Exam 5: Analytic Trigonometry

Use the graph of the function $f ( x ) = - 2 \cos ( x ) + \sin ( x )$ to approximate the maximum points of the graph in the interval $[ 0,2 \pi ]$ . Round your answer to one decimal.

Solve the following equation. $\tan x - \sqrt { 3 } = 0$

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

Verify the identity shown below. $\frac { \sin ^ { 2 } \theta - 1 } { \cos ^ { 2 } \theta - 1 } = \csc ^ { 2 } \theta - 1$

Find the exact value of the given expression using a sum or difference formula. $\cos \frac { 7 \pi } { 12 }$

Verify the identity shown below. $\left( 1 + \cot ^ { 2 } \theta \right) \tan ^ { 2 } \theta = \sec ^ { 2 } \theta$

Use fundamental identities to simplify the expression below and then determine which of the following is not equivalent. $\cot \beta \sec \beta$

The rate of change of the function $f ( x ) = - \csc x - \sin x$ is given by the expression $\csc x \cot x - \cos x \text {. }$ Which of the following is its simplification?

Use the sum-to-product formulas to write the given expression as a product. $\sin 9 \theta - \sin 7 \theta$

Solve the multi-angle equation below. $\cos \left( \frac { x } { 2 } \right) = - \frac { \sqrt { 3 } } { 2 }$

Write the given expression as the cosine of an angle. $\cos 30 ^ { \circ } \cos 50 ^ { \circ } + \sin 30 ^ { \circ } \sin 50 ^ { \circ }$

Find the exact value of $\cos ( u + v )$ given that $\sin u = \frac { 11 } { 61 }$ and $\cos v = - \frac { 40 } { 41 }$ . (Both $u$ and $v$ are in Quadrant II.)

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

Use the graph below of the function to approximate the solutions to $4 \cos ( 2 x ) - \cos ( x ) = 0$ in the interval $[ 0,2 \pi )$ . Round your answers to one decimal.

Find all solutions of the following equation on the interval $[ 0,2 \pi )$ . $\csc ^ { 2 } ( x ) - 2 = 0$

Find the exact value of the given expression using a sum or difference formula. $\sin 165 ^ { \circ }$

Find the exact value of $\cos ( u + v )$ given that $\sin u = \frac { 9 } { 41 }$ and $\cos v = - \frac { 15 } { 17 }$ . (Both $u$ and $v$ are in Quadrant II.)

Use the product-to-sum formula to write the given product as a sum or difference. $6 \sin \frac { \pi } { 8 } \cos \frac { \pi } { 8 }$

Simplify the given expression algebraically. $\cos \left( \frac { \pi } { 2 } + x \right)$

Solve the multi-angle equation below. $\sin ( 2 x ) = - \frac { \sqrt { 2 } } { 2 }$