{3} \csc \left(\frac{2}{5} x-\frac{\pi}{2}\right)$ $Graph the function. - y=\frac{2}{3} \csc \left(\frac{2}{5} x-\frac{\pi}{2}\right)$

Solve the problem. -A pendulum swinging through a central angle of $108 ^ { \circ }$ completes an arc of length $24 \mathrm {~cm}$ . What is the length of the pendulum? Round to the nearest hundredth.

Solve the problem. -A weight attached to a spring is pulled down 7 inches below the equilibrium position. Assuming that the frequency of the system is $\frac { 5 } { \pi }$ cycles per second, determine a trigonometric model that gives the position of the weight at time $t$ second.

Graph the function. - $y=\frac{1}{2} \sin \frac{1}{2} x$ $Graph the function. - y=\frac{1}{2} \sin \frac{1}{2} x$

Graph the function. - $y=-\frac{1}{2} \cos \left(x+\frac{\pi}{3}\right)$ $Graph the function. - y=-\frac{1}{2} \cos \left(x+\frac{\pi}{3}\right)$

Convert the degree measure to radians. Leave answer as a multiple of π. - $36 ^ { \circ }$

Graph the function. - $y=\csc \left(x-\frac{\pi}{4}\right)$ $Graph the function. - y=\csc \left(x-\frac{\pi}{4}\right) \$ \

Solve the problem. -The position of a weight attached to a spring is $s ( t ) = - 5 \cos 16 \pi t$ inches after $t$ seconds. What is the maximum height that the weight reaches above the equilibrium position and when does it first reach the maximum height?

Convert the degree measure to radians, correct to four decimal places. Use 3.1416 for π. - $16 ^ { \circ } 10 ^ { \prime }$

Solve the problem. -A weight attached to a spring is pulled down 4 inches below the equilibrium position. Assuming that the period of the system is $\frac { 1 } { 3 } \mathrm { sec }$ , what is the frequency of the system?

Solve the problem. -Suppose the tip of the minute hand of a clock is 8 in. from the center of the clock. Determine the distance traveled by the tip of the minute hand in $3 \frac { 1 } { 2 }$ hours. Give an exact answer.

Give the amplitude or period as requested. -Amplitude of $y = - 5 \sin x$

Solve the problem. -The position of a weight attached to a spring is $s ( t ) = - 5 \cos ( 6 \pi t )$ inches after $t$ seconds. What is the maximum height that the weight rises above the equilibrium position?