Exam 5: Analytic Trigonometry

Determine which of the following are trigonometric identities. $\frac { \cos ( 4 x ) - \cos ( 2 x ) } { 2 \tan ( 3 x ) } = - \tan ( x )$ $\frac { \cos ( 3 x ) - \cos ( x ) } { \sin ( 3 x ) - \sin ( x ) } = - \tan ( 2 x )$ $\frac { \cos ( 6 x ) - \cos ( 2 x ) } { \sin ( 4 x ) + \sin ( 2 x ) } = - \tan ( 3 x$

Verify the given identity. $\frac { \sin 5 x + \sin 3 x } { \cos 5 x + \cos 3 x } = \tan 4 x$

Approximate the solutions of the equation $2 \sin ^ { 2 } ( x ) = 2 \cos ( x ) + 1$ by considering its graph below. Round your answer to one decimal.

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

Determine the exact value of the following expression. $\cos \left( 120 ^ { \circ } - 0 ^ { \circ } \right)$

Write the given expression as the sine of an angle. $\sin 75 ^ { \circ } \cos 35 ^ { \circ } - \sin 35 ^ { \circ } \cos 75 ^ { \circ }$

Solve the following equation. $3 \csc ^ { 2 } ( x ) - 4 = 0$

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

Approximate the solutions of the equation $2 \sin ^ { 2 } ( x ) + 5 \sin ( x ) - 2 = 0$ by considering its graph below. Round your answer to one decimal.

Verify the identity shown below. $\frac { \tan \theta + 1 } { \sec \theta + \csc \theta } = \sin \theta$

Verify the given identity. $\frac { \sin u + \sin v } { \cos u + \cos v } = \tan \frac { 1 } { 2 } ( u + v )$

Find the exact value of $\tan ( u + v )$ given that $\sin u = - \frac { 7 } { 25 }$ and $\cos v = \frac { 12 } { 13 }$ . (Both $u$ and $v$ are in Quadrant IV.)

Multiply; then use fundamental identities to simplify the expression below and determine which of the following is not equivalent. $( \sin x + \cos x ) ( \sin x - \cos x )$

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 sum-to-product formulas to write the given expression as a product. $\cos 8 \theta - \cos 6 \theta$

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.

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

Determine which of the following are trigonometric identities. I. $\frac { \cos ( 4 x ) + \cos ( 2 x ) } { 2 \cot ( 3 x ) } = \cot ( x )$ II. $\frac { \cos ( 4 x ) + \cos ( x ) } { \sin ( 3 x ) - \sin ( x ) } = \cot ( 2 x )$ III. $\frac { \cos ( 6 x ) + \cos ( 2 x ) } { \sin ( 4 x ) + \sin ( 2 x ) } = \cot ( 3 x )$

Verify the identity shown below. $\sec ^ { 2 } \mu - \cot ^ { 2 } \left( \frac { \pi } { 2 } - \mu \right) = 1$

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