Question 1

Find the period of the function $y = \frac { 1 } { 3 } \sec \left( x - \frac { \pi } { 3 } \right)$ and sketch its graph.

Accepted Answer

period $= 2 \pi$

Question 2

Suppose that the terminal point determined by $\text { t }$ is the point $\left( - \frac { 1 } { - \sqrt { 5 } } , \frac { 2 } { - \sqrt { 6 } } \right)$ on the unit circle. Find the terminal point determined by
(a) $- \pi - t$ ,
(b) $\pi + t$ ,
(c) $2 \pi + t$ ,
(d) $t - 7 \pi$ .

Accepted Answer

(a) $\left( \frac { 1 } { \sqrt { 5 } } , \frac { 2 } { \sqrt { 5 } } \right)$
(b) $\left( \frac { 1 } { \sqrt { 5} } , - \frac { 2 } { - \sqrt { 6 } } \right)$
(c) $\left( - \frac { 1 } { - \sqrt { 5 } } , \frac { 2 } { - \sqrt { 5 } } \right)$
(d) $\left( \frac { 1 } { \sqrt { 5 } } , - \frac { 2 } { \sqrt {5} } \right)$

Question 3

On the unit circle, determine which quadrant contains the terminal point of the arc whose length is   .

Accepted Answer

First quadrant.

Question 4

Find the value of $\cos ^ { - 1 } \left( - \frac { 1 } { 2 } \right)$ if it is defined.

Accepted Answer

The answer of Find the value of \(\cos ^ {...

Question 5

Find the exact value of $\sin \left( \frac { 2 \pi } { 3 } \right)$ and $\cos \left( \frac { 2 \pi } { 3 } \right)$ .

Accepted Answer

The answer of Find the exact value of \(\sin \left(...

Question 6

Use a calculator to find an approximate value of each expression correct to five decimal places, if it is defined.
(a)
 $\cos ^ { - 1 } ( 0.34567 )$ 
(b)
 $\sin ^ { - 1 } ( - 0.34567 )$ 
(c)
 $\tan ^ { - 1 } ( 0.34567 )$

Accepted Answer

1.21784, -

Question 7

The carrier wave for an FM radio signal is modeled by the function $$y = a \sin \left( 2 \pi \left( 9.47 \times 10 ^ { 7 } \right) t \right)$$ ,
where t is measured in seconds. Find the period and frequency of the carrier wave.

Accepted Answer

period = @#LAT-DLM&

Question 8

Find the value of $\cos ^ { - 1 } \sqrt { 3 }$ if it is defined.

Accepted Answer

@#LAT-DLM& , so @#LAT-DLM& is

Question 9

Find a function that models the simple harmonic motion having the given properties. Sketch its graph. Assume that the displacement is zero at time t.
amplitude 24ft, period 2 seconds

Accepted Answer

The answer of Find a function that models the simple...

Question 10

Show that $\left( - \frac { 2 \sqrt { 2 } } { 3 } , - \frac { 1 } { 3 } \right)$ is on the unit circle.

Accepted Answer

The answer of Show that \(\left( - \frac { 2...

Question 11

Find the period of the function $y = \frac { 1 } { 5 } \cot x$ and sketch its graph.

Accepted Answer

The answer of Find the period of the function \(y...

Question 12

Find the period of the function $y = 4 \csc \left( x + \frac { \pi } { 2 } \right)$ and sketch its graph.

Accepted Answer

The answer of Find the period of the function \(y...

Question 13

Find the amplitude, period and phase shift of $y = 3 \cos ( 2 x )$ and sketch its graph.

Accepted Answer

Amplitude:

Question 14

Sketch the graph of $y = 3 - 2 \sin x$ .

Accepted Answer

The answer of Sketch the graph of \(y = 3...

Question 15

Find the exact value of $\sin \left( \frac { 19 \pi } { 6 } \right)$ and   .

Accepted Answer

The answer of Find the exact value of \(\sin \left(...

Question 16

Find the value of $\tan ^ { - 1 } \sqrt { 3 }$ if it is defined.

Accepted Answer

The answer of Find the value of \(\tan ^ {...

Question 17

Find the period of the function $y = 3 \sec \left( \frac { \pi } { 2 } \left( x - \frac { 1 } { 2 } \right) \right)$ and sketch its graph.

Accepted Answer

The answer of Find the period of the function \(y...

Question 18

Show that $$\left( - \frac { 1 } { 8 } , \frac { 3 , \sqrt { 7 } } { 8 } \right)$$ is on the unit circle.

Accepted Answer

The answer of Show that \[\left( - \frac { 1...

Question 19

Find the exact value of each expression, if it is defined.
(a)
 $\sin ^ { - 1 } ( - 2 )$ ( b)
 $\cos ^ { - 1 } ( - \sqrt { 3 } )$ 
(c)
 $\tan ^ { - 1 } ( - 1 )$

Accepted Answer

not define

Question 20

An initial amplitude k, damping constant c, and frequency f are given.
a.) Find a function of the form $y = k \mathcal { e } ^ { - c t } \cos \omega t$ that models damped harmonic motion.
b.) Graph the function. $k = 15 , c = 0.25 , f = 0.6$

Accepted Answer

The answer of An initial amplitude k, damping constant c,...

Find the period of the function $y = \frac { 1 } { 3 } \sec \left( x - \frac { \pi } { 3 } \right)$ and sketch its graph.

Suppose that the terminal point determined by $\text { t }$ is the point $\left( - \frac { 1 } { - \sqrt { 5 } } , \frac { 2 } { - \sqrt { 6 } } \right)$ on the unit circle. Find the terminal point determined by (a) $- \pi - t$ , (b) $\pi + t$ , (c) $2 \pi + t$ , (d) $t - 7 \pi$ .

On the unit circle, determine which quadrant contains the terminal point of the arc whose length is .

Find the value of $\cos ^ { - 1 } \left( - \frac { 1 } { 2 } \right)$ if it is defined.

Find the exact value of $\sin \left( \frac { 2 \pi } { 3 } \right)$ and $\cos \left( \frac { 2 \pi } { 3 } \right)$ .

Use a calculator to find an approximate value of each expression correct to five decimal places, if it is defined. (a) $\cos ^ { - 1 } ( 0.34567 )$ (b) $\sin ^ { - 1 } ( - 0.34567 )$ (c) $\tan ^ { - 1 } ( 0.34567 )$

The carrier wave for an FM radio signal is modeled by the function $y = a \sin \left( 2 \pi \left( 9.47 \times 10 ^ { 7 } \right) t \right)$ , where t is measured in seconds. Find the period and frequency of the carrier wave.

Find the value of $\cos ^ { - 1 } \sqrt { 3 }$ if it is defined.

Find a function that models the simple harmonic motion having the given properties. Sketch its graph. Assume that the displacement is zero at time t. amplitude 24ft, period 2 seconds

Show that $\left( - \frac { 2 \sqrt { 2 } } { 3 } , - \frac { 1 } { 3 } \right)$ is on the unit circle.

Find the period of the function $y = \frac { 1 } { 5 } \cot x$ and sketch its graph.

Find the period of the function $y = 4 \csc \left( x + \frac { \pi } { 2 } \right)$ and sketch its graph.

Find the amplitude, period and phase shift of $y = 3 \cos ( 2 x )$ and sketch its graph.

Sketch the graph of $y = 3 - 2 \sin x$ .

Find the exact value of $\sin \left( \frac { 19 \pi } { 6 } \right)$ and $Find the exact value of \sin \left( \frac { 19 \pi } { 6 } \right) and .$ .

Find the value of $\tan ^ { - 1 } \sqrt { 3 }$ if it is defined.

Find the period of the function $y = 3 \sec \left( \frac { \pi } { 2 } \left( x - \frac { 1 } { 2 } \right) \right)$ and sketch its graph.

Show that $\left( - \frac { 1 } { 8 } , \frac { 3 , \sqrt { 7 } } { 8 } \right)$ is on the unit circle.

Find the exact value of each expression, if it is defined. (a) $\sin ^ { - 1 } ( - 2 )$ ( b) $\cos ^ { - 1 } ( - \sqrt { 3 } )$ (c) $\tan ^ { - 1 } ( - 1 )$

An initial amplitude k, damping constant c, and frequency f are given. a.) Find a function of the form $y = k \mathcal { e } ^ { - c t } \cos \omega t$ that models damped harmonic motion. b.) Graph the function. $k = 15 , c = 0.25 , f = 0.6$

Fundamentals

Functions

Polynomial and Rational Functions

Exponential and Logarithmic Functions

Trigonometric Functions: Right Triangle Approach

Analytic Trigonometry

Polar Coordinates and Parametric Equations

Vectors in Two and Three Dimensions

Systems of Equations and Inequalities

Conic Sections

Sequences and Series

Limits: a Preview of Calculus

Filters

Exam 5: Trigonometric Functions: Unit Circle Approach

Find the period of the function y=13sec⁡(x−π3)y = \frac { 1 } { 3 } \sec \left( x - \frac { \pi } { 3 } \right)y=31​sec(x−3π​) and sketch its graph.

On the unit circle, determine which quadrant contains the terminal point of the arc whose length is .

Find the value of cos⁡−1(−12)\cos ^ { - 1 } \left( - \frac { 1 } { 2 } \right)cos−1(−21​) if it is defined.

Find the exact value of sin⁡(2π3)\sin \left( \frac { 2 \pi } { 3 } \right)sin(32π​) and cos⁡(2π3)\cos \left( \frac { 2 \pi } { 3 } \right)cos(32π​) .

The carrier wave for an FM radio signal is modeled by the function y=asin⁡(2π(9.47×107)t)y = a \sin \left( 2 \pi \left( 9.47 \times 10 ^ { 7 } \right) t \right)y=asin(2π(9.47×107)t) , where t is measured in seconds. Find the period and frequency of the carrier wave.

Find the value of cos⁡−13\cos ^ { - 1 } \sqrt { 3 }cos−13​ if it is defined.

Find a function that models the simple harmonic motion having the given properties. Sketch its graph. Assume that the displacement is zero at time t. amplitude 24ft, period 2 seconds

Show that (−223,−13)\left( - \frac { 2 \sqrt { 2 } } { 3 } , - \frac { 1 } { 3 } \right)(−322​​,−31​) is on the unit circle.

Find the period of the function y=15cot⁡xy = \frac { 1 } { 5 } \cot xy=51​cotx and sketch its graph.

Find the period of the function y=4csc⁡(x+π2)y = 4 \csc \left( x + \frac { \pi } { 2 } \right)y=4csc(x+2π​) and sketch its graph.

Find the amplitude, period and phase shift of y=3cos⁡(2x)y = 3 \cos ( 2 x )y=3cos(2x) and sketch its graph.

Sketch the graph of y=3−2sin⁡xy = 3 - 2 \sin xy=3−2sinx .

Find the exact value of sin⁡(19π6)\sin \left( \frac { 19 \pi } { 6 } \right)sin(619π​) and .

Find the value of tan⁡−13\tan ^ { - 1 } \sqrt { 3 }tan−13​ if it is defined.

Find the period of the function y=3sec⁡(π2(x−12))y = 3 \sec \left( \frac { \pi } { 2 } \left( x - \frac { 1 } { 2 } \right) \right)y=3sec(2π​(x−21​)) and sketch its graph.

Show that (−18,3,78)\left( - \frac { 1 } { 8 } , \frac { 3 , \sqrt { 7 } } { 8 } \right)(−81​,83,7​​) is on the unit circle.

Find the exact value of each expression, if it is defined. (a) sin⁡−1(−2)\sin ^ { - 1 } ( - 2 )sin−1(−2) ( b) cos⁡−1(−3)\cos ^ { - 1 } ( - \sqrt { 3 } )cos−1(−3​) (c) tan⁡−1(−1)\tan ^ { - 1 } ( - 1 )tan−1(−1)

Fundamentals

Functions

Polynomial and Rational Functions

Exponential and Logarithmic Functions

Trigonometric Functions: Right Triangle Approach

Analytic Trigonometry

Polar Coordinates and Parametric Equations

Vectors in Two and Three Dimensions

Systems of Equations and Inequalities

Conic Sections

Sequences and Series

Limits: a Preview of Calculus

Filters

Find the period of the function $y = \frac { 1 } { 3 } \sec \left( x - \frac { \pi } { 3 } \right)$ and sketch its graph.

Find the value of $\cos ^ { - 1 } \left( - \frac { 1 } { 2 } \right)$ if it is defined.

Find the exact value of $\sin \left( \frac { 2 \pi } { 3 } \right)$ and $\cos \left( \frac { 2 \pi } { 3 } \right)$ .

The carrier wave for an FM radio signal is modeled by the function $y = a \sin \left( 2 \pi \left( 9.47 \times 10 ^ { 7 } \right) t \right)$ , where t is measured in seconds. Find the period and frequency of the carrier wave.

Find the value of $\cos ^ { - 1 } \sqrt { 3 }$ if it is defined.

Show that $\left( - \frac { 2 \sqrt { 2 } } { 3 } , - \frac { 1 } { 3 } \right)$ is on the unit circle.

Find the period of the function $y = \frac { 1 } { 5 } \cot x$ and sketch its graph.

Find the period of the function $y = 4 \csc \left( x + \frac { \pi } { 2 } \right)$ and sketch its graph.

Find the amplitude, period and phase shift of $y = 3 \cos ( 2 x )$ and sketch its graph.

Sketch the graph of $y = 3 - 2 \sin x$ .

Find the exact value of $\sin \left( \frac { 19 \pi } { 6 } \right)$ and $Find the exact value of \sin \left( \frac { 19 \pi } { 6 } \right) and .$ .

Find the value of $\tan ^ { - 1 } \sqrt { 3 }$ if it is defined.

Find the period of the function $y = 3 \sec \left( \frac { \pi } { 2 } \left( x - \frac { 1 } { 2 } \right) \right)$ and sketch its graph.

Show that $\left( - \frac { 1 } { 8 } , \frac { 3 , \sqrt { 7 } } { 8 } \right)$ is on the unit circle.

Find the exact value of each expression, if it is defined. (a) $\sin ^ { - 1 } ( - 2 )$ ( b) $\cos ^ { - 1 } ( - \sqrt { 3 } )$ (c) $\tan ^ { - 1 } ( - 1 )$