Question 1

Use a graphing calculator to make a conjecture as to whether each equation is an identity.
-$\cos 2 x = \sin 2 x \tan x$

Accepted Answer

A)  Not an identity 
B)  Identity 
A)  Not an identity 
B)  Identity

Question 2

Verify that each equation is an identity.
-$$\frac { 1 - \cot ^ { 2 } \theta } { 1 + \cot ^ { 2 } \theta } + 1 = 2 \sin ^ { 2 } \theta$$

Accepted Answer

\[\begin{array} { l } 
\frac { 1 - \cot

Question 3

Verify that each equation is an identity.
-$$\sin ^ { 3 } x \cos ^ { 2 } x = \sin x \left( \cos ^ { 2 } x - \cos ^ { 4 } x \right)$$

Accepted Answer

The answer of Verify that each equation is an identity.
-\[\sin...

Question 4

For the graph of a circular function y = f(x), determine whether f(-x) = f(x) or f(-x) = -f(x) is true.
-

Accepted Answer

A)  $f ( - x ) = - f ( x )$ 
B)  $f ( - x ) = f ( x )$ 
A)  $f ( - x ) = - f ( x )$ 
B)  $f ( - x ) = f ( x )$

Question 5

Write in terms of the cofunction of a complementary angle.
-$$\sin \frac { \pi } { 9 }$$

Accepted Answer

A)  $\csc \frac { 7 \pi } { 18 }$ 
B)  $\cos \frac { 7 \pi } { 18 }$ 
C)  $\cos \frac { 8 \pi } { 9 }$ 
D)  $\csc \frac { 8 \pi } { 9 }$ 
A)  $\csc \frac { 7 \pi } { 18 }$ 
B)  $\cos \frac { 7 \pi } { 18 }$ 
C)  $\cos \frac { 8 \pi } { 9 }$ 
D)  $\csc \frac { 8 \pi } { 9 }$

Question 6

Solve the problem.
-Consider the formula $T = 591 - 76 \cos 2 \theta$, where $\theta$ is measured in degrees. To the nearest hundredth of a degree, what is the smallest positive value of $\theta$ for which the value of $T$ will be 567?

Accepted Answer

A)  $35.79 ^ { \circ }$ 
B)  $74.58 ^ { \circ }$ 
C)  $71.58 ^ { \circ }$ 
D)  $32.79 ^ { \circ }$ 
A)  $35.79 ^ { \circ }$ 
B)  $74.58 ^ { \circ }$ 
C)  $71.58 ^ { \circ }$ 
D)  $32.79 ^ { \circ }$

Question 7

Complete the sentence so the result is an identity. Let x be any real number.
-$$ـــــــ+ \tan ^ { 2 } x = \sec ^ { 2 } x$$

Accepted Answer

A)  $\sin ^ { 2 } x$ 
B)  $\cos ^ { 2 } x$ 
C)  1 
D)  $- 1$ 
A)  $\sin ^ { 2 } x$ 
B)  $\cos ^ { 2 } x$ 
C)  1 
D)  $- 1$

Question 8

Solve the equation for x, where x is restricted to the given interval.
-$y = \sin x - \pi$, for $x$ in $\left[ - \frac { \pi } { 2 } , \frac { \pi } { 2 } \right]$

Accepted Answer

A)  $x = \arcsin ( y - \pi )$ 
B)  $x = \arcsin ( y + \pi )$ 
C)  $x = \arcsin y + \pi$ 
D)  $x = \arcsin y - \pi$ 
A)  $x = \arcsin ( y - \pi )$ 
B)  $x = \arcsin ( y + \pi )$ 
C)  $x = \arcsin y + \pi$ 
D)  $x = \arcsin y - \pi$

Question 9

Solve the equation for solutions in the interval [0°, 360°). Round to the nearest degree.
-$\cos ^ { 2 } \frac { \theta } { 2 } = 1$

Accepted Answer

A)  $\left\{ 0 ^ { \circ } \right\}$ 
B)  $\left\{ 0 ^ { \circ } , 45 ^ { \circ } , 90 ^ { \circ } , 135 ^ { \circ } , 180 ^ { \circ } , 225 ^ { \circ } , 270 ^ { \circ } \right\}$ 
C)  $\left\{ 0 ^ { \circ } , 90 ^ { \circ } , 180 ^ { \circ } , 270 ^ { \circ } \right\}$ 
D)  $\left\{ 33 ^ { \circ } , 57 ^ { \circ } , 123 ^ { \circ } , 147 ^ { \circ } , 213 ^ { \circ } , 237 ^ { \circ } , 303 ^ { \circ } , 327 ^ { \circ } \right\}$ 
A)  $\left\{ 0 ^ { \circ } \right\}$ 
B)  $\left\{ 0 ^ { \circ } , 45 ^ { \circ } , 90 ^ { \circ } , 135 ^ { \circ } , 180 ^ { \circ } , 225 ^ { \circ } , 270 ^ { \circ } \right\}$ 
C)  $\left\{ 0 ^ { \circ } , 90 ^ { \circ } , 180 ^ { \circ } , 270 ^ { \circ } \right\}$ 
D)  $\left\{ 33 ^ { \circ } , 57 ^ { \circ } , 123 ^ { \circ } , 147 ^ { \circ } , 213 ^ { \circ } , 237 ^ { \circ } , 303 ^ { \circ } , 327 ^ { \circ } \right\}$

Question 10

Use a sum or difference identity to find the exact value.
-$\frac { \tan 20 ^ { \circ } + \tan 10 ^ { \circ } } { 1 - \tan 20 ^ { \circ } \tan 10 ^ { \circ } }$

Accepted Answer

A)  2 
B)  $\frac { \sqrt { 3 } } { 3 }$ 
C)  $\sqrt { 3 }$ 
D)  $\frac { 1 } { 2 }$ 
A)  2 
B)  $\frac { \sqrt { 3 } } { 3 }$ 
C)  $\sqrt { 3 }$ 
D)  $\frac { 1 } { 2 }$

Question 11

Solve the equation for exact solutions.
-$\arctan x = \operatorname { arccot } \frac { 6 } { 19 }$

Accepted Answer

A)  $\varnothing$ 
B)  $\{ 0 \}$ 
C)  $\left\{ \frac { 19 } { 6 } \right\}$ 
D)  $\left\{ \frac { 6 } { 19 } \right\}$ 
A)  $\varnothing$ 
B)  $\{ 0 \}$ 
C)  $\left\{ \frac { 19 } { 6 } \right\}$ 
D)  $\left\{ \frac { 6 } { 19 } \right\}$

Question 12

Verify that each equation is an identity.
-$$\cot ( 2 \theta ) = \frac { \csc ^ { 2 } \theta - 2 } { 2 \cot \theta }$$

Accepted Answer

The answer of Verify that each equation is an identity.
-\[\cot...

Question 13

Verify that the equation is an identity.
-$$\sin \left( \frac { 3 \pi } { 2 } - \theta \right) = - \cos \theta$$

Accepted Answer

The answer of Verify that the equation is an identity.
-\[\sin...

Question 14

Find the exact value of the expression using the provided information.
-Find $\sin ( \mathrm { s } - \mathrm { t } )$ given that $\sin \mathrm { s } = - \frac { 1 } { 2 }$, with $\mathrm { s }$ in quadrant IV, and $\sin t = \frac { 1 } { 4 }$, with $t$ in quadrant II.

Accepted Answer

A)  $\frac { - \sqrt { 15 } + \sqrt { 3 } } { 8 }$ 
B)  $\frac { \sqrt { 15 } - \sqrt { 3 } } { 8 }$ 
C)  $\frac { \sqrt { 15 } + \sqrt { 3 } } { 8 }$ 
D)  $\frac { - \sqrt { 15 } - \sqrt { 3 } } { 8 }$ 
A)  $\frac { - \sqrt { 15 } + \sqrt { 3 } } { 8 }$ 
B)  $\frac { \sqrt { 15 } - \sqrt { 3 } } { 8 }$ 
C)  $\frac { \sqrt { 15 } + \sqrt { 3 } } { 8 }$ 
D)  $\frac { - \sqrt { 15 } - \sqrt { 3 } } { 8 }$

Question 15

Verify that the equation is an identity.
-$$\sin ( \alpha - \beta ) \cos ( \alpha + \beta ) = \sin \alpha \cos \alpha - \sin \beta \cos \beta$$

Accepted Answer

\[\begin{array} { l } 
\sin ( \alpha - \

Question 16

Determine all solutions of the equation in radians.
-Find $\sin \frac { x } { 2 }$, given that $\sin x = \frac { 1 } { 4 }$ and $x$ terminates in $0 < x < \frac { \pi } { 2 }$.

Accepted Answer

A)  $\frac { \sqrt { 10 } } { 4 }$ 
B)  $\frac { \sqrt { 8 + 2 \sqrt { 15 } } } { 4 }$ 
C)  $\frac { \sqrt { 8 - 2 \sqrt { 15 } } } { 4 }$ 
D)  $\frac { \sqrt { 6 } } { 4 }$ 
A)  $\frac { \sqrt { 10 } } { 4 }$ 
B)  $\frac { \sqrt { 8 + 2 \sqrt { 15 } } } { 4 }$ 
C)  $\frac { \sqrt { 8 - 2 \sqrt { 15 } } } { 4 }$ 
D)  $\frac { \sqrt { 6 } } { 4 }$

Question 17

Find the exact value of the expression using the provided information.
-Find $\cos ( s + t )$ given that $\sin s = - \frac { 1 } { 2 }$, with $s$ in quadrant IV, and $\sin t = \frac { 1 } { 4 }$, with $t$ in quadrant II.

Accepted Answer

A)  $- \frac { 4 \sqrt { 3 } + \sqrt { 15 } } { 11 }$ 
B)  $\frac { 1 + 3 \sqrt { 5 } } { 8 }$ 
C)  $\frac { 1 - 3 \sqrt { 5 } } { 8 }$ 
D)  $\frac { 4 \sqrt { 3 } - \sqrt { 15 } } { 11 }$ 
A)  $- \frac { 4 \sqrt { 3 } + \sqrt { 15 } } { 11 }$ 
B)  $\frac { 1 + 3 \sqrt { 5 } } { 8 }$ 
C)  $\frac { 1 - 3 \sqrt { 5 } } { 8 }$ 
D)  $\frac { 4 \sqrt { 3 } - \sqrt { 15 } } { 11 }$

Question 18

Write the following as an algebraic expression in u, u > 0.
-$\sin ( \arctan \mathrm { u } )$

Accepted Answer

A)  $u \sqrt { u ^ { 2 } + 1 }$ 
B)  $\frac { u \sqrt { u ^ { 2 } - 1 } } { u ^ { 2 } - 1 }$ 
C)  $\frac { \mathrm { u } \sqrt { \mathrm { u } ^ { 2 } + 1 } } { \mathrm { u } ^ { 2 } + 1 }$ 
D)  $\frac { \sqrt { \mathrm { u } ^ { 2 } + 1 } } { \mathrm { u } ^ { 2 } + 1 }$ 
A)  $u \sqrt { u ^ { 2 } + 1 }$ 
B)  $\frac { u \sqrt { u ^ { 2 } - 1 } } { u ^ { 2 } - 1 }$ 
C)  $\frac { \mathrm { u } \sqrt { \mathrm { u } ^ { 2 } + 1 } } { \mathrm { u } ^ { 2 } + 1 }$ 
D)  $\frac { \sqrt { \mathrm { u } ^ { 2 } + 1 } } { \mathrm { u } ^ { 2 } + 1 }$

Question 19

Solve the equation for solutions in the interval [0°, 360°). Round to the nearest degree.
-$\sin 2 \theta + \cos 2 \theta = 1$

Accepted Answer

A)  $\left\{ 0 ^ { \circ } , 90 ^ { \circ } , 180 ^ { \circ } , 270 ^ { \circ } \right\}$ 
B)  $\left\{ 33 ^ { \circ } , 57 ^ { \circ } , 123 ^ { \circ } , 147 ^ { \circ } , 213 ^ { \circ } , 237 ^ { \circ } , 303 ^ { \circ } \right\}$ 
C)  $\left\{ 0 ^ { \circ } \right\}$ 
D)  $\left\{ 0 ^ { \circ } , 45 ^ { \circ } , 180 ^ { \circ } , 225 ^ { \circ } \right\}$ 
A)  $\left\{ 0 ^ { \circ } , 90 ^ { \circ } , 180 ^ { \circ } , 270 ^ { \circ } \right\}$ 
B)  $\left\{ 33 ^ { \circ } , 57 ^ { \circ } , 123 ^ { \circ } , 147 ^ { \circ } , 213 ^ { \circ } , 237 ^ { \circ } , 303 ^ { \circ } \right\}$ 
C)  $\left\{ 0 ^ { \circ } \right\}$ 
D)  $\left\{ 0 ^ { \circ } , 45 ^ { \circ } , 180 ^ { \circ } , 225 ^ { \circ } \right\}$

Question 20

Find the exact value of the expression using the provided information.
-Find $\tan ( \mathrm { s } - \mathrm { t } )$ given that $\sin \mathrm { s } = - \frac { 3 \sqrt { 13 } } { 13 }$, with $\mathrm { s }$ in quadrant $\mathrm { IV }$, and $\sin \mathrm { t } = - \frac { \sqrt { 10 } } { 10 }$, with $\mathrm { t }$ in quadrant IV.

Accepted Answer

A)  $- \frac { 11 } { 9 }$ 
B)  $\frac { 11 } { 3 }$ 
C)  $- \frac { 11 } { 3 }$ 
D)  $- \frac { 7 } { 9 }$ 
A)  $- \frac { 11 } { 9 }$ 
B)  $\frac { 11 } { 3 }$ 
C)  $- \frac { 11 } { 3 }$ 
D)  $- \frac { 7 } { 9 }$

Use a graphing calculator to make a conjecture as to whether each equation is an identity. - $\cos 2 x = \sin 2 x \tan x$

Verify that each equation is an identity. - $\frac { 1 - \cot ^ { 2 } \theta } { 1 + \cot ^ { 2 } \theta } + 1 = 2 \sin ^ { 2 } \theta$

Verify that each equation is an identity. - $\sin ^ { 3 } x \cos ^ { 2 } x = \sin x \left( \cos ^ { 2 } x - \cos ^ { 4 } x \right)$

For the graph of a circular function y = f(x), determine whether f(-x) = f(x) or f(-x) = -f(x) is true. -

Write in terms of the cofunction of a complementary angle. - $\sin \frac { \pi } { 9 }$

Solve the problem. -Consider the formula $T = 591 - 76 \cos 2 \theta$ , where $\theta$ is measured in degrees. To the nearest hundredth of a degree, what is the smallest positive value of $\theta$ for which the value of $T$ will be 567?

Complete the sentence so the result is an identity. Let x be any real number. - $ـــــــ+ \tan ^ { 2 } x = \sec ^ { 2 } x$

Solve the equation for x, where x is restricted to the given interval. - $y = \sin x - \pi$ , for $x$ in $\left[ - \frac { \pi } { 2 } , \frac { \pi } { 2 } \right]$

Solve the equation for solutions in the interval [0°, 360°). Round to the nearest degree. - $\cos ^ { 2 } \frac { \theta } { 2 } = 1$

Use a sum or difference identity to find the exact value. - $\frac { \tan 20 ^ { \circ } + \tan 10 ^ { \circ } } { 1 - \tan 20 ^ { \circ } \tan 10 ^ { \circ } }$

Solve the equation for exact solutions. - $\arctan x = \operatorname { arccot } \frac { 6 } { 19 }$

Verify that each equation is an identity. - $\cot ( 2 \theta ) = \frac { \csc ^ { 2 } \theta - 2 } { 2 \cot \theta }$

Verify that the equation is an identity. - $\sin \left( \frac { 3 \pi } { 2 } - \theta \right) = - \cos \theta$

Find the exact value of the expression using the provided information. -Find $\sin ( \mathrm { s } - \mathrm { t } )$ given that $\sin \mathrm { s } = - \frac { 1 } { 2 }$ , with $\mathrm { s }$ in quadrant IV, and $\sin t = \frac { 1 } { 4 }$ , with $t$ in quadrant II.

Verify that the equation is an identity. - $\sin ( \alpha - \beta ) \cos ( \alpha + \beta ) = \sin \alpha \cos \alpha - \sin \beta \cos \beta$

Determine all solutions of the equation in radians. -Find $\sin \frac { x } { 2 }$ , given that $\sin x = \frac { 1 } { 4 }$ and $x$ terminates in $0 < x < \frac { \pi } { 2 }$ .

Find the exact value of the expression using the provided information. -Find $\cos ( s + t )$ given that $\sin s = - \frac { 1 } { 2 }$ , with $s$ in quadrant IV, and $\sin t = \frac { 1 } { 4 }$ , with $t$ in quadrant II.

Write the following as an algebraic expression in u, u > 0. - $\sin ( \arctan \mathrm { u } )$

Solve the equation for solutions in the interval [0°, 360°). Round to the nearest degree. - $\sin 2 \theta + \cos 2 \theta = 1$

Equations and Inequalities

Graphs and Functions

Polynomial and Rational Functions

Inverse, Exponential, and Logarithmic Functions

Trigonometric Functions

The Circular Functions and Their Graphs

Applications of Trigonometry

Systems and Matrices

Analytic Geometry

Further Topics in Algebra

Review of Basic Concepts

Filters

Exam 7: Trigonometric Identities and Equations

Use a graphing calculator to make a conjecture as to whether each equation is an identity. - cos⁡2x=sin⁡2xtan⁡x\cos 2 x = \sin 2 x \tan xcos2x=sin2xtanx

Verify that each equation is an identity. - 1−cot⁡2θ1+cot⁡2θ+1=2sin⁡2θ\frac { 1 - \cot ^ { 2 } \theta } { 1 + \cot ^ { 2 } \theta } + 1 = 2 \sin ^ { 2 } \theta1+cot2θ1−cot2θ​+1=2sin2θ

Verify that each equation is an identity. - sin⁡3xcos⁡2x=sin⁡x(cos⁡2x−cos⁡4x)\sin ^ { 3 } x \cos ^ { 2 } x = \sin x \left( \cos ^ { 2 } x - \cos ^ { 4 } x \right)sin3xcos2x=sinx(cos2x−cos4x)

For the graph of a circular function y = f(x), determine whether f(-x) = f(x) or f(-x) = -f(x) is true. -

Write in terms of the cofunction of a complementary angle. - sin⁡π9\sin \frac { \pi } { 9 }sin9π​

Solve the problem. -Consider the formula T=591−76cos⁡2θT = 591 - 76 \cos 2 \thetaT=591−76cos2θ , where θ\thetaθ is measured in degrees. To the nearest hundredth of a degree, what is the smallest positive value of θ\thetaθ for which the value of TTT will be 567?

Complete the sentence so the result is an identity. Let x be any real number. - ـــــــ+tan⁡2x=sec⁡2xـــــــ+ \tan ^ { 2 } x = \sec ^ { 2 } xـــــــ+tan2x=sec2x

Solve the equation for x, where x is restricted to the given interval. - y=sin⁡x−πy = \sin x - \piy=sinx−π , for xxx in [−π2,π2]\left[ - \frac { \pi } { 2 } , \frac { \pi } { 2 } \right][−2π​,2π​]

Solve the equation for solutions in the interval [0°, 360°). Round to the nearest degree. - cos⁡2θ2=1\cos ^ { 2 } \frac { \theta } { 2 } = 1cos22θ​=1

Use a sum or difference identity to find the exact value. - tan⁡20∘+tan⁡10∘1−tan⁡20∘tan⁡10∘\frac { \tan 20 ^ { \circ } + \tan 10 ^ { \circ } } { 1 - \tan 20 ^ { \circ } \tan 10 ^ { \circ } }1−tan20∘tan10∘tan20∘+tan10∘​

Solve the equation for exact solutions. - arctan⁡x=arccot⁡619\arctan x = \operatorname { arccot } \frac { 6 } { 19 }arctanx=arccot196​

Verify that each equation is an identity. - cot⁡(2θ)=csc⁡2θ−22cot⁡θ\cot ( 2 \theta ) = \frac { \csc ^ { 2 } \theta - 2 } { 2 \cot \theta }cot(2θ)=2cotθcsc2θ−2​

Verify that the equation is an identity. - sin⁡(3π2−θ)=−cos⁡θ\sin \left( \frac { 3 \pi } { 2 } - \theta \right) = - \cos \thetasin(23π​−θ)=−cosθ

Verify that the equation is an identity. - sin⁡(α−β)cos⁡(α+β)=sin⁡αcos⁡α−sin⁡βcos⁡β\sin ( \alpha - \beta ) \cos ( \alpha + \beta ) = \sin \alpha \cos \alpha - \sin \beta \cos \betasin(α−β)cos(α+β)=sinαcosα−sinβcosβ

Determine all solutions of the equation in radians. -Find sin⁡x2\sin \frac { x } { 2 }sin2x​ , given that sin⁡x=14\sin x = \frac { 1 } { 4 }sinx=41​ and xxx terminates in 0<x<π20 < x < \frac { \pi } { 2 }0<x<2π​ .

Find the exact value of the expression using the provided information. -Find cos⁡(s+t)\cos ( s + t )cos(s+t) given that sin⁡s=−12\sin s = - \frac { 1 } { 2 }sins=−21​ , with sss in quadrant IV, and sin⁡t=14\sin t = \frac { 1 } { 4 }sint=41​ , with ttt in quadrant II.

Write the following as an algebraic expression in u, u > 0. - sin⁡(arctan⁡u)\sin ( \arctan \mathrm { u } )sin(arctanu)

Solve the equation for solutions in the interval [0°, 360°). Round to the nearest degree. - sin⁡2θ+cos⁡2θ=1\sin 2 \theta + \cos 2 \theta = 1sin2θ+cos2θ=1