Question 1

Simplify the expression.
-$$( 1 + \cot \theta ) ( 1 - \cot \theta ) - \csc ^ { 2 } \theta$$

Accepted Answer

A)  0 
B)  $- 2 \cot ^ { 2 } \theta$ 
C)  $2 \cot ^ { 2 } \theta$ 
D)  2 
A)  0 
B)  $- 2 \cot ^ { 2 } \theta$ 
C)  $2 \cot ^ { 2 } \theta$ 
D)  2

Question 2

Solve the problem.
-Factor and simplify: $\frac { 8 \sin ^ { 2 } \theta + 9 \sin \theta + 1 } { \sin ^ { 2 } \theta - 1 }$

Accepted Answer

The answer of Solve the problem.
-Factor and simplify: \(\frac {...

Question 3

Complete the identity.
-$$\frac { 1 } { \cos ^ { 2 } \theta } - \frac { 1 } { \cot ^ { 2 } \theta } = \text { ? }$$

Accepted Answer

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

Question 4

Find the exact value of the expression.
-$\cos \left( 2 \sin ^ { - 1 } \frac { 1 } { 4 } \right)$

Accepted Answer

A)  $\frac { 3 } { 8 }$ 
B)  $\frac { 5 } { 8 }$ 
C)  $\frac { 1 } { 8 }$ 
D)  $\frac { 7 } { 8 }$ 
A)  $\frac { 3 } { 8 }$ 
B)  $\frac { 5 } { 8 }$ 
C)  $\frac { 1 } { 8 }$ 
D)  $\frac { 7 } { 8 }$

Question 5

Find the exact value of the expression.
-$\sin \left( - \frac { 11 \pi } { 12 } \right)$

Accepted Answer

A)  $\frac { \sqrt { 2 } - \sqrt { 6 } } { 4 }$

В) $\frac { \sqrt { 2 } + \sqrt { 6 } } { 4 }$ 
C)  $\frac { \sqrt { 6 } - \sqrt { 2 } } { 4 }$ 
D)  $- \frac { \sqrt { 6 } + \sqrt { 2 } } { 4 }$ 
A)  $\frac { \sqrt { 2 } - \sqrt { 6 } } { 4 }$

В) $\frac { \sqrt { 2 } + \sqrt { 6 } } { 4 }$ 
C)  $\frac { \sqrt { 6 } - \sqrt { 2 } } { 4 }$ 
D)  $- \frac { \sqrt { 6 } + \sqrt { 2 } } { 4 }$

Question 6

Solve the problem.
-$$\cos ^ { 4 } x = \frac { 1 } { 0 } ( 3 + 4 \cos 2 x + \cos 4 x )$$

Accepted Answer

\[\begin{array} { l } 
\cos ^ { 4 } x =

Question 7

Find the exact value under the given conditions.
-$\sin \alpha = - \frac { 5 } { 13 } , \frac { 3 \pi } { 2 } < \alpha < 2 \pi ; \quad \tan \beta = - \frac { 7 } { 24 } , \frac { \pi } { 2 } < \beta < \pi \quad$ Find $\cos ( \alpha + \beta )$.

Accepted Answer

A)  $\frac { 204 } { 325 }$ 
B)  $- \frac { 253 } { 325 }$ 
C)  $\frac { 36 } { 325 }$ 
D)  $- \frac { 323 } { 325 }$ 
A)  $\frac { 204 } { 325 }$ 
B)  $- \frac { 253 } { 325 }$ 
C)  $\frac { 36 } { 325 }$ 
D)  $- \frac { 323 } { 325 }$

Question 8

Complete the identity.
-$\cos \left( \tan ^ { - 1 } v \right) = ?$

Accepted Answer

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

Question 9

Solve the problem.
-The path of a projectile fired at an inclination $\theta \text { (in degrees) }$ to the horizontal with an initial velocity $\mathrm { v } _ { 0 }$ is a
parabola. The range R of the projectile, that is, the horizontal distance that the projectile travels, is found by using the
formula $$\mathrm { R } = \frac { \mathrm { v } _ { 0 } ^ { 2 } } { \mathrm {~g} } \sin ( 2 \theta )$$ where g is the acceleration due to gravity. Suppose the projectile is fired with an initial velocity of 400 feet per seconds
and g = 32 feet per $\operatorname { second } \mathrm { d } ^ { 2 }$ . What angle $\theta , 0 ^ { \circ } \leq \theta < 90 ^ { \circ }$ would you select for the range to be 2500 feet? (There
should be two values of $\theta$ .)

Accepted Answer

The answer of Solve the problem.
-The path of a projectile...

Question 10

Express the sum or difference as a product of sines and/or cosines.
-$$\cos ( 10 \theta ) - \cos ( 4 \theta )$$

Accepted Answer

A)  $2 \cos ( 3 \theta )$ 
B)  $2 \cos ( 7 \theta ) \cos ( 3 \theta )$ 
C)  $- 2 \sin ( 7 \theta ) \sin ( 3 \theta )$ 
D)  $- 2 \cos ( 7 \theta ) \sin ( 3 \theta )$ 
A)  $2 \cos ( 3 \theta )$ 
B)  $2 \cos ( 7 \theta ) \cos ( 3 \theta )$ 
C)  $- 2 \sin ( 7 \theta ) \sin ( 3 \theta )$ 
D)  $- 2 \cos ( 7 \theta ) \sin ( 3 \theta )$

Question 11

Find the exact value of the expression. Do not use a calculator.
-$\sin \left[ \sin ^ { - 1 } ( - 0.3 ) \right]$

Accepted Answer

A)  2.7 
B)  -2.7 
C)  0.3 
D)  -0.3 
A)  2.7 
B)  -2.7 
C)  0.3 
D)  -0.3

Question 12

Use the information given about the angle θ, 0 ≤θ ≤ 2π, to find the exact value of the indicated trigonometric function.
-$\sin \theta = - \frac { 3 } { 5 } , \frac { 3 \pi } { 2 } < \theta < 2 \pi \quad$ Find $\sin \frac { \theta } { 2 }$.

Accepted Answer

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

Question 13

Find the exact value of the expression.
-$\cos \left( \sin ^ { - 1 } \frac { 1 } { 4 } \right)$

Accepted Answer

A)  $\frac { 2 \sqrt { 15 } } { 15 }$ 
B)  $\frac { 4 \sqrt { 15 } } { 15 }$ 
C)  $\frac { \sqrt { 15 } } { 4 }$ 
D)  $\frac { \sqrt { 15 } } { 2 }$ 
A)  $\frac { 2 \sqrt { 15 } } { 15 }$ 
B)  $\frac { 4 \sqrt { 15 } } { 15 }$ 
C)  $\frac { \sqrt { 15 } } { 4 }$ 
D)  $\frac { \sqrt { 15 } } { 2 }$

Question 14

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

Accepted Answer

A)  $\frac { \sec \alpha \sec \beta } { \cos ^ { 2 } \alpha - \sin ^ { 2 } \beta }$ 
B)  $\frac { 1 } { \cos ^ { 2 } \alpha - \cos ^ { 2 } \beta }$ 
C)  $\frac { \sec ^ { 2 } \alpha \sec ^ { 2 } \beta } { 1 - \tan ^ { 2 } \alpha \tan ^ { 2 } \beta }$ 
D)  $\sec ^ { 2 } \alpha - \sec ^ { 2 } \beta$ 
A)  $\frac { \sec \alpha \sec \beta } { \cos ^ { 2 } \alpha - \sin ^ { 2 } \beta }$ 
B)  $\frac { 1 } { \cos ^ { 2 } \alpha - \cos ^ { 2 } \beta }$ 
C)  $\frac { \sec ^ { 2 } \alpha \sec ^ { 2 } \beta } { 1 - \tan ^ { 2 } \alpha \tan ^ { 2 } \beta }$ 
D)  $\sec ^ { 2 } \alpha - \sec ^ { 2 } \beta$

Question 15

Solve the equation on the interval $$0 \leq \theta < 2 \pi$$
-$$\sin ^ { 2 } \theta - \cos ^ { 2 } \theta = 0$$

Accepted Answer

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

Question 16

Express the sum or difference as a product of sines and/or cosines.
-$\sin ( 2 \theta ) - \sin ( 4 \theta )$

Accepted Answer

A)  $2 \cos ( 2 \theta ) \cos ( 3 \theta )$ 
B)  $- 2 \sin \theta$ 
C)  $- 2 \sin \theta \cos ( 3 \theta )$ 
D)  $2 \sin ( 3 \theta ) \cos \theta$ 
A)  $2 \cos ( 2 \theta ) \cos ( 3 \theta )$ 
B)  $- 2 \sin \theta$ 
C)  $- 2 \sin \theta \cos ( 3 \theta )$ 
D)  $2 \sin ( 3 \theta ) \cos \theta$

Question 17

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

Accepted Answer

The answer of Establish the identity.
-\[\csc ^ { 4 }...

Question 18

Find the exact value of the expression.884:890
-$\sin \left[ 2 \cos ^ { - 1 } \left( - \frac { 3 } { 5 } \right) \right]$

Accepted Answer

A)  $- \frac { 12 } { 25 }$ 
B)  $- \frac { 24 } { 25 }$ 
C)  $\frac { 12 } { 25 }$ 
D)  $\frac { 24 } { 25 }$ 
A)  $- \frac { 12 } { 25 }$ 
B)  $- \frac { 24 } { 25 }$ 
C)  $\frac { 12 } { 25 }$ 
D)  $\frac { 24 } { 25 }$

Question 19

Use a calculator to solve the equation on the interval 0 $0 \leq x < 2 \pi$ . Round the answer to one decimal place if necessary.
-$x + 3 \sin x = 1$

Accepted Answer

The answer of Use a calculator to solve the equation...

Question 20

Solve the equation. Give a general formula for all the solutions.
-$\cos \theta - 1 = 0$

Accepted Answer

A)  $\theta = \frac { 3 \pi } { 2 } + 2 \mathrm { k } \pi$ 
B)  $\theta = \pi + 2 \mathrm { k } \pi$ 
C)  $\theta = 2 \mathrm { k } \pi$ 
D)  $\theta = \frac { \pi } { 2 } + 2 \mathrm { k } \pi$ 
A)  $\theta = \frac { 3 \pi } { 2 } + 2 \mathrm { k } \pi$ 
B)  $\theta = \pi + 2 \mathrm { k } \pi$ 
C)  $\theta = 2 \mathrm { k } \pi$ 
D)  $\theta = \frac { \pi } { 2 } + 2 \mathrm { k } \pi$

Simplify the expression. - $( 1 + \cot \theta ) ( 1 - \cot \theta ) - \csc ^ { 2 } \theta$

Solve the problem. -Factor and simplify: $\frac { 8 \sin ^ { 2 } \theta + 9 \sin \theta + 1 } { \sin ^ { 2 } \theta - 1 }$

Complete the identity. - $\frac { 1 } { \cos ^ { 2 } \theta } - \frac { 1 } { \cot ^ { 2 } \theta } = \text { ? }$

Find the exact value of the expression. - $\cos \left( 2 \sin ^ { - 1 } \frac { 1 } { 4 } \right)$

Find the exact value of the expression. - $\sin \left( - \frac { 11 \pi } { 12 } \right)$

Solve the problem. - $\cos ^ { 4 } x = \frac { 1 } { 0 } ( 3 + 4 \cos 2 x + \cos 4 x )$

Find the exact value under the given conditions. - $\sin \alpha = - \frac { 5 } { 13 } , \frac { 3 \pi } { 2 } < \alpha < 2 \pi ; \quad \tan \beta = - \frac { 7 } { 24 } , \frac { \pi } { 2 } < \beta < \pi \quad$ Find $\cos ( \alpha + \beta )$ .

Complete the identity. - $\cos \left( \tan ^ { - 1 } v \right) = ?$

Express the sum or difference as a product of sines and/or cosines. - $\cos ( 10 \theta ) - \cos ( 4 \theta )$

Find the exact value of the expression. Do not use a calculator. - $\sin \left[ \sin ^ { - 1 } ( - 0.3 ) \right]$

Use the information given about the angle θ, 0 ≤θ ≤ 2π, to find the exact value of the indicated trigonometric function. - $\sin \theta = - \frac { 3 } { 5 } , \frac { 3 \pi } { 2 } < \theta < 2 \pi \quad$ Find $\sin \frac { \theta } { 2 }$ .

Find the exact value of the expression. - $\cos \left( \sin ^ { - 1 } \frac { 1 } { 4 } \right)$

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

Solve the equation on the interval $0 \leq \theta < 2 \pi$ - $\sin ^ { 2 } \theta - \cos ^ { 2 } \theta = 0$

Express the sum or difference as a product of sines and/or cosines. - $\sin ( 2 \theta ) - \sin ( 4 \theta )$

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

Find the exact value of the expression.884:890 - $\sin \left[ 2 \cos ^ { - 1 } \left( - \frac { 3 } { 5 } \right) \right]$

Use a calculator to solve the equation on the interval 0 $0 \leq x < 2 \pi$ . Round the answer to one decimal place if necessary. - $x + 3 \sin x = 1$

Solve the equation. Give a general formula for all the solutions. - $\cos \theta - 1 = 0$

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Exam 6: Analytic Trigonometry

Simplify the expression. - (1+cot⁡θ)(1−cot⁡θ)−csc⁡2θ( 1 + \cot \theta ) ( 1 - \cot \theta ) - \csc ^ { 2 } \theta(1+cotθ)(1−cotθ)−csc2θ

Solve the problem. -Factor and simplify: 8sin⁡2θ+9sin⁡θ+1sin⁡2θ−1\frac { 8 \sin ^ { 2 } \theta + 9 \sin \theta + 1 } { \sin ^ { 2 } \theta - 1 }sin2θ−18sin2θ+9sinθ+1​

Complete the identity. - 1cos⁡2θ−1cot⁡2θ= ? \frac { 1 } { \cos ^ { 2 } \theta } - \frac { 1 } { \cot ^ { 2 } \theta } = \text { ? }cos2θ1​−cot2θ1​= ?

Find the exact value of the expression. - cos⁡(2sin⁡−114)\cos \left( 2 \sin ^ { - 1 } \frac { 1 } { 4 } \right)cos(2sin−141​)

Find the exact value of the expression. - sin⁡(−11π12)\sin \left( - \frac { 11 \pi } { 12 } \right)sin(−1211π​)

Solve the problem. - cos⁡4x=10(3+4cos⁡2x+cos⁡4x)\cos ^ { 4 } x = \frac { 1 } { 0 } ( 3 + 4 \cos 2 x + \cos 4 x )cos4x=01​(3+4cos2x+cos4x)

Complete the identity. - cos⁡(tan⁡−1v)=?\cos \left( \tan ^ { - 1 } v \right) = ?cos(tan−1v)=?

Express the sum or difference as a product of sines and/or cosines. - cos⁡(10θ)−cos⁡(4θ)\cos ( 10 \theta ) - \cos ( 4 \theta )cos(10θ)−cos(4θ)

Find the exact value of the expression. Do not use a calculator. - sin⁡[sin⁡−1(−0.3)]\sin \left[ \sin ^ { - 1 } ( - 0.3 ) \right]sin[sin−1(−0.3)]

Find the exact value of the expression. - cos⁡(sin⁡−114)\cos \left( \sin ^ { - 1 } \frac { 1 } { 4 } \right)cos(sin−141​)

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

Solve the equation on the interval 0≤θ<2π0 \leq \theta < 2 \pi0≤θ<2π - sin⁡2θ−cos⁡2θ=0\sin ^ { 2 } \theta - \cos ^ { 2 } \theta = 0sin2θ−cos2θ=0

Express the sum or difference as a product of sines and/or cosines. - sin⁡(2θ)−sin⁡(4θ)\sin ( 2 \theta ) - \sin ( 4 \theta )sin(2θ)−sin(4θ)

Establish the identity. - csc⁡4x−cot⁡4x=csc⁡2x+cot⁡2x\csc ^ { 4 } x - \cot ^ { 4 } x = \csc ^ { 2 } x + \cot ^ { 2 } xcsc4x−cot4x=csc2x+cot2x

Find the exact value of the expression.884:890 - sin⁡[2cos⁡−1(−35)]\sin \left[ 2 \cos ^ { - 1 } \left( - \frac { 3 } { 5 } \right) \right]sin[2cos−1(−53​)]

Use a calculator to solve the equation on the interval 0 0≤x<2π0 \leq x < 2 \pi0≤x<2π . Round the answer to one decimal place if necessary. - x+3sin⁡x=1x + 3 \sin x = 1x+3sinx=1

Solve the equation. Give a general formula for all the solutions. - cos⁡θ−1=0\cos \theta - 1 = 0cosθ−1=0