Question 1

Evaluate the following expression. ​
Cot2t (2 sec2t - 2)
​

Accepted Answer

A) 2 
B) 0 
C) 3 
D) 4 
E) 1 
A) 2 
B) 0 
C) 3 
D) 4 
E) 1 A

Question 2

Use the half-angle formulas to determine the exact value of the given trigonometric expression.  
  $\tan \frac { 3 \pi } { 8 }$

Accepted Answer

A)   $\tan \frac { 3 \pi } { 8 } = \sqrt { 2 } - 1$ 
B)   $\tan \frac { 3 \pi } { 8 } = \frac { \sqrt { 2 - \sqrt { 2 } } } { 4 }$ 
C)  $\tan \frac { 3 \pi } { 8 } = \sqrt { 2 + \sqrt { 2 } }$ 
D)   $\tan \frac { 3 \pi } { 8 } = \sqrt { 2 } + 1$ 
E)   $\tan \frac { 3 \pi } { 8 } = - \frac { \sqrt { 2 + \sqrt { 2 } } } { 4 }$ 
A)   $\tan \frac { 3 \pi } { 8 } = \sqrt { 2 } - 1$ 
B)   $\tan \frac { 3 \pi } { 8 } = \frac { \sqrt { 2 - \sqrt { 2 } } } { 4 }$ 
C)  $\tan \frac { 3 \pi } { 8 } = \sqrt { 2 + \sqrt { 2 } }$ 
D)   $\tan \frac { 3 \pi } { 8 } = \sqrt { 2 } + 1$ 
E)   $\tan \frac { 3 \pi } { 8 } = - \frac { \sqrt { 2 + \sqrt { 2 } } } { 4 }$ D

Question 3

Write the given expression as the cosine of an angle. 
 $\cos 45 ^ { \circ } \cos 40 ^ { \circ } + \sin 45 ^ { \circ } \sin 40 ^ { \circ }$

Accepted Answer

A)  $\cos \left( 45 ^ { \circ } \right)$ 
B)  $\cos \left( 5 ^ { \circ } \right)$ 
C)  $\cos \left( - 80 ^ { \circ } \right)$ 
D)  $\cos \left( 40 ^ { \circ } \right)$ 
E)  $\cos \left( 85 ^ { \circ } \right)$ 
A)  $\cos \left( 45 ^ { \circ } \right)$ 
B)  $\cos \left( 5 ^ { \circ } \right)$ 
C)  $\cos \left( - 80 ^ { \circ } \right)$ 
D)  $\cos \left( 40 ^ { \circ } \right)$ 
E)  $\cos \left( 85 ^ { \circ } \right)$ B

Question 4

Use the trigonometric substitution u = a csc $\theta $, where 0  0 to simplify the expression $\sqrt { u ^ { 2 } - a ^ { 2 } }$ .

Accepted Answer

A) a tan$\theta $ 
B) - a cot $\theta $ 
C) a csc $\theta $ 
D) a cos $\theta $ 
E) a cot $\theta $ 
A) a tan$\theta $ 
B) - a cot $\theta $ 
C) a csc $\theta $ 
D) a cos $\theta $ 
E) a cot $\theta $

Question 5

Convert the expression.   $\cos 4 \alpha$

Accepted Answer

A)   $\cos ^ { 2 } 2 \alpha - \sin ^ { 2 } 2 \alpha$ 
B)   $\cos ^ { 2 } 2 \alpha + \sin ^ { 2 } 2 \alpha$ 
C)   $\cos ^ { 4 } 2 \alpha + \sin ^ { 4 } 2 \alpha$ 
D)   $\cos ^ { 4 } 2 \alpha - \sin ^ { 4 } 2 \alpha$ 
E)   $\cos ^ { 4 } 2 \alpha - \sin ^ { 2 } 2 \alpha$ 
A)   $\cos ^ { 2 } 2 \alpha - \sin ^ { 2 } 2 \alpha$ 
B)   $\cos ^ { 2 } 2 \alpha + \sin ^ { 2 } 2 \alpha$ 
C)   $\cos ^ { 4 } 2 \alpha + \sin ^ { 4 } 2 \alpha$ 
D)   $\cos ^ { 4 } 2 \alpha - \sin ^ { 4 } 2 \alpha$ 
E)   $\cos ^ { 4 } 2 \alpha - \sin ^ { 2 } 2 \alpha$

Question 6

Which of the following expressions is equivalent to 3 sin2 α - 3 sin4 α? ​

Accepted Answer

A) 3 cos4 α + cos2 α 
B) -cos2 α - 3 cos4 α) 
C) 3 (cos2 α + cos4 α) 
D) 3 (cos2 α - cos4 α) 
E) -3 (cos2 α - cos4 α) 
A) 3 cos4 α + cos2 α 
B) -cos2 α - 3 cos4 α) 
C) 3 (cos2 α + cos4 α) 
D) 3 (cos2 α - cos4 α) 
E) -3 (cos2 α - cos4 α)

Question 7

Use the half-angle formulas to simplify the expression.  
  $\sqrt { \frac { 1 + \cos 8 x } { 2 } }$

Accepted Answer

A)  -  |cos x| 
B)  |cos x| 
C)   |cos 4x| 
D) - |cos 4x| 
E)  -  |cos 8x| 
A)  -  |cos x| 
B)  |cos x| 
C)   |cos 4x| 
D) - |cos 4x| 
E)  -  |cos 8x|

Question 8

Find all solutions of the following equation in the interval [0, 2$\pi $ ).   $2 \cos ^ { 3 } x = 2 \cos x$

Accepted Answer

A)  $0 , \frac { \pi } { 2 } , 2 \pi , \frac { 3 \pi } { 4 }$ 
B)   $0 , \frac { \pi } { 2 } , 3 \pi , \frac { 3 \pi } { 2 }$ 
C)   $0 , \frac { \pi } { 3 } , \pi , \frac { 2 \pi } { 3 }$ 
D)   $0 , \frac { \pi } { 4 } , \pi , \frac { 2 \pi } { 3 }$ 
E)   $0 , \frac { \pi } { 2 } , \pi , \frac { 3 \pi } { 2 }$ 
A)  $0 , \frac { \pi } { 2 } , 2 \pi , \frac { 3 \pi } { 4 }$ 
B)   $0 , \frac { \pi } { 2 } , 3 \pi , \frac { 3 \pi } { 2 }$ 
C)   $0 , \frac { \pi } { 3 } , \pi , \frac { 2 \pi } { 3 }$ 
D)   $0 , \frac { \pi } { 4 } , \pi , \frac { 2 \pi } { 3 }$ 
E)   $0 , \frac { \pi } { 2 } , \pi , \frac { 3 \pi } { 2 }$

Question 9

Evaluate the following expression.($\theta $ $\neq$ ($\pi$+2$\pi$n)/6, where n is a whole number)   
$\frac { \sec 3 \theta - 1 } { 1 - \cos 3 \theta }$

Accepted Answer

A) - cos 3$\theta $ 
B) cos 3$\theta $ 
C) sec 3$\theta $ 
D) - sec 3$\theta $ 
E) sec 3$\theta $  - cos 3$\theta $ 
A) - cos 3$\theta $ 
B) cos 3$\theta $ 
C) sec 3$\theta $ 
D) - sec 3$\theta $ 
E) sec 3$\theta $  - cos 3$\theta $

Question 10

Use the product-to-sum formulas to rewrite the product as a sum or difference.   $\sin \frac { \pi } { 3 } \cos \frac { \pi } { 6 }$

Accepted Answer

A)   $\frac { 1 } { 2 } \left( \sin \frac { \pi } { 2 } + \cos \frac { \pi } { 6 } \right)$ 
B)   $\frac { 1 } { 2 } \left( \sin \frac { \pi } { 2 } + \sin \frac { \pi } { 6 } \right)$ 
C)  $\frac { 1 } { 2 } \left( \sin \frac { \pi } { 2 } - \cos \frac { \pi } { 6 } \right)$ 
D)   $\frac { 1 } { 2 } \left( \sin \frac { \pi } { 2 } - \sin \frac { \pi } { 6 } \right)$ 
E)   $\left( \sin \frac { \pi } { 2 } + \cos \frac { \pi } { 6 } \right)$ 
A)   $\frac { 1 } { 2 } \left( \sin \frac { \pi } { 2 } + \cos \frac { \pi } { 6 } \right)$ 
B)   $\frac { 1 } { 2 } \left( \sin \frac { \pi } { 2 } + \sin \frac { \pi } { 6 } \right)$ 
C)  $\frac { 1 } { 2 } \left( \sin \frac { \pi } { 2 } - \cos \frac { \pi } { 6 } \right)$ 
D)   $\frac { 1 } { 2 } \left( \sin \frac { \pi } { 2 } - \sin \frac { \pi } { 6 } \right)$ 
E)   $\left( \sin \frac { \pi } { 2 } + \cos \frac { \pi } { 6 } \right)$

Question 11

Evaluate the following expression.(x ≠ π/2+πn, where n is a whole number) ​
Tan7x sec2x - tan7x
​

Accepted Answer

A) 9 tan9x 
B) tan2x 
C) tan9x 
D) 7 tan7x 
E) tan7x 
A) 9 tan9x 
B) tan2x 
C) tan9x 
D) 7 tan7x 
E) tan7x

Question 12

Use the sum-to-product formulas to rewrite the sum or difference as a product.   $\sin 3 \theta - \sin \theta$

Accepted Answer

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

Question 13

Evaluate the following expression. ​
4 sec y cos y
​

Accepted Answer

A) 5 
B) 3 
C) 6 
D) 7 
E) 4 
A) 5 
B) 3 
C) 6 
D) 7 
E) 4

Question 14

Which of the following expressions is equivalent to $\frac { 5 } { \tan \beta } + 5 \tan \beta$ ?
 ($\beta$  $\neq$  $\pi$/2+$\pi$n, where n is a whole number)

Accepted Answer

A)  $\frac { 5 \cos ^ { 2 } \beta } { \cot \beta }$ 
B)  $\frac { 5 \tan ^ { 2 } \beta } { \sec \beta }$ 
C)  $\frac { 5 \sec ^ { 2 } \beta } { \tan \beta }$ 
D)  $\frac { 5 \sec ^ { 2 } \beta } { \cot \beta }$ 
E)  $\frac { 5 \cos ^ { 2 } \beta } { \tan \beta }$ 
A)  $\frac { 5 \cos ^ { 2 } \beta } { \cot \beta }$ 
B)  $\frac { 5 \tan ^ { 2 } \beta } { \sec \beta }$ 
C)  $\frac { 5 \sec ^ { 2 } \beta } { \tan \beta }$ 
D)  $\frac { 5 \sec ^ { 2 } \beta } { \cot \beta }$ 
E)  $\frac { 5 \cos ^ { 2 } \beta } { \tan \beta }$

Question 15

Simplify the expression algebraically. 
 $\cos ( 6 x + 9 y ) + \cos ( 6 x - 9 y )$

Accepted Answer

A)  $\cos 6 x$ 
B)  $\cos 6 x \cos 9 y$ 
C)  $2 \cos 6 x \cos 9 y$ 
D)  $2 \cos 6 x$ 
E)  $2 \cos x \cos y$ 
A)  $\cos 6 x$ 
B)  $\cos 6 x \cos 9 y$ 
C)  $2 \cos 6 x \cos 9 y$ 
D)  $2 \cos 6 x$ 
E)  $2 \cos x \cos y$

Question 16

If x = 2 tan $\theta $, use trigonometric substitution to write $\sqrt { 4 + x ^ { 2 } }$ as a trigonometric function of $\theta $, where $0 < \theta < \frac { \pi } { 2 }$ .

Accepted Answer

A)  2 cos $\theta $ 
B)  2 tan $\theta $ 
C)  2 sin $\theta $ 
D)  2 sec $\theta $ 
E)  2 csc $\theta $ 
A)  2 cos $\theta $ 
B)  2 tan $\theta $ 
C)  2 sin $\theta $ 
D)  2 sec $\theta $ 
E)  2 csc $\theta $

Question 17

If $\sin x = \frac { 1 } { 2 } \text { and } \cos x = \frac { \sqrt { 3 } } { 2 }$ , evaluate the following function. 
cot x

Accepted Answer

A)  cot x = $\frac { 2 } { 3 }$ $\sqrt { 3 }$ 
B)  cot x = $\frac { 1 } { 3 }$ $\sqrt { 3 }$ 
C)  cot x = 2 
D)  cot x = $\frac { 1 } { 3 }$ 
E)  cot x = $\sqrt { 3 }$ 
A)  cot x = $\frac { 2 } { 3 }$ $\sqrt { 3 }$ 
B)  cot x = $\frac { 1 } { 3 }$ $\sqrt { 3 }$ 
C)  cot x = 2 
D)  cot x = $\frac { 1 } { 3 }$ 
E)  cot x = $\sqrt { 3 }$

Question 18

Find all solutions of the following equation in the interval [0, 2$\pi $). 
 $5 \sec x \csc x = 10 \csc x$

Accepted Answer

A)  $\frac { \pi } { 3 } , \frac { 5 \pi } { 3 }$ 
B)  $\frac { 5 \pi } { 2 } , \frac { 5 \pi } { 3 }$ 
C)   $\frac { 7 \pi } { 2 } , \frac { \pi } { 3 }$ 
D)   $\frac { 3 \pi } { 2 } , \frac { \pi } { 3 }$ 
E)   $2 \pi , \pi$ 
A)  $\frac { \pi } { 3 } , \frac { 5 \pi } { 3 }$ 
B)  $\frac { 5 \pi } { 2 } , \frac { 5 \pi } { 3 }$ 
C)   $\frac { 7 \pi } { 2 } , \frac { \pi } { 3 }$ 
D)   $\frac { 3 \pi } { 2 } , \frac { \pi } { 3 }$ 
E)   $2 \pi , \pi$

Question 19

Use the given values to evaluate (if possible) three trigonometric functions cos x, sin x, cot x. 
 $\tan x = \frac { 9 } { 40 } , \sec x = - \frac { 41 } { 40 }$

Accepted Answer

A)  $\begin{array} { l } 
\cos x = \frac { 40 } { 41 } \
\sin x = \frac { 9 } { 41 } \
\cot x = \frac { 40 } { 9 }
\end{array}$ 
B)   $\begin{array} { l } 
\cos x = - \frac { 40 } { 41 } \
\sin x = - \frac { 9 } { 41 } \
\cot x = - \frac { 40 } { 9 }
\end{array}$ 
C)   $\begin{array} { l } 
\cos x = - \frac { 40 } { 41 } \
\sin x = - \frac { 9 } { 41 } \
\cot x = \frac { 40 } { 9 }
\end{array}$ 
D)   $\begin{array} { l } 
\cos x = \frac { 40 } { 41 } \
\sin x = - \frac { 9 } { 41 } \
\cot x = - \frac { 40 } { 9 }
\end{array}$ 
E)   $\begin{array} { l } 
\cos x = - \frac { 40 } { 41 } \
\sin x = \frac { 9 } { 41 } \
\cot x = \frac { 40 } { 9 }
\end{array}$ 
A)  $\begin{array} { l } 
\cos x = \frac { 40 } { 41 } \
\sin x = \frac { 9 } { 41 } \
\cot x = \frac { 40 } { 9 }
\end{array}$ 
B)   $\begin{array} { l } 
\cos x = - \frac { 40 } { 41 } \
\sin x = - \frac { 9 } { 41 } \
\cot x = - \frac { 40 } { 9 }
\end{array}$ 
C)   $\begin{array} { l } 
\cos x = - \frac { 40 } { 41 } \
\sin x = - \frac { 9 } { 41 } \
\cot x = \frac { 40 } { 9 }
\end{array}$ 
D)   $\begin{array} { l } 
\cos x = \frac { 40 } { 41 } \
\sin x = - \frac { 9 } { 41 } \
\cot x = - \frac { 40 } { 9 }
\end{array}$ 
E)   $\begin{array} { l } 
\cos x = - \frac { 40 } { 41 } \
\sin x = \frac { 9 } { 41 } \
\cot x = \frac { 40 } { 9 }
\end{array}$

Question 20

A weight is attached to a spring suspended vertically from a ceiling.When a driving force is applied to the system, the weight moves vertically from its equilibrium position, and this motion is modeled by 
 $y = \frac { 1 } { 8 } \sin 2 t + \frac { 1 } { 6 } \cos 2 t$  
Where y is the distance from equilibrium (in feet) and t is the time (in seconds).
 
Use the identity $a \sin B \theta + b \cos B \theta = \sqrt { a ^ { 2 } + b ^ { 2 } } \sin ( B \theta + C )$ where $C = \arctan ( b / a ) , a > 0$ , to write the model in the form $y = \sqrt { a ^ { 2 } + b ^ { 2 } } \sin ( B t + C )$ .

Accepted Answer

A)  $y = \sin ( 2 t + 0.9273 )$ 
B)  $y = \frac { 24 } { 5 } \sin ( 2 t + 0.9273 )$ 
C)  $y = \frac { 24 } { 5 } \sin ( 2 t - 0.9273 )$ 
D)  $y = \sin ( 2 t - 0.9273 )$ 
E)  $y = \frac { 5 } { 24 } \sin ( 2 t + 0.9273 )$ 
A)  $y = \sin ( 2 t + 0.9273 )$ 
B)  $y = \frac { 24 } { 5 } \sin ( 2 t + 0.9273 )$ 
C)  $y = \frac { 24 } { 5 } \sin ( 2 t - 0.9273 )$ 
D)  $y = \sin ( 2 t - 0.9273 )$ 
E)  $y = \frac { 5 } { 24 } \sin ( 2 t + 0.9273 )$

Evaluate the following expression. Cot²t (2 sec²t - 2)

Use the half-angle formulas to determine the exact value of the given trigonometric expression. $\tan \frac { 3 \pi } { 8 }$

Write the given expression as the cosine of an angle. $\cos 45 ^ { \circ } \cos 40 ^ { \circ } + \sin 45 ^ { \circ } \sin 40 ^ { \circ }$

Use the trigonometric substitution u = a csc $\theta$ , where 0 < $\theta$ < $\pi$ /2 and a > 0 to simplify the expression $\sqrt { u ^ { 2 } - a ^ { 2 } }$ .

Convert the expression. $\cos 4 \alpha$

Which of the following expressions is equivalent to 3 sin² α - 3 sin⁴ α?

Use the half-angle formulas to simplify the expression. $\sqrt { \frac { 1 + \cos 8 x } { 2 } }$

Find all solutions of the following equation in the interval [0, 2 $\pi$ ). $2 \cos ^ { 3 } x = 2 \cos x$

Evaluate the following expression.( $\theta$ $\neq$ ( $\pi$ +2 $\pi$ n)/6, where n is a whole number) $\frac { \sec 3 \theta - 1 } { 1 - \cos 3 \theta }$

Use the product-to-sum formulas to rewrite the product as a sum or difference. $\sin \frac { \pi } { 3 } \cos \frac { \pi } { 6 }$

Evaluate the following expression.(x ≠ π/2+πn, where n is a whole number) Tan⁷x sec²x - tan⁷x

Use the sum-to-product formulas to rewrite the sum or difference as a product. $\sin 3 \theta - \sin \theta$

Evaluate the following expression. 4 sec y cos y

Which of the following expressions is equivalent to $\frac { 5 } { \tan \beta } + 5 \tan \beta$ ? ( $\beta$ $\neq$ $\pi$ /2+ $\pi$ n, where n is a whole number)

Simplify the expression algebraically. $\cos ( 6 x + 9 y ) + \cos ( 6 x - 9 y )$

If x = 2 tan $\theta$ , use trigonometric substitution to write $\sqrt { 4 + x ^ { 2 } }$ as a trigonometric function of $\theta$ , where $0 < \theta < \frac { \pi } { 2 }$ .

If $\sin x = \frac { 1 } { 2 } \text { and } \cos x = \frac { \sqrt { 3 } } { 2 }$ , evaluate the following function. cot x

Find all solutions of the following equation in the interval [0, 2 $\pi$ ). $5 \sec x \csc x = 10 \csc x$

Use the given values to evaluate (if possible) three trigonometric functions cos x, sin x, cot x. $\tan x = \frac { 9 } { 40 } , \sec x = - \frac { 41 } { 40 }$

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Exam 5: Analytic Trigonometry

Evaluate the following expression. ​ Cot2t (2 sec2t - 2) ​

Use the half-angle formulas to determine the exact value of the given trigonometric expression. tan⁡3π8\tan \frac { 3 \pi } { 8 }tan83π​

Write the given expression as the cosine of an angle. cos⁡45∘cos⁡40∘+sin⁡45∘sin⁡40∘\cos 45 ^ { \circ } \cos 40 ^ { \circ } + \sin 45 ^ { \circ } \sin 40 ^ { \circ }cos45∘cos40∘+sin45∘sin40∘

Use the trigonometric substitution u = a csc θ\theta θ , where 0 < θ\theta θ < π\pi π /2 and a > 0 to simplify the expression u2−a2\sqrt { u ^ { 2 } - a ^ { 2 } }u2−a2​ .

Convert the expression. cos⁡4α\cos 4 \alphacos4α