Question 1

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

Accepted Answer

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) ​ $1 - \cot ^ { 2 } \alpha$ 
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 2

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

Accepted Answer

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) ​ $\sin \left( - 258 ^ { \circ } \right) = - \sin 258 ^ { \circ } \approx - 0.9981$ 
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 3

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

Accepted Answer

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

Question 4

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

Accepted Answer

A) ​ $\cos ^ { 2 } \alpha$ 
B) ​ $\sin ^ { 2 } \left( \frac { \pi } { 2 } - \alpha \right)$ 
C) ​ $1 - \sin ^ { 2 } \alpha$ 
D) ​ $\cot ^ { 2 } \alpha$ 
E) ​ $\frac { 1 } { \sec ^ { 2 } \alpha }$ 
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 5

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

Accepted Answer

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) ​ $\begin{array} { l } 
\cot \theta = \frac { 1 } { 4 } \
\sec \theta = \sqrt { 17 } \
\cos \theta = - \frac { \sqrt { 17 } } { 17 }
\end{array}$ ​ 
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 6

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.​ $4 = \sqrt { 64 - x ^ { 2 } } , x = 8 \sin \theta$

Accepted Answer

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) ​ $8 \cos \theta = 4 ; \sin \theta = \frac { 1 } { 2 } ; \cos \theta = \pm \frac { \sqrt { 3 } } { 2 }$ 
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 7

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

Accepted Answer

A) ​ $\tan x + \cos x$ 
B) ​ $1 - \sin x$ 
C) ​ $\csc x + \cot x$ 
D) ​ $\cot x - \sin x$ 
E) ​ $\cot x \cos x + \tan x$ 
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 8

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

Accepted Answer

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) ​ $\begin{array} { l } 
\cos x = \frac { 2 \sqrt { 6 } } { 5 } \
\csc x = \frac { 1 } { 5 } \
\tan x = \frac { \sqrt { 6 } } { 12 }
\end{array}$ ​ 
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 9

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

Accepted Answer

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) ​ $\sec ^ { 2 } x ( 1 + 2 \sin x \cos x )$ 
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 10

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

Accepted Answer

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) ​ $\begin{array} { l } 
\cos \phi = - \frac { 4 } { 3 } \
\sin \phi = - \frac { \sqrt { 5 } } { 3 } \
\tan \phi = - \frac { \sqrt { 5 } } { 4 }
\end{array}$ 
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 11

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

Accepted Answer

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

Question 12

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

Accepted Answer

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

Question 13

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

Accepted Answer

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

Question 14

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

Accepted Answer

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

Question 15

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

Accepted Answer

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

Question 16

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

Accepted Answer

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

Question 17

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 18

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

Accepted Answer

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)  $- \csc x \cot x + \sec x$ 
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 19

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

Accepted Answer

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) ​ $\begin{array} { l } 
\tan \varphi = - \frac { 1 } { 5 } \
\csc \psi = - \sqrt { 10 }
\end{array}$ ​ 
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 20

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

Accepted Answer

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

Exam 31: Using Fundamental Identities

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

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

Rewrite the expression as a single logarithm and simplify the result.​ ln⁡∣sec⁡x∣+ln⁡∣sin⁡x∣\ln | \sec x | + \ln | \sin x |ln∣secx∣+ln∣sinx∣ ​

Factor;then use fundamental identities to simplify the expression below and determine which of the following is not equivalent.​ cot⁡2α−cot⁡2αcos⁡2α\cot ^ { 2 } \alpha - \cot ^ { 2 } \alpha \cos ^ { 2 } \alphacot2α−cot2αcos2α

Use the given values to evaluate (if possible)three trigonometric functions cot⁡θ,sec⁡θ,cos⁡θ\cot \theta , \sec \theta , \cos \thetacotθ,secθ,cosθ .​ tan⁡θ=4,cos⁡θ<0\tan \theta = 4 , \cos \theta < 0tanθ=4,cosθ<0 ​

Which of the following is equivalent to the given expression? cos⁡2x1+sin⁡x\frac { \cos ^ { 2 } x } { 1 + \sin x }1+sinxcos2x​

Use the given values to evaluate (if possible)three trigonometric functions cos x,csc x,tan x.​ sin⁡x=15\sin x = \frac { 1 } { 5 }sinx=51​ , cos⁡x>0\cos x > 0cosx>0 ​

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

Use the given values to evaluate (if possible)three trigonometric functions cos⁡ϕ,sin⁡ϕ,tan⁡ϕ\cos \phi , \sin \phi , \tan \phicosϕ,sinϕ,tanϕ .​ sec⁡ϕ=34,csc⁡ϕ=−355\sec \phi = \frac { 3 } { 4 } , \csc \phi = \frac { - 3 \sqrt { 5 } } { 5 }secϕ=43​,cscϕ=5−35​​ ​

If x = 6 sin θ,use trigonometric substitution to write 36−x2\sqrt { 36 - x ^ { 2 } }36−x2​ as a trigonometric function of θ,where 0<θ<π20 < \theta < \frac { \pi } { 2 }0<θ<2π​ .

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

Find the rate of change of the function f(x)=−x+cos⁡xf ( x ) = - x + \cos xf(x)=−x+cosx . ​

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

If x = 3 cot θ,use trigonometric substitution to write 9+x2\sqrt { 9 + x ^ { 2 } }9+x2​ as a trigonometric function of θ,where 0<θ<π0 < \theta < \pi0<θ<π . ​

Use the trigonometric substitution u = a tan θ,where 0 < θ < π/2 and to simplify the expression a2+u2\sqrt { a ^ { 2 } + u ^ { 2 } }a2+u2​ . ​

Use the given values to evaluate (if possible)three trigonometric functions cos x,sin x,cot x.​ tan⁡x=940,sec⁡x=−4140\tan x = \frac { 9 } { 40 } , \sec x = - \frac { 41 } { 40 }tanx=409​,secx=−4041​ ​

Find the rate of change of the function f(x)=sec⁡x+csc⁡xf ( x ) = \sec x + \csc xf(x)=secx+cscx . ​

Use the given values to evaluate (if possible)two trigonometric functions tan⁡φ\tan \varphitanφ and csc⁡ψ\csc \psicscψ .​ cot⁡φ=−5,sin⁡ψ=1010\cot \varphi = - 5 , \sin \psi = \frac { \sqrt { 10 } } { 10 }cotφ=−5,sinψ=1010​​ ​

Use the trigonometric substitution to rewrite the algebraic expression as a trigonometric function of θ,where 0<θ<π20 < \theta < \frac { \pi } { 2 }0<θ<2π​ . 16−x2,x=4sin⁡θ\sqrt { 16 - x ^ { 2 } } , x = 4 \sin \theta16−x2​,x=4sinθ ​

Rectangular Coordinates

Graphs of Equations

Linear Equations in Two Variables

Functions

Analyzing Graphs of Functions

A Library of Parent Functions

Transformations of Functions

Combinations of Functions Composite Functions

Inverse Functions

Mathematical Modeling and Variation

Quadratic Functions and Models

Polynomial Functions of Higher Degree

Polynomial and Synthetic Division

Complex Numbers

Zeros of Polynomial Functions

Rational Functions

Nonlinear Inequalities

Exponential Functions and Their Graphs

Logarithmic Functions and Their Graphs

Properties of Logarithms

Exponential and Logarithmic Equations

Exponential and Logarithmic Models

Radian and Degree Measure

Trigonometric Functions The Unit Circle

Right Triangle Trigonometry

Trigonometric Functions of Any Angle

Graphs of Sine and Cosine Functions

Graphs of Other Trigonometric Functions

Inverse Trigonometric Functions

Applications and Models

Verifying Trigonometric Equations

Solving Trigonometric Equations

Sum and Difference Formulas

Multiple Angle and Product to Sum Formulas

Law of Sines

Law of Cosines

Vectors in the Plane

Vectors and Dot Products

Trigonometric Form of a Complex Number

Linear and Nonlinear Systems of Equations

Two Variable Linear Systems

Multivariable Linear Systems

Partial Fractions

Systems of Inequalities

Linear Programming

Matrices and Systems of Equations

Operations With Matrices

The Inverse of a Square Matrix

The Determinant of a Square Matrix

Applications of Matrices and Determinants

Sequences and Series

Arithmetic Sequences and Partial Sums

Geometric Sequences and Series

Mathematical Induction

The Binomial Theorem

Counting Principles

Probability

Lines

Introduction to Conics Parabolas

Ellipses

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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