Question 1

Evaluate the following expression. $\frac { \tan ^ { 2 } \theta } { \sec \theta }$

Accepted Answer

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

Question 2

Find the expression as the sine or cosine of an angle. $$\cos 6 x \cos 3 y + \sin 6 x \sin 3 y$$

Accepted Answer

A)  $\cos ( 6 x - 3 y )$ 
B)  $\sin ( 6 x - 3 y )$ 
C)  $\cos ( 6 x + 3 y )$ 
D)  $\sin ( 3 x - 6 y )$ 
E)  $\cos ( 3 x - 6 y )$ 
A)  $\cos ( 6 x - 3 y )$ 
B)  $\sin ( 6 x - 3 y )$ 
C)  $\cos ( 6 x + 3 y )$ 
D)  $\sin ( 3 x - 6 y )$ 
E)  $\cos ( 3 x - 6 y )$

Question 3

Use the given values to evaluate (if possible) three trigonometric functions $\operatorname { cosec } \theta$ $\tan \theta , \cos \theta$ $\sin \theta = - 3 , \quad \cot \theta = 0$

Accepted Answer

A)  $\csc \theta = - \frac { 1 } { 3 }$
$\tan \theta$ is undefined.
$\cos \theta = - 2 \sqrt { 2 }$ 
B)  $\csc \theta = - \frac { 1 } { 3 }$
$\tan \theta$ is undefined.
$\cos \theta = 0$ 
C)  $\csc \theta = \frac { 1 } { 3 }$
$\tan \theta$ is undefined.
$\cos \theta = - 2 \sqrt { 2 }$ 
D)  $\csc \theta = \frac { 1 } { 3 }$
$\tan \theta$ is undefined.
$\cos \theta = 2 \sqrt { 2 }$ 
E)  $\csc \theta$ is undefined.
$\tan \theta = \frac { 1 } { 3 }$
$\cos \theta = - 2 \sqrt { 2 }$ 
A)  $\csc \theta = - \frac { 1 } { 3 }$
$\tan \theta$ is undefined.
$\cos \theta = - 2 \sqrt { 2 }$ 
B)  $\csc \theta = - \frac { 1 } { 3 }$
$\tan \theta$ is undefined.
$\cos \theta = 0$ 
C)  $\csc \theta = \frac { 1 } { 3 }$
$\tan \theta$ is undefined.
$\cos \theta = - 2 \sqrt { 2 }$ 
D)  $\csc \theta = \frac { 1 } { 3 }$
$\tan \theta$ is undefined.
$\cos \theta = 2 \sqrt { 2 }$ 
E)  $\csc \theta$ is undefined.
$\tan \theta = \frac { 1 } { 3 }$
$\cos \theta = - 2 \sqrt { 2 }$

Question 4

Use the trigonometric substitution to select the algebraic expression as a trigonomet- $$\text { ric function of } \theta \text {, where } 0 < \theta < \frac { \pi } { 2 } \text {. }$$ $$\sqrt { 16 x ^ { 2 } + 25 } , 4 x = 5 \tan \theta$$

Accepted Answer

A)  $5 \sec \theta$ 
B)  $16 \tan \theta$ 
C)  $16 \sec \theta$ 
D)  $- 4 \sec \theta$ 
E)  $4 \tan \theta$ 
A)  $5 \sec \theta$ 
B)  $16 \tan \theta$ 
C)  $16 \sec \theta$ 
D)  $- 4 \sec \theta$ 
E)  $4 \tan \theta$

Question 5

Use the figure to find the exact value of the trigonometric function.

Accepted Answer

A) $\frac { 9 } { 17 }$ 
B)  $\frac { 17 } { 8 }$ 
C)  $\frac { 8 } { 17 }$ 
D)  $\frac { 17 } { 9 }$ 
E)  $\frac { 8 } { 9 }$ 
A) $\frac { 9 } { 17 }$ 
B)  $\frac { 17 } { 8 }$ 
C)  $\frac { 8 } { 17 }$ 
D)  $\frac { 17 } { 9 }$ 
E)  $\frac { 8 } { 9 }$

Question 6

Use the sum-to-product formulas to select the sum or difference as a product. $$\sin 8 \theta + \sin 6 \theta$$

Accepted Answer

A)  $2 \sin 7 \theta \cos \theta$ 
B)  $2 \sin 7 \theta \sin \theta$ 
C)  $2 \cos 7 \theta \sin \theta$ 
D)  $2 \sin 8 \theta \cos 6 \theta$ 
E)  $2 \cos 7 \theta \cos \theta$ 
A)  $2 \sin 7 \theta \cos \theta$ 
B)  $2 \sin 7 \theta \sin \theta$ 
C)  $2 \cos 7 \theta \sin \theta$ 
D)  $2 \sin 8 \theta \cos 6 \theta$ 
E)  $2 \cos 7 \theta \cos \theta$

Question 7

Use the law of Cosines to solve the given triangle. Round your answer to two decimal places. $$a = 11 , b = 15 , c = 21$$

Accepted Answer

A)  $\quad A \approx 43.16 ^ { \circ } , B \approx 43.16 ^ { \circ } , C \approx 93.68 ^ { \circ }$ 
B)  $A \approx 30.11 ^ { \circ } , B \approx 106.73 ^ { \circ } , C \approx 43.16 ^ { \circ }$ 
C)  $A \approx 43.16 ^ { \circ } , B \approx 30.11 ^ { \circ } , C \approx 106.73 ^ { \circ }$ 
D)  $A \approx 30.11 ^ { \circ } , B \approx 43.16 ^ { \circ } , C \approx 106.73 ^ { \circ }$ 
E)  $A \approx 106.73 ^ { \circ } , B \approx 43.16 ^ { \circ } , C \approx 30.11 ^ { \circ }$ 
A)  $\quad A \approx 43.16 ^ { \circ } , B \approx 43.16 ^ { \circ } , C \approx 93.68 ^ { \circ }$ 
B)  $A \approx 30.11 ^ { \circ } , B \approx 106.73 ^ { \circ } , C \approx 43.16 ^ { \circ }$ 
C)  $A \approx 43.16 ^ { \circ } , B \approx 30.11 ^ { \circ } , C \approx 106.73 ^ { \circ }$ 
D)  $A \approx 30.11 ^ { \circ } , B \approx 43.16 ^ { \circ } , C \approx 106.73 ^ { \circ }$ 
E)  $A \approx 106.73 ^ { \circ } , B \approx 43.16 ^ { \circ } , C \approx 30.11 ^ { \circ }$

Question 8

Use inverse functions where needed to find all solutions (if they exist) of the given equation on the interval $[ 0,2 \pi )$ $\cos ^ { 2 } x - 5 \sin x + 5 = 0$

Accepted Answer

A)  solution does not exist 
B)  $x = 0 , \frac { \pi } { 3 } , \frac { 2 \pi } { 3 }$ 
C)  $x = \frac { \pi } { 2 }$ 
D)  $x = \frac { 3 \pi } { 4 } , \frac { 7 \pi } { 4 }$ 
E)  $x = \frac { \pi } { 6 } , \frac { 3 \pi } { 2 }$ 
A)  solution does not exist 
B)  $x = 0 , \frac { \pi } { 3 } , \frac { 2 \pi } { 3 }$ 
C)  $x = \frac { \pi } { 2 }$ 
D)  $x = \frac { 3 \pi } { 4 } , \frac { 7 \pi } { 4 }$ 
E)  $x = \frac { \pi } { 6 } , \frac { 3 \pi } { 2 }$

Question 9

Evaluate the expression. $$\cos 8 \alpha$$

Accepted Answer

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

Question 10

Evaluate the following expression. $( 1 + \sin \alpha ) ( 1 - \sin \alpha )$

Accepted Answer

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

Question 11

Use the trigonometric substitution to select the algebraic expression as a trigonometric function of$\text {  } \theta \text {, where } 0 < \theta < \frac { \pi } { 2 } \text {. }$ $\sqrt { 25 x ^ { 2 } + 36 } , 5 x = 6 \tan \theta$

Accepted Answer

A)  $5 \tan \theta$ 
B)  $25 \sec \theta$ 
C)  $25 \tan \theta$ 
D)  $- 5 \sec \theta$ 
E)  $6 \sec \theta$ 
A)  $5 \tan \theta$ 
B)  $25 \sec \theta$ 
C)  $25 \tan \theta$ 
D)  $- 5 \sec \theta$ 
E)  $6 \sec \theta$

Question 12

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

Accepted Answer

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

Question 13

The table shows the average daily high temperatures in Houston H (in degrees Fahr- enheit) for month t, with $t = 1$ corresponding to January. $\begin{array} { | l | l | } 
\hline \text { Month } , t & \text { Houston, } H \
\hline 1 & 62.4 \
\hline 2 & 66.4 \
\hline 3 & 73.4 \
\hline 4 & 79.4 \
\hline 5 & 85.4 \
\hline 6 & 90.4 \
\hline 7 & 93.4 \
\hline 8 & 93.5 \
\hline 9 & 89.4 \
\hline 10 & 82.4 \
\hline 11 & 72.4 \
\hline 12 & 64.4 \
\hline
\end{array}$
Select the correct scatter plot from the above data.

Accepted Answer

A)   
B) 
  
C) 
  
D) 
  
E) 
  
A)   
B) 
  
C) 
  
D) 
  
E)

Question 14

Determine whether the Law of Sines or the Law of Cosines is needed to solve the tri- angle. Then solve the triangle. $a = 17 , b = 19 , C = 65 ^ { \circ }$

Accepted Answer

A)  Law of Cosine; $A \approx 52.52 ^ { \circ } , B \approx 62.48 ^ { \circ } , c \approx 19.42$ 
B)  Law of Sines; $A \approx 52.52 ^ { \circ } , B \approx 65 ^ { \circ } , c \approx 19$ 
C)  Law of Sines; $A \approx 65 ^ { \circ } , B \approx 52.52 ^ { \circ } , c \approx 19$ 
D)  Law of Sines; No solution 
E)  Law of Cosine; No solution 
A)  Law of Cosine; $A \approx 52.52 ^ { \circ } , B \approx 62.48 ^ { \circ } , c \approx 19.42$ 
B)  Law of Sines; $A \approx 52.52 ^ { \circ } , B \approx 65 ^ { \circ } , c \approx 19$ 
C)  Law of Sines; $A \approx 65 ^ { \circ } , B \approx 52.52 ^ { \circ } , c \approx 19$ 
D)  Law of Sines; No solution 
E)  Law of Cosine; No solution

Question 15

Use the Heron's formula to find the area of the triangle. Round your answer upto two decimal places. $$a = 1 , b = \frac { 1 } { 2 } , c = \frac { 3 } { 5 }$$

Accepted Answer

A)  $0.71$ 
B)  $0.11$ 
C)  $2.35$ 
D)  $0.73$ 
E)  $0.55$ 
A)  $0.71$ 
B)  $0.11$ 
C)  $2.35$ 
D)  $0.73$ 
E)  $0.55$

Question 16

Use a graphing utility to select the correct graph $y _ { 1 } \text { and } y _ { 2 }$ in the same viewing window. Use the graphs to determine whether $y _ { 1 } = y _ { 2 }$ Explain your reasoning. $$y _ { 1 } = \cos ( x + 6 ) , y _ { 2 } = \cos x + \cos 6$$

Accepted Answer

A)

No, $y _ { 1 } \neq y _ { 2 }$ because their graphs are different. 
B) 
 
No, $y _ { 1 } \neq y _ { 2 }$ because their graphs are Same. 
C) 
 
Yes, $y _ { 1 } = y _ { 2 }$ because their graphs are different. 
D) 
 
Yes, $y _ { 1 } = y _ { 2 }$ because their graphs are 
same. 
E)  
 
 No, $y _ { 1 } = y _ { 2 }$ because their graphs are  different. 
A)

No, $y _ { 1 } \neq y _ { 2 }$ because their graphs are different. 
B) 
 
No, $y _ { 1 } \neq y _ { 2 }$ because their graphs are Same. 
C) 
 
Yes, $y _ { 1 } = y _ { 2 }$ because their graphs are different. 
D) 
 
Yes, $y _ { 1 } = y _ { 2 }$ because their graphs are 
same. 
E)  
 
 No, $y _ { 1 } = y _ { 2 }$ because their graphs are  different.

Question 17

Evaluate the following expression. $$\frac { 3 + 3 \sin 4 \theta } { \cos 4 \theta } + \frac { 3 \cos 4 \theta } { 1 + \sin 4 \theta }$$

Accepted Answer

A)  $- 4 \sec 4 \theta$ 
B)  $- 6 \sec 4 \theta$ 
C)  $6 \sec 4 \theta$ 
D)  $4 \sec 4 \theta$ 
E)  $\sec 6 \theta$ 
A)  $- 4 \sec 4 \theta$ 
B)  $- 6 \sec 4 \theta$ 
C)  $6 \sec 4 \theta$ 
D)  $4 \sec 4 \theta$ 
E)  $\sec 6 \theta$

Question 18

Determine whether the Law of Sines or the Law of Cosines is needed to solve the tri- angle. Then solve the triangle. $$a = 17 , b = 19 , C = 60 ^ { \circ }$$

Accepted Answer

A)  Law of Sines; $A \approx 60 ^ { \circ } , B \approx 54.5 ^ { \circ } , c \approx 19$ 
B)  Law of Sines; $A \approx 54.5 ^ { \circ } , B \approx 60 ^ { \circ } , c \approx 19$ 
C)  Law of Cosine; $A \approx 54.5 ^ { \circ } , B \approx 65.5 ^ { \circ } , c \approx 18.08$ 
D)  Law of Sines; No solution 
E)  Law of Cosine; No solution 
A)  Law of Sines; $A \approx 60 ^ { \circ } , B \approx 54.5 ^ { \circ } , c \approx 19$ 
B)  Law of Sines; $A \approx 54.5 ^ { \circ } , B \approx 60 ^ { \circ } , c \approx 19$ 
C)  Law of Cosine; $A \approx 54.5 ^ { \circ } , B \approx 65.5 ^ { \circ } , c \approx 18.08$ 
D)  Law of Sines; No solution 
E)  Law of Cosine; No solution

Question 19

Given $a = 8 , b = 7 , \text { and } c = 6$ , use the Law of Cosines to solve the triangle for the value of C. Round answer to two decimal places.

Accepted Answer

A)  $46.57 ^ { \circ }$ 
B)  $75.52 ^ { \circ }$ 
C)  $57.91 ^ { \circ }$ 
D)  $80.44 ^ { \circ }$ 
E)  $60.33 ^ { \circ }$ 
A)  $46.57 ^ { \circ }$ 
B)  $75.52 ^ { \circ }$ 
C)  $57.91 ^ { \circ }$ 
D)  $80.44 ^ { \circ }$ 
E)  $60.33 ^ { \circ }$

Question 20

Determine the angle $\theta$ in the design of the streetlight shown in the following figure. $a = 4 , b = 5 \frac { 1 } { 2 } , c = 3$

Accepted Answer

A)  $97.6 ^ { \circ }$ 
B)  $112.6 ^ { \circ }$ 
C)  $92.6 ^ { \circ }$ 
D)  $102.6 ^ { \circ }$ 
E)  $107.6 ^ { \circ }$ 
A)  $97.6 ^ { \circ }$ 
B)  $112.6 ^ { \circ }$ 
C)  $92.6 ^ { \circ }$ 
D)  $102.6 ^ { \circ }$ 
E)  $107.6 ^ { \circ }$

Evaluate the following expression. $\frac { \tan ^ { 2 } \theta } { \sec \theta }$

Find the expression as the sine or cosine of an angle. $\cos 6 x \cos 3 y + \sin 6 x \sin 3 y$

Use the given values to evaluate (if possible) three trigonometric functions $\operatorname { cosec } \theta$ $\tan \theta , \cos \theta$ $\sin \theta = - 3 , \quad \cot \theta = 0$

Use the trigonometric substitution to select the algebraic expression as a trigonomet- $\text { ric function of } \theta \text {, where } 0 < \theta < \frac { \pi } { 2 } \text {. }$ $\sqrt { 16 x ^ { 2 } + 25 } , 4 x = 5 \tan \theta$

Use the figure to find the exact value of the trigonometric function.

Use the sum-to-product formulas to select the sum or difference as a product. $\sin 8 \theta + \sin 6 \theta$

Use the law of Cosines to solve the given triangle. Round your answer to two decimal places. $a = 11 , b = 15 , c = 21$

Use inverse functions where needed to find all solutions (if they exist) of the given equation on the interval $[ 0,2 \pi )$ $\cos ^ { 2 } x - 5 \sin x + 5 = 0$

Evaluate the expression. $\cos 8 \alpha$

Evaluate the following expression. $( 1 + \sin \alpha ) ( 1 - \sin \alpha )$

Use the trigonometric substitution to select the algebraic expression as a trigonometric function of $\text { } \theta \text {, where } 0 < \theta < \frac { \pi } { 2 } \text {. }$ $\sqrt { 25 x ^ { 2 } + 36 } , 5 x = 6 \tan \theta$

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

Determine whether the Law of Sines or the Law of Cosines is needed to solve the tri- angle. Then solve the triangle. $a = 17 , b = 19 , C = 65 ^ { \circ }$

Use the Heron's formula to find the area of the triangle. Round your answer upto two decimal places. $a = 1 , b = \frac { 1 } { 2 } , c = \frac { 3 } { 5 }$

Use a graphing utility to select the correct graph $y _ { 1 } \text { and } y _ { 2 }$ in the same viewing window. Use the graphs to determine whether $y _ { 1 } = y _ { 2 }$ Explain your reasoning. $y _ { 1 } = \cos ( x + 6 ) , y _ { 2 } = \cos x + \cos 6$

Evaluate the following expression. $\frac { 3 + 3 \sin 4 \theta } { \cos 4 \theta } + \frac { 3 \cos 4 \theta } { 1 + \sin 4 \theta }$

Determine whether the Law of Sines or the Law of Cosines is needed to solve the tri- angle. Then solve the triangle. $a = 17 , b = 19 , C = 60 ^ { \circ }$

Given $a = 8 , b = 7 , \text { and } c = 6$ , use the Law of Cosines to solve the triangle for the value of C. Round answer to two decimal places. $Given a = 8 , b = 7 , \text { and } c = 6 , use the Law of Cosines to solve the triangle for the value of C. Round answer to two decimal places.$

Determine the angle $\theta$ in the design of the streetlight shown in the following figure. $a = 4 , b = 5 \frac { 1 } { 2 } , c = 3$ $Determine the angle \theta in the design of the streetlight shown in the following figure. a = 4 , b = 5 \frac { 1 } { 2 } , c = 3$

Functions and Their Graphs

Polynomial and Rational Functions

Exponential and Logarithmic Functions

Trigonometry

Topics in Analytic Geometry

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Exam 5: Analytic Trigonometry

Evaluate the following expression. tan⁡2θsec⁡θ\frac { \tan ^ { 2 } \theta } { \sec \theta }secθtan2θ​

Find the expression as the sine or cosine of an angle. cos⁡6xcos⁡3y+sin⁡6xsin⁡3y\cos 6 x \cos 3 y + \sin 6 x \sin 3 ycos6xcos3y+sin6xsin3y

Use the given values to evaluate (if possible) three trigonometric functions cosec⁡θ\operatorname { cosec } \thetacosecθ tan⁡θ,cos⁡θ\tan \theta , \cos \thetatanθ,cosθ sin⁡θ=−3,cot⁡θ=0\sin \theta = - 3 , \quad \cot \theta = 0sinθ=−3,cotθ=0

Use the figure to find the exact value of the trigonometric function.

Use the sum-to-product formulas to select the sum or difference as a product. sin⁡8θ+sin⁡6θ\sin 8 \theta + \sin 6 \thetasin8θ+sin6θ

Use the law of Cosines to solve the given triangle. Round your answer to two decimal places. a=11,b=15,c=21a = 11 , b = 15 , c = 21a=11,b=15,c=21

Use inverse functions where needed to find all solutions (if they exist) of the given equation on the interval [0,2π)[ 0,2 \pi )[0,2π) cos⁡2x−5sin⁡x+5=0\cos ^ { 2 } x - 5 \sin x + 5 = 0cos2x−5sinx+5=0

Evaluate the expression. cos⁡8α\cos 8 \alphacos8α

Evaluate the following expression. (1+sin⁡α)(1−sin⁡α)( 1 + \sin \alpha ) ( 1 - \sin \alpha )(1+sinα)(1−sinα)

Use the half-angle formulas to determine the exact value of the given trigonometric expression. cot⁡3π8\cot \frac { 3 \pi } { 8 }cot83π​

Determine whether the Law of Sines or the Law of Cosines is needed to solve the tri- angle. Then solve the triangle. a=17,b=19,C=65∘a = 17 , b = 19 , C = 65 ^ { \circ }a=17,b=19,C=65∘

Use the Heron's formula to find the area of the triangle. Round your answer upto two decimal places. a=1,b=12,c=35a = 1 , b = \frac { 1 } { 2 } , c = \frac { 3 } { 5 }a=1,b=21​,c=53​

Evaluate the following expression. 3+3sin⁡4θcos⁡4θ+3cos⁡4θ1+sin⁡4θ\frac { 3 + 3 \sin 4 \theta } { \cos 4 \theta } + \frac { 3 \cos 4 \theta } { 1 + \sin 4 \theta }cos4θ3+3sin4θ​+1+sin4θ3cos4θ​

Determine whether the Law of Sines or the Law of Cosines is needed to solve the tri- angle. Then solve the triangle. a=17,b=19,C=60∘a = 17 , b = 19 , C = 60 ^ { \circ }a=17,b=19,C=60∘

Given a=8,b=7, and c=6a = 8 , b = 7 , \text { and } c = 6a=8,b=7, and c=6 , use the Law of Cosines to solve the triangle for the value of C. Round answer to two decimal places.

Determine the angle θ\thetaθ in the design of the streetlight shown in the following figure. a=4,b=512,c=3a = 4 , b = 5 \frac { 1 } { 2 } , c = 3a=4,b=521​,c=3

Functions and Their Graphs

Polynomial and Rational Functions

Exponential and Logarithmic Functions

Trigonometry

Topics in Analytic Geometry

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Evaluate the following expression. $\frac { \tan ^ { 2 } \theta } { \sec \theta }$

Find the expression as the sine or cosine of an angle. $\cos 6 x \cos 3 y + \sin 6 x \sin 3 y$

Use the given values to evaluate (if possible) three trigonometric functions $\operatorname { cosec } \theta$ $\tan \theta , \cos \theta$ $\sin \theta = - 3 , \quad \cot \theta = 0$

Use the sum-to-product formulas to select the sum or difference as a product. $\sin 8 \theta + \sin 6 \theta$

Use the law of Cosines to solve the given triangle. Round your answer to two decimal places. $a = 11 , b = 15 , c = 21$

Use inverse functions where needed to find all solutions (if they exist) of the given equation on the interval $[ 0,2 \pi )$ $\cos ^ { 2 } x - 5 \sin x + 5 = 0$

Evaluate the expression. $\cos 8 \alpha$

Evaluate the following expression. $( 1 + \sin \alpha ) ( 1 - \sin \alpha )$

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

Determine whether the Law of Sines or the Law of Cosines is needed to solve the tri- angle. Then solve the triangle. $a = 17 , b = 19 , C = 65 ^ { \circ }$

Use the Heron's formula to find the area of the triangle. Round your answer upto two decimal places. $a = 1 , b = \frac { 1 } { 2 } , c = \frac { 3 } { 5 }$

Evaluate the following expression. $\frac { 3 + 3 \sin 4 \theta } { \cos 4 \theta } + \frac { 3 \cos 4 \theta } { 1 + \sin 4 \theta }$

Determine whether the Law of Sines or the Law of Cosines is needed to solve the tri- angle. Then solve the triangle. $a = 17 , b = 19 , C = 60 ^ { \circ }$

Given $a = 8 , b = 7 , \text { and } c = 6$ , use the Law of Cosines to solve the triangle for the value of C. Round answer to two decimal places. $Given a = 8 , b = 7 , \text { and } c = 6 , use the Law of Cosines to solve the triangle for the value of C. Round answer to two decimal places.$

Determine the angle $\theta$ in the design of the streetlight shown in the following figure. $a = 4 , b = 5 \frac { 1 } { 2 } , c = 3$ $Determine the angle \theta in the design of the streetlight shown in the following figure. a = 4 , b = 5 \frac { 1 } { 2 } , c = 3$