Question 1

Complete the identity.
-$$\frac { ( \sin \theta + \cos \theta ) ^ { 2 } } { 1 + 2 \sin \theta \cos \theta } = \text { ? }$$

Accepted Answer

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

Question 2

Use the information given about the angle $$\theta , 0 \leq \theta \leq 2 \pi$$ , to find the exact value of the indicated trigonometric function.
-$\cos \theta = \frac { \sqrt { 5 } } { 5 } , \quad 0 < \theta < \frac { \pi } { 2 } \quad$ Find $\sin ( 2 \theta )$

Accepted Answer

A)  $\frac { 2 } { 5 }$ 
B)  $\frac { 4 } { 5 }$ 
C)  $\frac { 1 } { 5 }$ 
D)  $\frac { 3 } { 5 }$ 
A)  $\frac { 2 } { 5 }$ 
B)  $\frac { 4 } { 5 }$ 
C)  $\frac { 1 } { 5 }$ 
D)  $\frac { 3 } { 5 }$

Question 3

Express the sum or difference as a product of sines and/or cosines.
-$\cos ( 4 \theta ) - \cos ( 6 \theta )$

Accepted Answer

A)  $2 \sin ( 5 \theta ) \sin \theta$ 
B)  $\cos ( - 2 \theta )$ 
C)  $- 2 \cos ( 5 \theta ) \sin \theta$ 
D)  $- 2 \sin ( 5 \theta ) \sin \theta$ 
A)  $2 \sin ( 5 \theta ) \sin \theta$ 
B)  $\cos ( - 2 \theta )$ 
C)  $- 2 \cos ( 5 \theta ) \sin \theta$ 
D)  $- 2 \sin ( 5 \theta ) \sin \theta$

Question 4

Complete the identity.
-$\cos ( \alpha + \beta ) \cos ( \alpha - \beta ) =$ ?

Accepted Answer

A)  $\cos \left( \alpha ^ { 2 } \right) \cos \left( \beta ^ { 2 } \right) + \sin \left( \alpha ^ { 2 } \right) \sin \left( \beta ^ { 2 } \right)$ 
B)  $2 - \sin ^ { 2 } \alpha - \sin ^ { 2 } \beta$ 
C)  $\cos ^ { 2 } \beta - 2 \sin ^ { 2 } \alpha \sin ^ { 2 } \beta$ 
D)  $\cos ^ { 2 } \beta - \sin ^ { 2 } \alpha$ 
A)  $\cos \left( \alpha ^ { 2 } \right) \cos \left( \beta ^ { 2 } \right) + \sin \left( \alpha ^ { 2 } \right) \sin \left( \beta ^ { 2 } \right)$ 
B)  $2 - \sin ^ { 2 } \alpha - \sin ^ { 2 } \beta$ 
C)  $\cos ^ { 2 } \beta - 2 \sin ^ { 2 } \alpha \sin ^ { 2 } \beta$ 
D)  $\cos ^ { 2 } \beta - \sin ^ { 2 } \alpha$

Question 5

Find the exact value of the expression.
-$\sin \frac { \pi } { 12 }$

Accepted Answer

A)  $\frac { \sqrt { 2 } ( \sqrt { 3 } - 1 ) } { 4 }$ 
B)  $\sqrt { 2 } ( \sqrt { 3 } - 1 )$ 
C)  $- \sqrt { 2 } ( \sqrt { 3 } - 1 )$ 
D)  $- \frac { \sqrt { 2 } ( \sqrt { 3 } - 1 ) } { 4 }$ 
A)  $\frac { \sqrt { 2 } ( \sqrt { 3 } - 1 ) } { 4 }$ 
B)  $\sqrt { 2 } ( \sqrt { 3 } - 1 )$ 
C)  $- \sqrt { 2 } ( \sqrt { 3 } - 1 )$ 
D)  $- \frac { \sqrt { 2 } ( \sqrt { 3 } - 1 ) } { 4 }$

Question 6

Simplify the expression.
-$$\cos ^ { 2 } 4 x - \sin ^ { 2 } 4 x$$

Accepted Answer

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

Question 7

Solve the equation on the interval $$0 \leq \theta < 2 \pi$$
-$$\tan ( 2 \theta ) - \tan \theta = 0$$

Accepted Answer

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

Question 8

Use the information given about the angle θ, 0 ≤θ ≤ 2π, to find the exact value of the indicated trigonometric function.
-$\sin \theta = \frac { 1 } { 4 } , \quad \tan \theta > 0 \quad$ Find $\cos \frac { \theta } { 2 }$

Accepted Answer

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

Question 9

Find the exact value of the expression.
-$\sin 10 ^ { \circ } \cos 50 ^ { \circ } + \cos 10 ^ { \circ } \sin 50 ^ { \circ }$

Accepted Answer

A)  $\frac { 1 } { 2 }$ 
B)  $\frac { \sqrt { 3 } } { 3 }$ 
C)  $\frac { \sqrt { 3 } } { 2 }$ 
D)  $\frac { 1 } { 6 }$ 
A)  $\frac { 1 } { 2 }$ 
B)  $\frac { \sqrt { 3 } } { 3 }$ 
C)  $\frac { \sqrt { 3 } } { 2 }$ 
D)  $\frac { 1 } { 6 }$

Question 10

Solve the equation on the interval $$0 \leq \theta < 2 \pi$$
-$\cot \left( 2 \theta - \frac { \pi } { 2 } \right) = 1$

Accepted Answer

A)  $\frac { \pi } { 4 } , \frac { 5 \pi } { 4 } , \frac { 9 \pi } { 4 }$, and $\frac { 13 \pi } { 4 }$ 
B)  $\frac { 3 \pi } { 8 } , \frac { 7 \pi } { 8 } , \frac { 11 \pi } { 8 }$, and $\frac { 15 \pi } { 8 }$ 
C)  $\frac { 3 \pi } { 8 }$ 
D)  $\frac { 3 \pi } { 8 } , \frac { 7 \pi } { 8 }$ 
A)  $\frac { \pi } { 4 } , \frac { 5 \pi } { 4 } , \frac { 9 \pi } { 4 }$, and $\frac { 13 \pi } { 4 }$ 
B)  $\frac { 3 \pi } { 8 } , \frac { 7 \pi } { 8 } , \frac { 11 \pi } { 8 }$, and $\frac { 15 \pi } { 8 }$ 
C)  $\frac { 3 \pi } { 8 }$ 
D)  $\frac { 3 \pi } { 8 } , \frac { 7 \pi } { 8 }$

Question 11

Find the exact value of the expression.
-$$\frac { \tan 10 ^ { \circ } + \tan 20 ^ { \circ } } { 1 - \tan 10 ^ { \circ } \tan 20 ^ { \circ } }$$

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 12

Use a calculator to find the value of the expression rounded to two decimal places.
-$\sin ^ { - 1 } \left( - \frac { 1 } { 7 } \right)$

Accepted Answer

A)  -8.21 
B)  -0.14 
C)  1.71 
D)  98.21 
A)  -8.21 
B)  -0.14 
C)  1.71 
D)  98.21

Question 13

Complete the identity.
-$\csc ( \alpha - \beta ) = ?$

Accepted Answer

A)  $\frac { \sec \alpha \sec \beta } { \tan \alpha \tan \beta - 1 }$ 
B)  $\frac { \sec \alpha \sec \beta } { \tan \alpha - \tan \beta }$ 
C)  $\csc \alpha \sec \beta - \sec \alpha \csc \beta$ 
D)  $\csc \alpha - \csc \beta$ 
A)  $\frac { \sec \alpha \sec \beta } { \tan \alpha \tan \beta - 1 }$ 
B)  $\frac { \sec \alpha \sec \beta } { \tan \alpha - \tan \beta }$ 
C)  $\csc \alpha \sec \beta - \sec \alpha \csc \beta$ 
D)  $\csc \alpha - \csc \beta$

Question 14

Solve the problem.
-The altitude of a projectile in feet (neglecting air resistance) is given by $y = ( \tan \theta ) x - \frac { 16 } { v ^ { 2 } \cos ^ { 2 } \theta } x ^ { 2 }$ where x is the horizontal distance covered in feet and v is the initial velocity of the projectile at an angle θ from the horizontal. Find the firing angle (in degrees) of a projectile fired at an initial velocity of 100 feet per second so
That it strikes the ground 312.5 feet from the firing point.

Accepted Answer

A)  30° 
B)  45° 
C)  50° 
D)  22.5° 
A)  30° 
B)  45° 
C)  50° 
D)  22.5°

Question 15

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 16

Find the exact value of the expression. Do not use a calculator.
-$$\tan ^ { - 1 } \left[ \tan \left( - \frac { \pi } { 8 } \right) \right]$$

Accepted Answer

A)  $- 8 \pi$ 
B)  $\frac { \pi } { 8 }$ 
C)  $8 \pi$ 
D)  $- \frac { \pi } { 8 }$ 
A)  $- 8 \pi$ 
B)  $\frac { \pi } { 8 }$ 
C)  $8 \pi$ 
D)  $- \frac { \pi } { 8 }$

Question 17

Complete the identity.
-$\ln | \cot x | = ?$

Accepted Answer

A)  $\frac { \ln | \sin x | } { \ln | \cos x | }$ 
B)  $\frac { \ln | \cos x | } { \ln | \sin x | }$ 
C)  $\ln | \sin x | - \ln | \cos x |$ 
D)  $\ln | \cos x | - \ln | \sin x |$ 
A)  $\frac { \ln | \sin x | } { \ln | \cos x | }$ 
B)  $\frac { \ln | \cos x | } { \ln | \sin x | }$ 
C)  $\ln | \sin x | - \ln | \cos x |$ 
D)  $\ln | \cos x | - \ln | \sin x |$

Question 18

Complete the identity.
-$\csc ( 2 \theta ) - \sec ( 2 \theta ) ( \tan \theta - 1 ) =$ ?

Accepted Answer

A)  $\frac { \sin \theta + \cos \theta } { \sin ( 2 \theta ) ( \sin \theta - \cos \theta ) }$ 
B)  $\frac { \csc \theta - \sec \theta } { \sin ( 2 \theta ) ( \sin \theta + \cos \theta ) }$ 
C)  $\frac { \csc \theta + \sec \theta } { \sin ( 2 \theta ) ( \sin \theta - \cos \theta ) }$ 
D)  $\frac { \cos \theta - \sin \theta } { \sin ( 2 \theta ) ( \sin \theta + \cos \theta ) }$ 
A)  $\frac { \sin \theta + \cos \theta } { \sin ( 2 \theta ) ( \sin \theta - \cos \theta ) }$ 
B)  $\frac { \csc \theta - \sec \theta } { \sin ( 2 \theta ) ( \sin \theta + \cos \theta ) }$ 
C)  $\frac { \csc \theta + \sec \theta } { \sin ( 2 \theta ) ( \sin \theta - \cos \theta ) }$ 
D)  $\frac { \cos \theta - \sin \theta } { \sin ( 2 \theta ) ( \sin \theta + \cos \theta ) }$

Question 19

Solve the equation on the interval $$0 \leq \theta < 2 \pi$$
-$\csc ( 3 \theta ) = 0$

Accepted Answer

A)  $\frac { \pi } { 8 } , \frac { 9 \pi } { 8 }$ 
B)  $\frac { \pi } { 4 } , \frac { 3 \pi } { 4 } , \frac { 5 \pi } { 4 } , \frac { 7 \pi } { 4 }$ 
C)  $0 , \frac { 2 \pi } { 3 } , \pi , \frac { 4 \pi } { 3 }$ 
D)  No solution 
A)  $\frac { \pi } { 8 } , \frac { 9 \pi } { 8 }$ 
B)  $\frac { \pi } { 4 } , \frac { 3 \pi } { 4 } , \frac { 5 \pi } { 4 } , \frac { 7 \pi } { 4 }$ 
C)  $0 , \frac { 2 \pi } { 3 } , \pi , \frac { 4 \pi } { 3 }$ 
D)  No solution

Question 20

Solve the problem.
-Find $\cos \frac { \theta } { 2 }$, given that $\cos \theta = \frac { 1 } { 4 }$ and $\theta$ terminates in $0 < \theta < \pi / 2$.

Accepted Answer

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

Complete the identity. - $\frac { ( \sin \theta + \cos \theta ) ^ { 2 } } { 1 + 2 \sin \theta \cos \theta } = \text { ? }$

Use the information given about the angle $\theta , 0 \leq \theta \leq 2 \pi$ , to find the exact value of the indicated trigonometric function. - $\cos \theta = \frac { \sqrt { 5 } } { 5 } , \quad 0 < \theta < \frac { \pi } { 2 } \quad$ Find $\sin ( 2 \theta )$

Express the sum or difference as a product of sines and/or cosines. - $\cos ( 4 \theta ) - \cos ( 6 \theta )$

Complete the identity. - $\cos ( \alpha + \beta ) \cos ( \alpha - \beta ) =$ ?

Find the exact value of the expression. - $\sin \frac { \pi } { 12 }$

Simplify the expression. - $\cos ^ { 2 } 4 x - \sin ^ { 2 } 4 x$

Solve the equation on the interval $0 \leq \theta < 2 \pi$ - $\tan ( 2 \theta ) - \tan \theta = 0$

Use the information given about the angle θ, 0 ≤θ ≤ 2π, to find the exact value of the indicated trigonometric function. - $\sin \theta = \frac { 1 } { 4 } , \quad \tan \theta > 0 \quad$ Find $\cos \frac { \theta } { 2 }$

Find the exact value of the expression. - $\sin 10 ^ { \circ } \cos 50 ^ { \circ } + \cos 10 ^ { \circ } \sin 50 ^ { \circ }$

Solve the equation on the interval $0 \leq \theta < 2 \pi$ - $\cot \left( 2 \theta - \frac { \pi } { 2 } \right) = 1$

Find the exact value of the expression. - $\frac { \tan 10 ^ { \circ } + \tan 20 ^ { \circ } } { 1 - \tan 10 ^ { \circ } \tan 20 ^ { \circ } }$

Use a calculator to find the value of the expression rounded to two decimal places. - $\sin ^ { - 1 } \left( - \frac { 1 } { 7 } \right)$

Complete the identity. - $\csc ( \alpha - \beta ) = ?$

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

Find the exact value of the expression. Do not use a calculator. - $\tan ^ { - 1 } \left[ \tan \left( - \frac { \pi } { 8 } \right) \right]$

Complete the identity. - $\ln | \cot x | = ?$

Complete the identity. - $\csc ( 2 \theta ) - \sec ( 2 \theta ) ( \tan \theta - 1 ) =$ ?

Solve the equation on the interval $0 \leq \theta < 2 \pi$ - $\csc ( 3 \theta ) = 0$

Solve the problem. -Find $\cos \frac { \theta } { 2 }$ , given that $\cos \theta = \frac { 1 } { 4 }$ and $\theta$ terminates in $0 < \theta < \pi / 2$ .

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Exponential and Logarithmic Functions

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Exam 6: Analytic Trigonometry

Complete the identity. - (sin⁡θ+cos⁡θ)21+2sin⁡θcos⁡θ= ? \frac { ( \sin \theta + \cos \theta ) ^ { 2 } } { 1 + 2 \sin \theta \cos \theta } = \text { ? }1+2sinθcosθ(sinθ+cosθ)2​= ?

Express the sum or difference as a product of sines and/or cosines. - cos⁡(4θ)−cos⁡(6θ)\cos ( 4 \theta ) - \cos ( 6 \theta )cos(4θ)−cos(6θ)

Complete the identity. - cos⁡(α+β)cos⁡(α−β)=\cos ( \alpha + \beta ) \cos ( \alpha - \beta ) =cos(α+β)cos(α−β)= ?

Find the exact value of the expression. - sin⁡π12\sin \frac { \pi } { 12 }sin12π​

Simplify the expression. - cos⁡24x−sin⁡24x\cos ^ { 2 } 4 x - \sin ^ { 2 } 4 xcos24x−sin24x

Solve the equation on the interval 0≤θ<2π0 \leq \theta < 2 \pi0≤θ<2π - tan⁡(2θ)−tan⁡θ=0\tan ( 2 \theta ) - \tan \theta = 0tan(2θ)−tanθ=0

Use the information given about the angle θ, 0 ≤θ ≤ 2π, to find the exact value of the indicated trigonometric function. - sin⁡θ=14,tan⁡θ>0\sin \theta = \frac { 1 } { 4 } , \quad \tan \theta > 0 \quadsinθ=41​,tanθ>0 Find cos⁡θ2\cos \frac { \theta } { 2 }cos2θ​

Find the exact value of the expression. - sin⁡10∘cos⁡50∘+cos⁡10∘sin⁡50∘\sin 10 ^ { \circ } \cos 50 ^ { \circ } + \cos 10 ^ { \circ } \sin 50 ^ { \circ }sin10∘cos50∘+cos10∘sin50∘

Solve the equation on the interval 0≤θ<2π0 \leq \theta < 2 \pi0≤θ<2π - cot⁡(2θ−π2)=1\cot \left( 2 \theta - \frac { \pi } { 2 } \right) = 1cot(2θ−2π​)=1

Find the exact value of the expression. - tan⁡10∘+tan⁡20∘1−tan⁡10∘tan⁡20∘\frac { \tan 10 ^ { \circ } + \tan 20 ^ { \circ } } { 1 - \tan 10 ^ { \circ } \tan 20 ^ { \circ } }1−tan10∘tan20∘tan10∘+tan20∘​

Use a calculator to find the value of the expression rounded to two decimal places. - sin⁡−1(−17)\sin ^ { - 1 } \left( - \frac { 1 } { 7 } \right)sin−1(−71​)

Complete the identity. - csc⁡(α−β)=?\csc ( \alpha - \beta ) = ?csc(α−β)=?

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. Do not use a calculator. - tan⁡−1[tan⁡(−π8)]\tan ^ { - 1 } \left[ \tan \left( - \frac { \pi } { 8 } \right) \right]tan−1[tan(−8π​)]

Complete the identity. - ln⁡∣cot⁡x∣=?\ln | \cot x | = ?ln∣cotx∣=?

Complete the identity. - csc⁡(2θ)−sec⁡(2θ)(tan⁡θ−1)=\csc ( 2 \theta ) - \sec ( 2 \theta ) ( \tan \theta - 1 ) =csc(2θ)−sec(2θ)(tanθ−1)= ?

Solve the equation on the interval 0≤θ<2π0 \leq \theta < 2 \pi0≤θ<2π - csc⁡(3θ)=0\csc ( 3 \theta ) = 0csc(3θ)=0

Solve the problem. -Find cos⁡θ2\cos \frac { \theta } { 2 }cos2θ​ , given that cos⁡θ=14\cos \theta = \frac { 1 } { 4 }cosθ=41​ and θ\thetaθ terminates in 0<θ<π/20 < \theta < \pi / 20<θ<π/2 .