Deck 31: Using Fundamental Identities

Question 1

Use the given values to evaluate (if possible)three trigonometric functions cos x,sin x,tan x.​ $\sec x = 4 , \sin x > 0$ ​ ​

A)​ $\begin{array} { l } \cos x = \frac { 1 } { 4 } \\\\\sin x = - \frac { \sqrt { 15 } } { 4 } \\\\\tan x = \sqrt { 15 }\end{array}$ ​
B)​ $\begin{array} { l } \cos x = \frac { 1 } { 4 } \\\\\sin x = \frac { \sqrt { 15 } } { 4 } \\\\\tan x = \sqrt { 15 }\end{array}$ ​
C)​ $\begin{array} { l } \cos x = \frac { 1 } { 4 } \\\sin x = - \frac { \sqrt { 15 } } { 4 } \\\tan x = - \sqrt { 15 }\end{array}$ ​
D)​ $\begin{array} { l } \cos x = - \frac { 1 } { 4 } \\\\\sin x = \frac { \sqrt { 15 } } { 4 } \\\\\tan x = \sqrt { 15 }\end{array}$ ​
E)​ $\begin{array} { l } \cos x = - \frac { 1 } { 4 } \\\sin x = - \frac { \sqrt { 15 } } { 4 } \\\tan x = - \sqrt { 15 }\end{array}$

Answer 1

​ $\begin{array} { l } \cos x = \frac { 1 } { 4 } \\\\\sin x = \frac { \sqrt { 15 } } { 4 } \\\\\tan x = \sqrt { 15 }\end{array}$ ​

Question 2

Use the trigonometric substitution to rewrite the algebraic expression as a trigonometric function of θ,where $0 < x < \frac { \pi } { 2 }$ .​ $\sqrt { 16 - x ^ { 2 } } , x = 4 \cos \theta$ ​

A)16 cos θ
B)16 sin θ
C)4 sin θ
D)- 4 sin θ
E)4 cos θ

Answer 2

4 sin θ

Question 3

Perform the multiplication and use the fundamental identities to simplify.​ $( \sin x - \cos x ) ^ { 2 }$ ​

A) $1 + \sin x + \cos x$ ​
B) $2 - 2 \sin x \cos x$ ​
C) $1 - 2 \sin x \cos x$ ​
D) $1 + 2 \sin x + \cos x$ ​
E) $$1 - \sin x \cos x$$ ​

Answer 3

$1 - 2 \sin x \cos x$ ​

Question 4

Use the given values to evaluate (if possible)three trigonometric functions cos x,csc x,tan x.​ $\sin x = \frac { 1 } { 5 }$ , $\cos x > 0$ ​

A)​ $\begin{array} { l } \cos x = - \frac { 2 \sqrt { 6 } } { 5 } \\\csc x = 5 \\\tan x = \frac { \sqrt { 6 } } { 12 }\end{array}$ ​
B)​ $\begin{array} { l } \cos x = \frac { 2 \sqrt { 6 } } { 5 } \\\csc x = 5 \\\tan x = - \frac { \sqrt { 6 } } { 12 }\end{array}$ ​
C)​ $\begin{array} { l } \cos x = - \frac { 2 \sqrt { 6 } } { 5 } \\\csc x = - 5 \\\tan x = - \frac { \sqrt { 6 } } { 12 }\end{array}$ ​
D)​ $\begin{array} { l } \cos x = \frac { 2 \sqrt { 6 } } { 5 } \\\csc x = 5 \\\tan x = \frac { \sqrt { 6 } } { 12 }\end{array}$ ​
E)​ $\begin{array} { l } \cos x = \frac { 2 \sqrt { 6 } } { 5 } \\\csc x = \frac { 1 } { 5 } \\\tan x = \frac { \sqrt { 6 } } { 12 }\end{array}$ ​

Question 5

Use the trigonometric substitution to rewrite the algebraic expression as a trigonometric function of θ,where $0 < \theta < \frac { \pi } { 2 }$ . $\sqrt { 36 - 9 x ^ { 2 } } , x = 2 \cos \theta$ ​

A)2 cos θ
B)36 sin θ
C)- 2 sin θ
D)36 cos θ
E)6 sin θ

Question 6

Use the given values to evaluate (if possible)three trigonometric functions csc x,tan x,cot x.​ $\cos \left( \frac { \pi } { 2 } - x \right) = \frac { 8 } { 17 } , \cos x = \frac { 15 } { 17 }$ ​

A)​ $\begin{array} { l } \csc x = - \frac { 8 } { 5 } \\\\\tan x = - \frac { 8 } { 15 } \\\\\cot x = - \frac { 15 } { 8 }\end{array}$ ​
B)​ $\begin{array} { l } \csc x = - \frac { 15 } { 8 } \\\\\tan x = \frac { 8 } { 15 } \\\\\cot x = \frac { 15 } { 8 }\end{array}$ ​
C)​ $\begin{array} { l } \csc x = \frac { 15 } { 8 } \\\tan x = - \frac { 8 } { 15 } \\\cot x = \frac { 15 } { 8 }\end{array}$ ​
D)​ $\begin{array} { l } \csc x = \frac { 17 } { 8 } \\\\\tan x = \frac { 8 } { 15 } \\\\\cot x = \frac { 15 } { 8 }\end{array}$ ​
E)​ $\begin{array} { l } \csc x = \frac { 5 } { 8 } \\\\\tan x = \frac { 8 } { 15 } \\\\\cot x = - \frac { 15 } { 8 }\end{array}$ ​

Question 7

​Use the given values to evaluate (if possible)three trigonometric functions $\csc \theta , \tan \theta , \cos \theta$ . ​​ $\sin \theta = - 1 , \cot \theta = 0$ ​

A)​ $$\begin{array} { l } \csc \theta~ is~ undefined.\\\tan \theta = - 1 \\\cos \theta = 0\end{array}$$ ​
B)​ $$\csc \theta = - 1\\\tan \theta~ is~ undefined.\\\cos \theta = 1$$ ​
C) $$\csc \theta = - 1\\\tan \theta ~is~ undefined.\\\cos \theta = 0$$ ​
D)​ $$\csc \theta = - 1\\\tan \theta ~is~ undefined.\\\cos \theta = - 1$$ ​
E)​ $$\csc \theta = 1\\\tan \theta~ is~ undefined.\\\cos \theta = - 1$$ ​

Question 8

Use the trigonometric substitution to rewrite the algebraic expression as a trigonometric function of θ,where $0 < \theta < \frac { \pi } { 2 }$ . $\sqrt { 16 - x ^ { 2 } } , x = 4 \sin \theta$ ​

A)4 sin θ
B)- 4 cos θ
C)4 cos θ
D)16 sin θ
E)16 cos θ

Question 9

Use the given values to evaluate (if possible)three trigonometric functions cos x,csc x,tan x.​ $\sec x = \sqrt { 3 } , \sin x = - \frac { \sqrt { 6 } } { 3 }$

A)​ $\begin{array} { l } \cos x = \frac { \sqrt { 3 } } { 3 } \\\csc x = - \frac { \sqrt { 6 } } { 2 }\end{array}$ tan x = - $\sqrt { 2 }$
​
B)​ $\begin{array} { l } \cos x = - \frac { \sqrt { 3 } } { 3 } \\\csc x = - \frac { \sqrt { 6 } } { 2 }\end{array}$ tan x = - ​ $\sqrt { 2 }$
C)​ $\begin{array} { l } \cos x = \frac { \sqrt { 6 } } { 6 } \\\csc x = \frac { \sqrt { 6 } } { 2 }\end{array}$ ​tan x = $\sqrt { 2 }$
​
D)​ $\begin{array} { l } \cos x = \frac { \sqrt { 3 } } { 3 } \\\csc x = - \frac { \sqrt { 6 } } { 2 }\end{array}$ ​tan x = $\sqrt { 2 }$
​
E)​ $\begin{array} { l } \cos x = \frac { \sqrt { 6 } } { 6 } \\\csc x = \frac { \sqrt { 6 } } { 2 }\end{array}$ ​tan x = - $\sqrt { 2 }$
​

Question 10

Use the given values to evaluate (if possible)three trigonometric functions $\cot \theta , \sec \theta , \cos \theta$ .​ $\tan \theta = 4 , \cos \theta < 0$ ​

A)​ $\begin{aligned}\cot \theta & = \frac { 1 } { 4 } \\\sec \theta & = - \sqrt { 17 } \\\cos \theta & = \frac { \sqrt { 17 } } { 17 }\end{aligned}$ ​
B)​ $\begin{array} { l } \cot \theta = 4 \\\sec \theta = \sqrt { 17 } \\\cos \theta = \frac { \sqrt { 17 } } { 17 }\end{array}$ ​
C)​ $$\begin{array} { l } \cot \theta = \frac { 1 } { 4 } \\\sec \theta = - \sqrt { 17 } \\\cos \theta = - \frac { \sqrt { 17 } } { 17 }\end{array}$$ ​
D)​ $$\begin{array} { l } \cot \theta = - \frac { 1 } { 4 } \\\sec \theta = - \sqrt { 17 } \\\cos \theta = - \frac { \sqrt { 17 } } { 17 }\end{array}$$ ​
E)​ $\begin{array} { l } \cot \theta = \frac { 1 } { 4 } \\\sec \theta = \sqrt { 17 } \\\cos \theta = - \frac { \sqrt { 17 } } { 17 }\end{array}$ ​

Question 11

By using a graphing utility to complete the following table.Round your answer to four decimal places. $$\begin{array} { | l | l | l | l | l | l | l | l | } \hline x & 0.2 & 0.4 & 0.6 & 0.8 & 1 & 1.2 & 1.4 \\\hline y _ { 1 } & \cdots & \cdots & \cdots & \cdots & \cdots & \cdots & \cdots \\\hline y _ { 2 } & \cdots & \cdots & \cdots & \ldots & \ldots & \cdots & \cdots \\\hline\end{array}$$ $y _ { 1 } = \cos \left( \frac { \pi } { 2 } - x \right) , y _ { 2 } = \sin x$

A) $$\begin{array} { | l | l | l | l | l | l | l | l | } \hline x & 0.2 & 0.4 & 0.6 & 0.8 & 1 & 1.2 & 1.4 \\\hline y _ { 1 } & - 0.1987 & 0.3894 & 0.5646 & 0.7174 & 0.8415 & 0.9320 & 0.9854 \\\hline y _ { 2 } & - 0.1987 & 0.3894 & 0.5646 & 0.7174 & 0.8415 & 0.9320 & 0.9854 \\\hline\end{array}$$
B) $$\begin{array} { | l | l | l | l | l | l | l | l | } \hline x & 0.2 & 0.4 & 0.6 & 0.8 & 1 & 1.2 & 1.4 \\\hline y _ { 1 } & 0.1987 & - 0.3894 & 0.5646 & 0.7174 & 0.8415 & 0.9320 & 0.9854 \\\hline y _ { 2 } & 0.1987 & - 0.3894 & 0.5646 & 0.7174 & 0.8415 & 0.9320 & 0.9854 \\\hline\end{array}$$
C) $$\begin{array} { | l | l | l | l | l | l | l | l | } \hline x & 0.2 & 0.4 & 0.6 & 0.8 & 1 & 1.2 & 1.4 \\\hline y _ { 1 } & 0.1987 & 0.3894 & - 0.5646 & 0.7174 & 0.8415 & 0.9320 & 0.9854 \\\hline y _ { 2 } & 0.1987 & 0.3894 & - 0.5646 & 0.7174 & 0.8415 & 0.9320 & 0.9854 \\\hline\end{array}$$
D) $$\begin{array} { | l | l | l | l | l | l | l | l | } \hline x & 0.2 & 0.4 & 0.6 & 0.8 & 1 & 1.2 & 1.4 \\\hline y _ { 1 } & 0.1987 & 0.3894 & 0.5646 & - 0.7174 & 0.8415 & 0.9320 & 0.9854 \\\hline y _ { 2 } & 0.1987 & 0.3894 & 0.5646 & - 0.7174 & 0.8415 & 0.9320 & 0.9854 \\\hline\end{array}$$
E) $$\begin{array} { | l | l | l | l | l | l | l | l | } \hline x & 0.2 & 0.4 & 0.6 & 0.8 & 0.8415 & 0.9320 & 0.9854 \\\hline y _ { 1 } & 0.1987 & 0.3894 & 0.5646 & 0.7174 & 0.8415 & 0.9320 & 0.9854 \\\hline y _ { 2 } & 0.1987 & 0.3894 & 0.5646 & 0.7174 & 0.8415 & 0.9320 & 0.9854 \\\hline\end{array}$$

Question 12

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

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

Question 13

By using a graphing utility to complete the following table.Round your answer to four decimal places. $$\begin{array} { | l | l | l | l | l | l | l | l | } \hline x & 0.2 & 0.4 & 0.6 & 0.8 & 1 & 1.2 & 1.4 \\\hline y _ { 1 } & \cdots & \cdots & \ldots & \cdots & \cdots & \cdots & \cdots \\\hline y _ { 2 } & \cdots & \cdots & \ldots & \ldots & \cdots & \cdots & \cdots \\\hline\end{array}$$ $y _ { 1 } = \sec x - \cos x , y _ { 2 } = \sin x \tan x$

A) $$\begin{array} { | l | l | l | l | l | l | l | l | } \hline x & 0.2 & 0.4 & 0.6 & 0.8 & 1 & 1.2 & 1.4 \\\hline y _ { 1 } & 0.0403 & 0.1646 & 0.3863 & 0.7386 & 1.3105 & 2.3973 & 5.7135 \\\hline y _ { 2 } & 0.0403 & 0.1646 & 0.3863 & 0.7386 & 1.3105 & 2.3973 & 5.7135 \\\hline\end{array}$$
B) $$\begin{array} { | l | l | l | l | l | l | l | l | } \hline x & 0.2 & 0.4 & 0.6 & 0.8 & 1 & 1.2 & 1.4 \\\hline y _ { 1 } & 0.0403 & 0.1646 & - 0.3863 & 0.7386 & 1.3105 & 2.3973 & 5.7135 \\\hline y _ { 2 } & 0.0403 & 0.1646 & - 0.3863 & 0.7386 & 1.3105 & 2.3973 & 5.7135 \\\hline\end{array}$$
C) $$\begin{array} { | l | l | l | l | l | l | l | l | } \hline x & 0.2 & 0.4 & 0.6 & 0.8 & 1 & 1.2 & 1.4 \\\hline y _ { 1 } & 0.0403 & - 0.1646 & 0.3863 & 0.7386 & 1.3105 & 2.3973 & 5.7135 \\\hline y _ { 2 } & 0.0403 & - 0.1646 & 0.3863 & 0.7386 & 1.3105 & 2.3973 & 5.7135 \\\hline\end{array}$$
D) $$\begin{array} { | l | l | l | l | l | l | l | l | } \hline x & 0.2 & 0.4 & 0.6 & 0.8 & 1 & 1.2 & 1.4 \\\hline y _ { 1 } & - 0.0403 & 0.1646 & 0.3863 & 0.7386 & 1.3105 & 2.3973 & 5.7135 \\\hline y _ { 2 } & - 0.0403 & 0.1646 & 0.3863 & 0.7386 & 1.3105 & 2.3973 & 5.7135 \\\hline\end{array}$$
E) $$\begin{array} { | l | l | l | l | l | l | l | l | } \hline x & 0.2 & 0.4 & 0.6 & 0.8 & 1 & 1.2 & 1.4 \\\hline y _ { 1 } & 0.0403 & 0.1646 & 0.3863 & - 0.7386 & 1.3105 & 2.3973 & 5.7135 \\\hline y _ { 2 } & 0.0403 & 0.1646 & 0.3863 & - 0.7386 & 1.3105 & 2.3973 & 5.7135 \\\hline\end{array}$$

Question 14

Use the given values to evaluate (if possible)three trigonometric functions $\cos \phi , \sin \phi , \tan \phi$ .​ $\sec \phi = \frac { 3 } { 4 } , \csc \phi = \frac { - 3 \sqrt { 5 } } { 5 }$ ​

A)​ $\begin{array} { l } \cos \phi = \frac { 4 } { 3 } \\\sin \phi = - \frac { \sqrt { 5 } } { 3 } \\\tan \phi = \frac { \sqrt { 5 } } { 4 }\end{array}$ ​
B)​ $\begin{array} { l } \cos \phi = \frac { 4 } { 3 } \\\sin \phi = \frac { \sqrt { 5 } } { 3 } \\\tan \phi = - \frac { \sqrt { 5 } } { 4 }\end{array}$ ​
C)​ $\begin{array} { l } \cos \phi = \frac { 4 } { 3 } \\\sin \phi = - \frac { \sqrt { 5 } } { 3 } \\\tan \phi = - \frac { \sqrt { 5 } } { 4 }\end{array}$ ​
D)​ $\begin{array} { l } \cos \phi = \frac { 4 } { 3 } \\\sin \phi = \frac { \sqrt { 5 } } { 3 } \\\tan \phi = \frac { \sqrt { 5 } } { 4 }\end{array}$ ​
E)​ $\begin{array} { l } \cos \phi = - \frac { 4 } { 3 } \\\sin \phi = - \frac { \sqrt { 5 } } { 3 } \\\tan \phi = - \frac { \sqrt { 5 } } { 4 }\end{array}$

Question 15

Use the given values to evaluate (if possible)two trigonometric functions $\tan \varphi$ and $\csc \psi$ .​ $\cot \varphi = - 5 , \sin \psi = \frac { \sqrt { 10 } } { 10 }$ ​

A)​ $\begin{array} { l } \tan \varphi = - \frac { 1 } { 5 } \\\csc \psi = \sqrt { 10 }\end{array}$ ​
B)​ $\begin{array} { l } \tan \varphi = \frac { 1 } { 5 } \\\csc \psi = \sqrt { 10 }\end{array}$ ​
C)​ $\begin{array} { l } \tan \varphi = \frac { 1 } { 5 } \\\csc \psi = - \sqrt { 10 }\end{array}$ ​
D)​ $\begin{array} { l } \tan \varphi = - 5 \\\csc \psi = \sqrt { 10 }\end{array}$ ​
E)​ $\begin{array} { l } \tan \varphi = - \frac { 1 } { 5 } \\\csc \psi = - \sqrt { 10 }\end{array}$ ​

Question 16

By using a graphing utility to complete the following table.Round your answer to two decimal places. $$\begin{array} { | l | l | l | l | l | l | l | l | } \hline x & 0.2 & 0.4 & 0.6 & 0.8 & 1 & 1.2 & 1.4 \\\hline y _ { 1 } & - & - & - & - & - & - & - \\\hline y _ { 2 } & - & - & - & - & - & - & - \\\hline\end{array}$$ $y _ { 1 } = \sec ^ { 4 } x - \sec ^ { 2 } x , y _ { 2 } = \tan ^ { 2 } x \sec ^ { 2 } x$

A) $$\begin{array} { | l | l | l | l | l | l | l | l | } \hline x & 0.2 & 0.4 & 0.6 & 0.8 & 1 & 1.2 & 1.4 \\\hline y _ { 1 } & - 0.04 & 0.21 & 0.69 & 2.18 & 8.31 & 50.39 & 1,163.61 \\\hline y _ { 2 } & - 0.04 & 0.21 & 0.69 & 2.18 & 8.31 & 50.39 & 1,163.61 \\\hline\end{array}$$
B) $$\begin{array} { | l | l | l | l | l | l | l | l | } \hline x & 0.2 & 0.4 & 0.6 & 0.8 & 1 & 1.2 & 1.4 \\\hline y _ { 1 } & 0.04 & 0.21 & 0.69 & 2.18 & 8.31 & 50.39 & 1,163.61 \\\hline y _ { 2 } & 0.04 & 0.21 & 0.69 & 2.18 & 8.31 & 50.39 & 1,163.61 \\\hline\end{array}$$
C) $$\begin{array} { | l | l | l | l | l | l | l | l | } \hline x & 0.2 & 0.4 & 0.6 & 0.8 & 1 & 1.2 & 1.4 \\\hline y _ { 1 } & 0.04 & 0.21 & 0.69 & - 2.18 & 8.31 & 50.39 & 1,163.61 \\\hline y _ { 2 } & 0.04 & 0.21 & 0.69 & - 2.18 & 8.31 & 50.39 & 1,163.61 \\\hline\end{array}$$
D) $$\begin{array} { | l | l | l | l | l | l | l | l | } \hline x & 0.2 & 0.4 & 0.6 & 0.8 & 1 & 1.2 & 1.4 \\\hline y _ { 1 } & 0.04 & 0.21 & - 0.69 & 2.18 & 8.31 & 50.39 & 1,163.61 \\\hline y _ { 2 } & 0.04 & 0.21 & - 0.69 & 2.18 & 8.31 & 50.39 & 1,163.61 \\\hline\end{array}$$
E) $$\begin{array} { | l | l | l | l | l | l | l | l | } \hline x & 0.2 & 0.4 & 0.6 & 0.8 & 1 & 1.2 & 1.4 \\\hline y _ { 1 } & 0.04 & 0.21 & 0.69 & 2.18 & 8.31 & 50.39 & - 1,163.61 \\\hline y _ { 2 } & 0.04 & 0.21 & 0.69 & 2.18 & 8.31 & 50.39 & - 1,163.61 \\\hline\end{array}$$

Question 17

Use the fundamental identities to simplify the expression.​ $\frac { \sin \theta } { \tan \theta }$ ​

A)cot θ
B)​cot θ
C)​sec θ
D)​cos θ
E)​csc θ

Question 18

By using a graphing utility to complete the following table.Round your answer to four decimal places. $$\begin{array} { | l | l | l | l | l | l | l | l | } \hline x & 0.8 & 1.0 & 1.2 & 1.4 & 1.6 & 1.8 & 2.0 \\\hline y _ { 1 } & \cdots & \cdots & \cdots & \cdots & \cdots & \cdots & \cdots \\\hline y _ { 2 } & \ldots & \ldots & \ldots & \ldots & \ldots & \ldots & \cdots \\\hline\end{array}$$ $y _ { 1 } = \frac { \cos x } { 1 - \sin x } , y _ { 2 } = \frac { 1 + \sin x } { \cos x }$

A) $$\begin{array} { | l | l | l | l | l | l | l | l | } \hline x & 0.8 & 1.0 & 1.2 & 1.4 & 1.6 & 1.8 & 2.0 \\\hline y _ { 1 } & 2.4650 & 3.4082 & 5.3319 & - 11.6814 & - 68.4797 & - 8.6876 & - 4.5880 \\\hline y _ { 2 } & 2.4650 & 3.4082 & 5.3319 & - 11.6814 & - 68.4797 & - 8.6876 & - 4.5880 \\\hline\end{array}$$
B) $$\begin{array} { | l | l | l | l | l | l | l | l | } \hline x & 0.8 & 1.0 & 1.2 & 1.4 & 1.6 & 1.8 & 2.0 \\\hline y _ { 1 } & - 2.4650 & 3.4082 & 5.3319 & 11.6814 & - 68.4797 & - 8.6876 & - 4.5880 \\\hline y _ { 2 } & - 2.4650 & 3.4082 & 5.3319 & 11.6814 & - 68.4797 & - 8.6876 & - 4.5880 \\\hline\end{array}$$
C) $$\begin{array} { | l | l | l | l | l | l | l | l | } \hline x & 0.8 & 1.0 & 1.2 & 1.4 & 1.6 & 1.8 & 2.0 \\\hline y & 2.4650 & 3.4082 & 5.3319 & 11.6814 & - - 68.4797 & - 8.6876 & - 4.5880 \\\hline y & 2.4650 & 3.4082 & 5.3319 & 11.6814 & - - 68.4797 & - 8.6876 & - 4.5880 \\\hline\end{array}$$
D) $$\begin{array} { | l | l | l | l | l | l | l | l | } \hline x & 0.8 & 1.0 & 1.2 & 1.4 & 1.6 & 1.8 & 2.0 \\\hline y _ { 1 } & 2.4650 & 3.4082 & - 5.3319 & 11.6814 & - 68.4797 & - 8.6876 & - 4.5880 \\\hline y _ { 2 } & 2.4650 & 3.4082 & - 5.3319 & 11.6814 & - 68.4797 & - 8.6876 & - 4.5880 \\\hline\end{array}$$
E) $$\begin{array} { | l | l | l | l | l | l | l | l | } \hline x & 0.8 & 1.0 & 1.2 & 1.4 & 1.6 & 1.8 & 2.0 \\\hline y _ { 1 } & 2.4650 & 3.4082 & 5.3319 & 11.6814 & - 68.4797 & - 8.6876 & - 4.5880 \\\hline y _ { 2 } & 2.4650 & 3.4082 & 5.3319 & 11.6814 & - 68.4797 & - 8.6876 & - 4.5880 \\\hline\end{array}$$

Question 19

Use the fundamental identities to simplify the expression. ​
Cot θ sec θ
​

A)tan θ
B)​sec θ cot θ
C)csc θ
D)cot θ
E)​cos θ

Question 20

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

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 21

Use the trigonometric substitution to rewrite the algebraic expression as a trigonometric function of θ,where $- \frac { \pi } { 2 } < \theta < \frac { \pi } { 2 }$ .Then find sin θ and cos θ.​ $4 = \sqrt { 64 - x ^ { 2 } } , x = 8 \sin \theta$

A)​ $64 \sin \theta = 4 ; \sin \theta = \pm \frac { \sqrt { 3 } } { 2 } ; \cos \theta = \frac { 1 } { 2 }$
B)​ $64 \cos \theta = 4 ; \sin \theta = \pm \frac { \sqrt { 3 } } { 2 } ; \cos \theta = \frac { 1 } { 2 }$
C)​ $8 \sin \theta = 4 ; \sin \theta = \pm \frac { \sqrt { 3 } } { 2 } ; \cos \theta = \frac { 1 } { 2 }$
D)​ $8 \cos \theta = 4 ; \sin \theta = \pm \frac { \sqrt { 3 } } { 2 } ; \cos \theta = \frac { 1 } { 2 }$
E)​ $8 \cos \theta = 4 ; \sin \theta = \frac { 1 } { 2 } ; \cos \theta = \pm \frac { \sqrt { 3 } } { 2 }$

Question 22

Use fundamental identities to simplify the expression below and then determine which of the following is not equivalent. $\cos \alpha ( \sec \alpha - \cos \alpha )$

A)​ $\frac { \csc ^ { 2 } \alpha - \sec ^ { 2 } \alpha + \tan ^ { 2 } \alpha } { \csc ^ { 2 } \alpha }$
B)​ $1 - \sin ^ { 2 } \alpha$
C)​ $\frac { \csc ^ { 2 } \alpha - 1 } { \csc ^ { 2 } \alpha }$
D)​ $1 - \cos ^ { 2 } \alpha$
E)​ $1 - \cot ^ { 2 } \alpha$

Question 23

Find the rate of change of the function $$f ( x ) = - x + \cos x$$ . ​

A)​ $- \sin x + 1$
B)​ $\sin x - 1$
C)​ $- \sin x - 1$
D)​ $- \sin x - x$
E)​ $\sin x + 1$

Question 24

Find the rate of change of the function $f ( x ) = - \csc x - \cos x$ . ​

A)​ $\sin x + \sec x$
B)​ $\sin x + 1$
C)​ $\sin x - \tan x \sec x$
D)​ $\sin x \sin x$
E)​ $\csc x \cot x - \sin x$

Question 25

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

A)a tan θ
B)- a cot θ
C)a csc θ
D)a cos θ
E)a cot θ

Question 26

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

A)25 tan θ
B)5 tan θ
C)25 sec θ
D)6 sec θ
E)- 5 sec θ

Question 27

Rewrite the expression as a single logarithm and simplify the result.​ $\ln | \cos x | - \ln | \sin x |$ ​

A)​ $\ln | \tan x |$
B) $\ln | \cos x |$
C) $\ln | \sin x |$
D) $\ln | \cot x |$
E) $\ln | \csc x |$

Question 28

Use a calculator to demonstrate the identity for the value of θ.​ $\sin ( - \theta ) = - \sin \theta , \theta = 258 ^ { \circ }$ ​

A)​ $\sin \left( - 258 ^ { \circ } \right) = - \sin 258 ^ { \circ } \approx 0.9781$
B)​ $\sin \left( - 258 ^ { \circ } \right) = - \sin 258 ^ { \circ } \approx 1.0081$
C)​ $\sin \left( - 258 ^ { \circ } \right) = - \sin 258 ^ { \circ } \approx 0.9981$
D)​ $\sin \left( - 258 ^ { \circ } \right) = - \sin 258 ^ { \circ } \approx - 0.9781$
E)​ $\sin \left( - 258 ^ { \circ } \right) = - \sin 258 ^ { \circ } \approx - 0.9981$

Question 29

Use the trigonometric substitution to rewrite the algebraic expression as a trigonometric function of θ,where $- \frac { \pi } { 2 } < x < \frac { \pi } { 2 }$ .Then find sin θ and cos θ. $4 = \sqrt { 16 - x ^ { 2 } } , x = 4 \sin \theta$

A) $16 \cos \theta = 4 ; \sin \theta = 0 ; \cos \theta = 1$
B)​ $4 \cos \theta = 4 ; \sin \theta = 0 ; \cos \theta = 1$
C)​ $16 \sin \theta = 4 ; \sin \theta = 0 ; \cos \theta = 1$
D)​ $4 \cos \theta = 4 ; \sin \theta = 1 ; \cos \theta = 0$
E)​ $4 \sin \theta = 4 ; \sin \theta = 0 ; \cos \theta = 1$

Question 30

Find the rate of change of the function $f ( x ) = \sec x + \csc x$ . ​

A) $- \csc x \cot x + \tan x \sec x$
B) $\csc x \cot x - \csc x \cot x$
C) $- \csc x \cot x + 1$
D) $- \csc x \cot x - \tan x \sec x$
E) $- \csc x \cot x + \sec x$

Question 31

Use the trigonometric substitution u = a sin θ,where - π/2 < θ < π/2 and a > 0 to simplify the expression $\sqrt { a ^ { 2 } - u ^ { 2 } }$ . ​

A)​a csc θ
B)​a sin θ
C)​a tan θ
D)​a cos θ
E)​- a cos θ

Question 32

Use the trigonometric substitution to rewrite the algebraic expression as a trigonometric function of θ,where $0 < \theta < \frac { \pi } { 2 }$ . $\sqrt { 11 - x ^ { 2 } } , x = \sqrt { 11 } \sin \theta$ ​

A) $- \sqrt { 11 } \cos \theta$
B) $\sqrt { 11 } \cos \theta$
C) $\sqrt { 11 } \sin \theta$
D) $11 \cos \theta$
E) $11 \sin \theta$

Question 33

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

A)- 5 sec θ
B)5 tan θ
C)25 sec θ
D)25 tan θ
E)5 sec θ

Question 34

Use the trigonometric substitution to rewrite the algebraic expression as a trigonometric function of θ,where $0 < x < \frac { \pi } { 2 }$ . $\sqrt { 6 - x ^ { 2 } } , x = \sqrt { 6 } \sin \theta$ ​

A)6 sin θ
B) $- \sqrt { 6 } \cos \theta$
C) $\sqrt { 6 } \sin \theta$
D) $\sqrt { 6 } \cos \theta$
E)6 cos θ

Question 35

Rewrite the expression as a single logarithm and simplify the result.​ $\ln | \sec x | + \ln | \sin x |$ ​

A) $\ln | \cos x |$
B) $\ln | \csc x |$
C) $\ln | \cot x |$
D) $\ln | \sin x |$
E) $\ln | \tan x |$

Question 36

Use the trigonometric substitution u = a tan θ,where 0 < θ < π/2 and to simplify the expression $\sqrt { a ^ { 2 } + u ^ { 2 } }$ . ​

A)​- a sec θ
B)​a sec θ
C)​a tan θ
D)​a sin θ
E)​a csc θ

Question 37

Use the trigonometric substitution to rewrite the algebraic expression as a trigonometric function of θ,where $- \frac { \pi } { 2 } < \theta < \frac { \pi } { 2 }$ .Then find sin θ and cos θ.​ $3 \sqrt { 3 } = \sqrt { 81 - 9 x ^ { 2 } } , x = 3 \cos \theta$ ​

A)​ $\sin \theta = 3 ; \sin \theta = \frac { \sqrt { 3 } } { 3 } ; \cos \theta = \frac { \sqrt { 6 } } { 3 }$
B)​ $3 \sin \theta = 3 ; \sin \theta = 0 ; \cos \theta = 1$
C)​ $\sin \theta = \sqrt { 3 } ; \sin \theta = \frac { \sqrt { 3 } } { 3 } ; \cos \theta = \frac { \sqrt { 6 } } { 3 }$
D)​ $9 \sin \theta = 3 \sqrt { 3 } ; \sin \theta = \frac { \sqrt { 3 } } { 3 } ; \cos \theta = \frac { \sqrt { 6 } } { 3 }$
E)​ $3 \cos \theta = 3 ; \sin \theta = 1 ; \cos \theta = 0$

Question 38

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

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 39

Use the trigonometric substitution to rewrite the algebraic expression as a trigonometric function of θ,where $0 < x < \frac { \pi } { 2 }$ . $\sqrt { x ^ { 2 } - 16 } , x = 4 \sec \theta$ ​

A)4 tan θ
B)16 sec θ
C)- 4 tan θ
D)4 sec θ
E)16 tan θ

Question 40

The forces acting on an object weighing W units on an inclined plane positioned at an angle of θ with the horizontal (see figure)are modeled by​ $\mu W \sin \theta = W \tan \theta$ ​ where μ is the coefficient of friction.Solve the equation for μ and simplify the result.​ ​

A) $$\mu = \tan \theta$$
B) $\mu = \csc \theta$
C) $\mu = \sin \theta$
D)​ $\mu = \sec \theta$
E) $\mu = \cot \theta$

Question 41

Factor;then use fundamental identities to simplify the expression below and determine which of the following is not equivalent.​ $\cot ^ { 2 } \alpha - \cot ^ { 2 } \alpha \cos ^ { 2 } \alpha$

A)​ $\cos ^ { 2 } \alpha$
B)​ $\sin ^ { 2 } \left( \frac { \pi } { 2 } - \alpha \right)$
C)​ $1 - \sin ^ { 2 } \alpha$
D)​ $\cot ^ { 2 } \alpha$
E)​ $\frac { 1 } { \sec ^ { 2 } \alpha }$

Question 42

If x = 6 sin θ,use trigonometric substitution to write $\sqrt { 36 - x ^ { 2 } }$ as a trigonometric function of θ,where $0 < \theta < \frac { \pi } { 2 }$ .

A)​6 csc θ
B)​6 sec θ
C)​6 sin θ
D)​6 cos θ
E)​6 tan θ

Question 43

Factor;then use fundamental identities to simplify the expression below and determine which of the following is not equivalent. ​​ $$\cot ^ { 2 } \alpha + \cot ^ { 2 } \alpha \cdot \tan ^ { 2 } \alpha$$

A)​ $\csc ^ { 2 } \alpha$
B)​ $1 + \cot ^ { 2 } \alpha$
C)​ $\sec ^ { 2 } \alpha$
D)​ $1 / \cos ^ { 2 } \left( \frac { \pi } { 2 } - \alpha \right)$
E)​ $\frac { 1 } { 1 - \cos ^ { 2 } \alpha }$

Question 44

Rewrite $\ln | \sin \theta | - \ln | \cos \theta |$ as a single logarithm and then simplify the result. ​

A)​ $\ln | \cos \theta |$
B)​ $\ln | \csc \theta |$
C)​ $\ln | \sec \theta |$
D)​ $\ln | \tan \theta |$
E)​ $\ln | \sin \theta |$

Question 45

Which of the following is equivalent to the given expression? $$\frac { \cos ^ { 2 } x } { 1 + \sin x }$$

A)​ $\tan x + \cos x$
B)​ $1 - \sin x$
C)​ $\csc x + \cot x$
D)​ $\cot x - \sin x$
E)​ $\cot x \cos x + \tan x$

Question 46

Use fundamental identities to simplify the expression below and then determine which of the following is not equivalent.​ $$\sin \alpha ( \csc \alpha - \sin \alpha )$$

A)​ $1 - \tan ^ { 2 } \alpha$
B)​ $\frac { \csc ^ { 2 } \alpha - 1 } { \csc ^ { 2 } \alpha }$
C)​ $\frac { \csc ^ { 2 } \alpha - \sec ^ { 2 } \alpha + \tan ^ { 2 } \alpha } { \csc ^ { 2 } \alpha }$
D)​ $\cos ^ { 2 } \alpha$

Question 47

Multiply;then use fundamental identities to simplify the expression below and determine which of the following is not equivalent.​ $$( \tan x + 1 ) ^ { 2 }$$ ​

A)​ $\tan ^ { 2 } x + 1$
B)​ $\sec ^ { 2 } x + 2 \tan x$
C)​ $\frac { 1 + 2 \sin x \cos x } { \cos ^ { 2 } x }$
D)​ $\tan ^ { 2 } x + 2 \tan x + 1$
E)​ $\sec ^ { 2 } x ( 1 + 2 \sin x \cos x )$

Question 48

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

A)​2 csc θ
B)​2 sec θ
C)​2 cos θ
D)​2 tan θ
E)​2 sin θ

Question 49

If x = 4 cot θ,use trigonometric substitution to write $\sqrt { 16 + x ^ { 2 } }$ as a trigonometric function of θ,where 0 < θ < π.

A)​4 cos θ
B)​4 csc θ
C)​4 cot θ
D)​4 sec θ
E)​4 sin θ

Question 50

Use fundamental identities to simplify the expression below and then determine which of the following is not equivalent.​ $\sin \left( \frac { \pi } { 2 } - x \right) \csc x$ ​

A)​-1
B)​ $\frac { \cos x } { \sin x }$
C)​cot x
D)​1/ tan x
E)​cos x csc x

Question 51

Use fundamental identities to simplify the expression below and then determine which of the following is not equivalent. $\sin \left( \frac { \pi } { 2 } - x \right) \csc x$

A)​ $\tan x$
B)​ $\cos \mathrm { x } \csc \mathrm { x }$
C)​ $\cot x$
D)​ $\frac { \cos x } { \sin x }$
E)​ $\frac { 1 } { \tan x }$

Question 52

Multiply;then use fundamental identities to simplify the expression below and determine which of the following is not equivalent.​ $( \cot x + 1 ) ^ { 2 }$

A)​ $\cot ^ { 2 } x + 1$
B)​ $\cot ^ { 2 } x + 2 \cot x$
C)​ $\cot ^ { 2 } x + 2 \cot x + 1$
D)​ $\csc ^ { 2 } x ( 1 + 2 \sin x \cos x )$
E)​ $\frac { 1 + 2 \sin x \cos x } { \sin ^ { 2 } x }$

Question 53

Factor;then use fundamental identities to simplify the expression below and determine which of the following is not equivalent. $\sin ^ { 3 } x - \sin ^ { 2 } x - \sin x + 1$ ​

A)​ $\frac { 1 + \cos ^ { 2 } x } { \sin }$
B)​ $\cos ^ { 2 } x ( 1 - \sin x )$
C)​ $\frac { \cos ^ { 4 } x } { 1 + \sin x }$
D)​ $\cos ^ { 2 } x - \cos ^ { 2 } x \sin x$
E)​ $\frac { 1 - \sin x } { \sec ^ { 2 } x }$

Question 54

Factor;then use fundamental identities to simplify the expression below and determine which of the following is not equivalent.​ $$\sin ^ { 3 } x - \sin ^ { 2 } x - \sin x + 1$$ ​

A)​ $\cos ^ { 2 } x - \cos ^ { 2 } x \sin x$
B)​ $\frac { 1 - \sin x } { \sec ^ { 2 } x }$
C)​ $\frac { \cos ^ { 4 } x } { 1 + \sin x }$
D)​ $\cos ^ { 2 } x ( 1 - \sin x )$
E)​ $\frac { 1 + \cos ^ { 2 } x } { \sin x }$

Question 55

If x = 5 sin θ,use trigonometric substitution to write $\sqrt { 25 - x ^ { 2 } }$ as a trigonometric function of θ,where $- \frac { \pi } { 2 } < \theta < \frac { \pi } { 2 }$ .

A)​5 sin θ
B)​5 cos θ
C)​5 tan θ
D)​5 csc θ
E)​5 sec θ

Question 56

Which of the following is equivalent to the given expression? $\frac { \cos ^ { 2 } x } { 1 + \sin x }$ ​

A)​ $\cot x \cos x + \tan x$
B)​ $\csc x + \cot x$
C)​ $1 - \sin x$
D)​ $\tan x + \cos x$
E)​ $\tan x \cot x - \sin x$

Question 57

Use a graphing utility to determine which of the trigonometric functions is equal to the following expression. $\frac { \csc x - \sin x } { \cot x }$

A)​ y = cos x

B)​ y = sin x
2
- 2π
2π $X _ { \text {scl } } = \frac { \pi } { 2 }$
- 2
C)​ y = csc x

D)​ y = cot x
4
- 2π
2π $X _ { \text {scl } } = \frac { \pi } { 2 }$
- 4
E)​ y = tan x

Question 58

If x = 3 cot θ,use trigonometric substitution to write $\sqrt { 9 + x ^ { 2 } }$ as a trigonometric function of θ,where $0 < \theta < \pi$ . ​

A)​3 sec θ
B)​3 csc θ
C)​3 cos θ
D)​3 sin θ
E)​3 cot θ

Question 59

Rewrite $\ln | \sec \theta | + \ln | \sin \theta |$ as a single logarithm and then simplify the result.

A)​ln |sin θ|
B)​ln |tan θ|
C)​ln |cos θ|
D)​ln |sec θ|
E)​ln |csc θ|

Question 60

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

A)​2 cos θ
B)​2 tan θ
C)​2 sin θ
D)​2 sec θ
E)​2 csc θ