Exam 5: Analytic Trigonometry

Use the figure below to determine the exact value of the given function. $\sin 2 \theta$

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.

Use the cofunction identities to evaluate the expression below without the aid of a calculator. $\sin ^ { 2 } 62 ^ { \circ } + \sin ^ { 2 } 30 ^ { \circ } + \sin ^ { 2 } 28 ^ { \circ } + \sin ^ { 2 } 60 ^ { \circ }$

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

Use the half-angle formulas to determine the exact value of the following. $\cos \left( 22.5 ^ { \circ } \right)$

Use the product-to-sum formulas to write the expression below as a sum or difference. $\sin ( 8 \theta ) \cos ( 6 \theta )$

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.)

Use a double angle formula to rewrite the following expression. $- 16 \sin ( x ) \cos ( x )$

Use a double-angle formula to find the exact value of $\cos 2 u$ when $\sin u = \frac { 8 } { 17 }$ , where $\frac { \pi } { 2 } < u < \pi$ .

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

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.

Expand the expression below and use fundamental trigonometric identities to simplify. $( \sin ( \omega ) + \cos ( \omega ) ) ^ { 2 }$

Which of the following is a solution to the given equation? $\cot x + 1 = 0$

Use a double-angle formula to find the exact value of $\cos 2 u$ when $\sin u = \frac { 5 } { 13 }$ , where $\frac { \pi } { 2 } < u < \pi$ .

Verify the identity shown below. $\frac { \tan \alpha + \cot \beta } { \tan \alpha \cot \beta } = \tan \beta + \cot \alpha$

Solve the following trigonometric equation on the interval $[ 0,2 \pi )$ . $\sin ( x ) + \cos ( x ) = 0$

Add or subtract as indicated; then use fundamental identities to simplify the expression below and determine which of the following is not equivalent. $\sin x + 1 + \frac { 1 } { \sin x - 1 }$

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

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

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