Question 1

Find the inverse function $\mathrm { f } ^ { - 1 }$ of the function f.
-$$f ( x ) = - 7 \cos ( 4 x )$$

Accepted Answer

A)  $f ^ { - 1 } ( x ) = \frac { 1 } { 4 } \cos ^ { - 1 } \left( \frac { x } { 7 } \right)$ 
B)  $f ^ { - 1 } ( x ) = - 7 \cos ^ { - 1 } ( 4 x )$ 
C)  $\mathrm { f } ^ { - 1 } ( \mathrm { x } ) = \frac { 1 } { 7 } \cos ^ { - 1 } \left( \frac { \mathrm { x } } { 4 } \right)$ 
D)  $\mathrm { f } ^ { - 1 } ( \mathrm { x } ) = - \frac { 1 } { 4 } \cos ^ { - 1 } \left( \frac { \mathrm { x } } { 7 } \right)$ 
A)  $f ^ { - 1 } ( x ) = \frac { 1 } { 4 } \cos ^ { - 1 } \left( \frac { x } { 7 } \right)$ 
B)  $f ^ { - 1 } ( x ) = - 7 \cos ^ { - 1 } ( 4 x )$ 
C)  $\mathrm { f } ^ { - 1 } ( \mathrm { x } ) = \frac { 1 } { 7 } \cos ^ { - 1 } \left( \frac { \mathrm { x } } { 4 } \right)$ 
D)  $\mathrm { f } ^ { - 1 } ( \mathrm { x } ) = - \frac { 1 } { 4 } \cos ^ { - 1 } \left( \frac { \mathrm { x } } { 7 } \right)$

Question 2

Establish the identity.
-$$\cot x \sec ^ { 4 } x = \cot x + 2 \tan x + \tan ^ { 3 } x$$

Accepted Answer

The answer of Establish the identity.
-\[\cot x \sec ^ {...

Question 3

Solve the equation on the interval $$0 \leq \theta < 2 \pi$$
-$$2 \cos ( 2 \theta ) = \sqrt { 3 }$$

Accepted Answer

A)  $\frac { \pi } { 2 }$ 
B)  $\frac { 3 \pi } { 2 }$ 
C)  $\frac { \pi } { 6 } , \frac { 11 \pi } { 6 }$ 
D)  $\frac { \pi } { 12 } , \frac { 11 \pi } { 12 } , \frac { 13 \pi } { 12 } , \frac { 23 \pi } { 12 }$ 
A)  $\frac { \pi } { 2 }$ 
B)  $\frac { 3 \pi } { 2 }$ 
C)  $\frac { \pi } { 6 } , \frac { 11 \pi } { 6 }$ 
D)  $\frac { \pi } { 12 } , \frac { 11 \pi } { 12 } , \frac { 13 \pi } { 12 } , \frac { 23 \pi } { 12 }$

Question 4

Establish the identity.
-$$\cos \left( x + \frac { \pi } { 2 } \right) = - \sin x$$

Accepted Answer

The answer of Establish the identity.
-\[\cos \left( x + \frac...

Question 5

Find the exact value of the expression. Do not use a calculator.
-$\cos ^ { - 1 } \left[ \cos \left( - \frac { 3 \pi } { 5 } \right) \right]$

Accepted Answer

A)  $- \frac { 3 \pi } { 5 }$ 
B)  $\frac { 2 \pi } { 5 }$ 
C)  $\frac { 3 \pi } { 5 }$ 
D)  $- \frac { 2 \pi } { 5 }$ 
A)  $- \frac { 3 \pi } { 5 }$ 
B)  $\frac { 2 \pi } { 5 }$ 
C)  $\frac { 3 \pi } { 5 }$ 
D)  $- \frac { 2 \pi } { 5 }$

Question 6

Simplify the expression.
-$\sin 8 x \cos 8 x$

Accepted Answer

A)  $\cos 8 x$ 
B)  $\cos 4 x$ 
C)  $2 \sin 4 x$ 
D)  $\frac { 1 } { 2 } \sin 16 x$ 
A)  $\cos 8 x$ 
B)  $\cos 4 x$ 
C)  $2 \sin 4 x$ 
D)  $\frac { 1 } { 2 } \sin 16 x$

Question 7

Complete the identity.
-$$\csc ^ { 2 } \frac { \theta } { 2 } - \sec ^ { 2 } \frac { \theta } { 2 } = ?$$

Accepted Answer

A)  $2 \sin ^ { 2 } \theta \sec \theta$ 
B)  $\frac { 2 } { 1 - \sec \theta }$ 
C)  0 
D)  $4 \cot \theta \csc \theta$ 
A)  $2 \sin ^ { 2 } \theta \sec \theta$ 
B)  $\frac { 2 } { 1 - \sec \theta }$ 
C)  0 
D)  $4 \cot \theta \csc \theta$

Question 8

Express the product as a sum containing only sines or cosines.
-$$\sin \frac { \theta } { 2 } \cos \frac { 5 \theta } { 2 }$$

Accepted Answer

A)  $\frac { 1 } { 4 } \sin \cos ( 5 \theta )$ 
B)  $\frac { 1 } { 4 } [ \cos ( 6 \theta ) - \sin ( 4 \theta ) ]$ 
C)  $\frac { 1 } { 2 } [ \cos ( 3 \theta ) + \sin ( 2 \theta ) ]$ 
D)  $\frac { 1 } { 2 } [ \sin ( 3 \theta ) - \sin ( 2 \theta ) ]$ 
A)  $\frac { 1 } { 4 } \sin \cos ( 5 \theta )$ 
B)  $\frac { 1 } { 4 } [ \cos ( 6 \theta ) - \sin ( 4 \theta ) ]$ 
C)  $\frac { 1 } { 2 } [ \cos ( 3 \theta ) + \sin ( 2 \theta ) ]$ 
D)  $\frac { 1 } { 2 } [ \sin ( 3 \theta ) - \sin ( 2 \theta ) ]$

Question 9

Establish the identity.
-$$\frac { \cot ^ { 2 } x } { \csc x + 1 } = \frac { 1 - \sin x } { \sin x }$$

Accepted Answer

The answer of Establish the identity.
-\[\frac { \cot ^ {...

Question 10

Find the exact value of the expression.
-$$\frac { \tan 40 ^ { \circ } + \tan 110 ^ { \circ } } { 1 - \tan 40 ^ { \circ } \tan 110 ^ { \circ } }$$

Accepted Answer

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

Question 11

Use the Half-angle Formulas to find the exact value of the trigonometric function.
-$\cos 75 ^ { \circ }$

Accepted Answer

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

Question 12

Find the exact value of the expression.
-$\cos \left( \sin ^ { - 1 } \frac { 1 } { 3 } - \tan ^ { - 1 } \frac { 1 } { 2 } \right)$

Accepted Answer

A)  $\frac { 2 \sqrt { 3 } + 4 } { 3 \sqrt { 5 } }$ 
B)  $\frac { 2 \sqrt { 6 } } { 5 }$ 
C)  $\frac { 4 \sqrt { 10 } + \sqrt { 5 } } { 15 }$ 
D)  $\frac { 2 \sqrt { 3 } + 1 } { 5 }$ 
A)  $\frac { 2 \sqrt { 3 } + 4 } { 3 \sqrt { 5 } }$ 
B)  $\frac { 2 \sqrt { 6 } } { 5 }$ 
C)  $\frac { 4 \sqrt { 10 } + \sqrt { 5 } } { 15 }$ 
D)  $\frac { 2 \sqrt { 3 } + 1 } { 5 }$

Question 13

Use a graphing utility to solve the equation on the interval 0 $0 ^ { \circ } \leq x < 360 ^ { \circ }$ . Express the solution(s) rounded to one decimal
place.
-$3 \cos ^ { 2 } x + 2 \cos x = 1$

Accepted Answer

A)  49.8°, 130.2°, 229.8°, 310.2° 
B)  51.8°, 128.2° 
C)  70.5°, 180.0°, 289.5° 
D)  103.2°, 145.2°, 283.2°, 325.2° 
A)  49.8°, 130.2°, 229.8°, 310.2° 
B)  51.8°, 128.2° 
C)  70.5°, 180.0°, 289.5° 
D)  103.2°, 145.2°, 283.2°, 325.2°

Question 14

Express the product as a sum containing only sines or cosines.
-$2 \cos ( 7 \theta ) \cos \theta$

Accepted Answer

A)  $\cos ( 10 \theta ) + \sin ( 4 \theta )$ 
B)  $\cos ( 14 \theta ) + \cos ( 2 \theta )$ 
C)  $\cos ( 8 \theta ) + \cos ( 6 \theta )$ 
D)  $\cos ( 8 \theta ) + \sin ( 6 \theta )$ 
A)  $\cos ( 10 \theta ) + \sin ( 4 \theta )$ 
B)  $\cos ( 14 \theta ) + \cos ( 2 \theta )$ 
C)  $\cos ( 8 \theta ) + \cos ( 6 \theta )$ 
D)  $\cos ( 8 \theta ) + \sin ( 6 \theta )$

Question 15

Complete the identity.
-$\sin \theta \cos ^ { 3 } \theta + \sin ^ { 3 } \theta \cos \theta = ?$

Accepted Answer

A)  $\frac { 1 } { 2 } \sin ( 2 \theta ) \cos ( 2 \theta )$ 
B)  $\frac { 1 } { 2 } \sin ( 2 \theta )$ 
C)  $\frac { 1 } { 4 } \sin ^ { 2 } ( 2 \theta )$ 
D)  $\frac { 1 } { 6 } \sin ( 3 \theta ) \sin ( 2 \theta )$ 
A)  $\frac { 1 } { 2 } \sin ( 2 \theta ) \cos ( 2 \theta )$ 
B)  $\frac { 1 } { 2 } \sin ( 2 \theta )$ 
C)  $\frac { 1 } { 4 } \sin ^ { 2 } ( 2 \theta )$ 
D)  $\frac { 1 } { 6 } \sin ( 3 \theta ) \sin ( 2 \theta )$

Question 16

Solve the problem.
-Find $\cos \frac { \theta } { 2 }$, given that $\cos \theta = - \frac { 3 } { 5 }$ and $\theta$ terminates in $90 ^ { \circ } < \theta < 180 ^ { \circ }$.

Accepted Answer

A)  $- \frac { \sqrt { 5 } } { 5 }$ 
B)  $\frac { \sqrt { 30 } } { 10 }$ 
C)  $\frac { \sqrt { 5 } } { 5 }$ 
D)  $- \frac { \sqrt { 30 } } { 10 }$ 
A)  $- \frac { \sqrt { 5 } } { 5 }$ 
B)  $\frac { \sqrt { 30 } } { 10 }$ 
C)  $\frac { \sqrt { 5 } } { 5 }$ 
D)  $- \frac { \sqrt { 30 } } { 10 }$

Question 17

Use a graphing utility to solve the equation on the interval 0 $0 ^ { \circ } \leq x < 360 ^ { \circ }$ . Express the solution(s) rounded to one decimal
place.
-$\sin ^ { 2 } x - 8 \sin x - 4 = 0$

Accepted Answer

A)  28.2°, 151.8°, 208.2°, 331.8° 
B)  28.2°, 151.8° 
C)  208.2°, 331.8° 
D)  No solution 
A)  28.2°, 151.8°, 208.2°, 331.8° 
B)  28.2°, 151.8° 
C)  208.2°, 331.8° 
D)  No solution

Question 18

Find the exact value of the expression.
-$\tan 75 ^ { \circ }$

Accepted Answer

A)  $\sqrt { 3 } + 2$ 
B)  $\sqrt { 3 } - 2$ 
C)  $- \sqrt { 3 } - 2$ 
D)  $- \sqrt { 3 } + 2$ 
A)  $\sqrt { 3 } + 2$ 
B)  $\sqrt { 3 } - 2$ 
C)  $- \sqrt { 3 } - 2$ 
D)  $- \sqrt { 3 } + 2$

Question 19

Complete the identity.
-$$\frac { \sin ^ { 2 } \frac { \theta } { 2 } } { \sin ^ { 2 } \theta } = ?$$

Accepted Answer

A)  $\frac { 1 } { 2 - 2 \cos \theta }$ 
B)  $\frac { 1 } { 2 + 2 \cos \theta }$ 
C)  $\cos ^ { 2 } \frac { \theta } { 2 }$ 
D)  $\sin ^ { 2 } \frac { \theta } { 2 }$ 
A)  $\frac { 1 } { 2 - 2 \cos \theta }$ 
B)  $\frac { 1 } { 2 + 2 \cos \theta }$ 
C)  $\cos ^ { 2 } \frac { \theta } { 2 }$ 
D)  $\sin ^ { 2 } \frac { \theta } { 2 }$

Question 20

Complete the identity.
-$\sin ^ { 2 } \theta + \tan ^ { 2 } \theta + \cos ^ { 2 } \theta =$ ?

Accepted Answer

A)  $\tan ^ { 2 } \theta$ 
B)  $\sin \theta$ 
C)  $\cos ^ { 3 } \theta$ 
D)  $\sec ^ { 2 } \theta$ 
A)  $\tan ^ { 2 } \theta$ 
B)  $\sin \theta$ 
C)  $\cos ^ { 3 } \theta$ 
D)  $\sec ^ { 2 } \theta$

Find the inverse function $\mathrm { f } ^ { - 1 }$ of the function f. - $f ( x ) = - 7 \cos ( 4 x )$

Establish the identity. - $\cot x \sec ^ { 4 } x = \cot x + 2 \tan x + \tan ^ { 3 } x$

Solve the equation on the interval $0 \leq \theta < 2 \pi$ - $2 \cos ( 2 \theta ) = \sqrt { 3 }$

Establish the identity. - $\cos \left( x + \frac { \pi } { 2 } \right) = - \sin x$

Find the exact value of the expression. Do not use a calculator. - $\cos ^ { - 1 } \left[ \cos \left( - \frac { 3 \pi } { 5 } \right) \right]$

Simplify the expression. - $\sin 8 x \cos 8 x$

Complete the identity. - $\csc ^ { 2 } \frac { \theta } { 2 } - \sec ^ { 2 } \frac { \theta } { 2 } = ?$

Express the product as a sum containing only sines or cosines. - $\sin \frac { \theta } { 2 } \cos \frac { 5 \theta } { 2 }$

Establish the identity. - $\frac { \cot ^ { 2 } x } { \csc x + 1 } = \frac { 1 - \sin x } { \sin x }$

Find the exact value of the expression. - $\frac { \tan 40 ^ { \circ } + \tan 110 ^ { \circ } } { 1 - \tan 40 ^ { \circ } \tan 110 ^ { \circ } }$

Use the Half-angle Formulas to find the exact value of the trigonometric function. - $\cos 75 ^ { \circ }$

Find the exact value of the expression. - $\cos \left( \sin ^ { - 1 } \frac { 1 } { 3 } - \tan ^ { - 1 } \frac { 1 } { 2 } \right)$

Use a graphing utility to solve the equation on the interval 0 $0 ^ { \circ } \leq x < 360 ^ { \circ }$ . Express the solution(s) rounded to one decimal place. - $3 \cos ^ { 2 } x + 2 \cos x = 1$

Express the product as a sum containing only sines or cosines. - $2 \cos ( 7 \theta ) \cos \theta$

Complete the identity. - $\sin \theta \cos ^ { 3 } \theta + \sin ^ { 3 } \theta \cos \theta = ?$

Solve the problem. -Find $\cos \frac { \theta } { 2 }$ , given that $\cos \theta = - \frac { 3 } { 5 }$ and $\theta$ terminates in $90 ^ { \circ } < \theta < 180 ^ { \circ }$ .

Use a graphing utility to solve the equation on the interval 0 $0 ^ { \circ } \leq x < 360 ^ { \circ }$ . Express the solution(s) rounded to one decimal place. - $\sin ^ { 2 } x - 8 \sin x - 4 = 0$

Find the exact value of the expression. - $\tan 75 ^ { \circ }$

Complete the identity. - $\frac { \sin ^ { 2 } \frac { \theta } { 2 } } { \sin ^ { 2 } \theta } = ?$

Complete the identity. - $\sin ^ { 2 } \theta + \tan ^ { 2 } \theta + \cos ^ { 2 } \theta =$ ?

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Exam 6: Analytic Trigonometry

Find the inverse function f−1\mathrm { f } ^ { - 1 }f−1 of the function f. - f(x)=−7cos⁡(4x)f ( x ) = - 7 \cos ( 4 x )f(x)=−7cos(4x)

Establish the identity. - cot⁡xsec⁡4x=cot⁡x+2tan⁡x+tan⁡3x\cot x \sec ^ { 4 } x = \cot x + 2 \tan x + \tan ^ { 3 } xcotxsec4x=cotx+2tanx+tan3x

Solve the equation on the interval 0≤θ<2π0 \leq \theta < 2 \pi0≤θ<2π - 2cos⁡(2θ)=32 \cos ( 2 \theta ) = \sqrt { 3 }2cos(2θ)=3​

Establish the identity. - cos⁡(x+π2)=−sin⁡x\cos \left( x + \frac { \pi } { 2 } \right) = - \sin xcos(x+2π​)=−sinx

Find the exact value of the expression. Do not use a calculator. - cos⁡−1[cos⁡(−3π5)]\cos ^ { - 1 } \left[ \cos \left( - \frac { 3 \pi } { 5 } \right) \right]cos−1[cos(−53π​)]

Simplify the expression. - sin⁡8xcos⁡8x\sin 8 x \cos 8 xsin8xcos8x

Complete the identity. - csc⁡2θ2−sec⁡2θ2=?\csc ^ { 2 } \frac { \theta } { 2 } - \sec ^ { 2 } \frac { \theta } { 2 } = ?csc22θ​−sec22θ​=?

Express the product as a sum containing only sines or cosines. - sin⁡θ2cos⁡5θ2\sin \frac { \theta } { 2 } \cos \frac { 5 \theta } { 2 }sin2θ​cos25θ​

Establish the identity. - cot⁡2xcsc⁡x+1=1−sin⁡xsin⁡x\frac { \cot ^ { 2 } x } { \csc x + 1 } = \frac { 1 - \sin x } { \sin x }cscx+1cot2x​=sinx1−sinx​

Find the exact value of the expression. - tan⁡40∘+tan⁡110∘1−tan⁡40∘tan⁡110∘\frac { \tan 40 ^ { \circ } + \tan 110 ^ { \circ } } { 1 - \tan 40 ^ { \circ } \tan 110 ^ { \circ } }1−tan40∘tan110∘tan40∘+tan110∘​

Use the Half-angle Formulas to find the exact value of the trigonometric function. - cos⁡75∘\cos 75 ^ { \circ }cos75∘

Find the exact value of the expression. - cos⁡(sin⁡−113−tan⁡−112)\cos \left( \sin ^ { - 1 } \frac { 1 } { 3 } - \tan ^ { - 1 } \frac { 1 } { 2 } \right)cos(sin−131​−tan−121​)

Use a graphing utility to solve the equation on the interval 0 0∘≤x<360∘0 ^ { \circ } \leq x < 360 ^ { \circ }0∘≤x<360∘ . Express the solution(s) rounded to one decimal place. - 3cos⁡2x+2cos⁡x=13 \cos ^ { 2 } x + 2 \cos x = 13cos2x+2cosx=1

Express the product as a sum containing only sines or cosines. - 2cos⁡(7θ)cos⁡θ2 \cos ( 7 \theta ) \cos \theta2cos(7θ)cosθ

Complete the identity. - sin⁡θcos⁡3θ+sin⁡3θcos⁡θ=?\sin \theta \cos ^ { 3 } \theta + \sin ^ { 3 } \theta \cos \theta = ?sinθcos3θ+sin3θcosθ=?

Solve the problem. -Find cos⁡θ2\cos \frac { \theta } { 2 }cos2θ​ , given that cos⁡θ=−35\cos \theta = - \frac { 3 } { 5 }cosθ=−53​ and θ\thetaθ terminates in 90∘<θ<180∘90 ^ { \circ } < \theta < 180 ^ { \circ }90∘<θ<180∘ .

Use a graphing utility to solve the equation on the interval 0 0∘≤x<360∘0 ^ { \circ } \leq x < 360 ^ { \circ }0∘≤x<360∘ . Express the solution(s) rounded to one decimal place. - sin⁡2x−8sin⁡x−4=0\sin ^ { 2 } x - 8 \sin x - 4 = 0sin2x−8sinx−4=0

Find the exact value of the expression. - tan⁡75∘\tan 75 ^ { \circ }tan75∘

Complete the identity. - sin⁡2θ2sin⁡2θ=?\frac { \sin ^ { 2 } \frac { \theta } { 2 } } { \sin ^ { 2 } \theta } = ?sin2θsin22θ​​=?

Complete the identity. - sin⁡2θ+tan⁡2θ+cos⁡2θ=\sin ^ { 2 } \theta + \tan ^ { 2 } \theta + \cos ^ { 2 } \theta =sin2θ+tan2θ+cos2θ= ?