Exam 6: Analytic Trigonometry

Solve the problem. -A water wheel rotates through the angle $\theta$ , the water level L behind the wheel changes according to the equation $\mathrm { L } = 1 - \sin \theta - 2 \cos ^ { 2 } \theta$ where L is measured in inches. Determine the values of $\theta$ for which the water level is zero. Find the exact values. Do not use a calculator.

Solve the equation on the interval $0 \leq \theta < 2 \pi$ - $2 \sqrt { 3 } \sin ( 4 \theta ) = 3$

Solve the equation on the interval $0 \leq \theta < 2 \pi$ - $3 \cot ^ { 2 } \theta - 4 \csc \theta = 1$

Find the domain of the function f and of its inverse function $\mathrm { f } ^ { - 1 }$ . - $f ( x ) = 3 \sin ( 5 x )$

Establish the identity. - $\frac { \cos ( x - y ) } { \cos ( x + y ) } = \frac { 1 + \tan x \tan y } { 1 - \tan x \tan y }$

Establish the identity. - $\frac { \cot x } { 1 + \csc x } = \frac { \csc x - 1 } { \cot x }$

Find the exact value under the given conditions. - $\tan \alpha = \frac { 5 } { 12 } , \pi < \alpha < \frac { 3 \pi } { 2 } ; \quad \cos \beta = - \frac { 4 } { 5 } , \frac { \pi } { 2 } < \beta < \pi \quad$ Find $\sin ( \alpha + \beta )$ .

Establish the identity. - $( \sin x ) ( \tan x \cos x - \cot x \cos x ) = 1 - 2 \cos ^ { 2 } x$

Find the exact value of the composition. - $\sin \left( \tan ^ { - 1 } 2 \right)$

Solve the equation on the interval $0 \leq \theta < 2 \pi \text {. }$ - $\sin \theta = \sqrt { 2 } - \cos \theta$

Express the sum or difference as a product of sines and/or cosines. - $\sin ( 6 \theta ) - \sin ( 4 \theta )$

Solve the problem. - $\text { Multiply } \frac { \sin \theta } { 1 - \cos \theta } \text { by } \frac { 1 + \cos \theta } { 1 + \cos \theta }$

Establish the identity. - $\frac { \cos t } { 1 + \sin t } + \frac { 1 + \sin t } { \cos t } = 2 \sec t$

Establish the identity. - $1 + \sec ^ { 2 } x \sin ^ { 2 } x = \sec ^ { 2 } x$

Use a graphing utility to solve the equation on the interval 0 $0 ^ { \circ } \leq x < 360 ^ { \circ }$ . Express the solution(s) rounded to one decimal place. - $\sin ^ { 2 } x + 8 \sin x + 16 = 0$

Complete the identity. - $\frac { 7 + 7 \sin \theta } { - 5 \cos \theta } = \text { ? }$

Establish the identity. - $\cos x \csc x \tan x = 1$

Establish the identity. - $\sin \left( x + \frac { \pi } { 2 } \right) = \cos x$

Complete the identity. - $\sin \theta [ \sin ( 2 \theta ) + \sin ( 4 \theta ) ] = ?$

Establish the identity. - $\tan ^ { 5 } x = \tan x \left( \sec ^ { 4 } x - 2 \sec ^ { 2 } x + 1 \right)$