Exam 5: Analytic Trigonometry

Use the formula $a \sin B \theta + b \cos B \theta = \sqrt { a ^ { 2 } + b ^ { 2 } } \cos ( B \theta - C )$ , where $C = \arctan ( a / b ) , a = 3 , b = 7 , B = 2$ to rewrite the trigonometric expression in the following form. $y = \sqrt { a ^ { 2 } + b ^ { 2 } } \cos ( B \theta - C )$ $a \sin B \theta + b \cos B \theta$

Find the exact value of $\cos ( u + v )$ given that $\sin u = \frac { 8 } { 17 }$ and $\cos v = - \frac { 60 } { 61 }$ .(Both u and v are in Quadrant II.)

Find the exact solutions of the given equation in the interval $[ 0,2 \pi )$ . sin 4x= -2sin 2x

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

Use the formula $a \sin B \theta + b \cos B \theta = \sqrt { a ^ { 2 } + b ^ { 2 } } \sin ( B \theta - C )$ , where $C = \arctan ( a / b ) , a = 2 , b = 8 , B = 1$ , to rewrite the trigonometric expression in the following form. $y = \sqrt { a ^ { 2 } + b ^ { 2 } } \sin ( B \theta - C )$ $a \sin B \theta + b \cos B \theta$

Find all solutions of the following equation in the interval [0, 2 $\pi$ ). $4 \tan x + 4 \sqrt { 3 } = 0$

Simplify the expression algebraically. $\sin ( 7 x + 7 y ) + \sin ( 7 x - 7 y )$

Evaluate the following expression.( $\theta$ $\neq$ $\pi$ n, where n is a whole number) $\frac { 5 \cos 4 \theta \cot 4 \theta } { 1 - \sin 4 \theta } - 5$

Evaluate the following expression.(a $\neq$ $\pi$ /2+ $\pi$ n, where n is a whole number) $\frac { 5 \tan ^ { 2 } \theta } { \sec \theta }$

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

Use the product-to-sum formulas to rewrite the product as a sum or difference. $4 \cos \frac { \pi } { 2 } \sin \frac { 5 \pi } { 4 }$

Simplify the following expression algebraically. $7 \cos ( \pi + x )$

Solve the multiple-angle equation. $\sin \frac { x } { 2 } = \frac { \sqrt { 2 } } { 2 }$

Use the given values to evaluate (if possible) three trigonometric functions cos x, sin x, tan x. $\sec x = 4 , \sin x > 0$

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

Solve the following equation. $2 \tan ^ { 2 } 3 x = 6$

Use a graphing utility to graph the function. $f ( x ) = 2 ( \sin x \cos x )$

Use the formula $a \sin B \theta + b \cos B \theta = \sqrt { a ^ { 2 } + b ^ { 2 } } \sin ( B \theta + C )$ , where $C = \arctan ( b / a ) , a = 1 , b = 3 , B = 2$ , to rewrite the trigonometric expression in the following form. $y = \sqrt { a ^ { 2 } + b ^ { 2 } } \sin ( B \theta + C )$ $a \sin B \theta + b \cos B \theta$

Find all solutions of the following equation in the interval [0, 2 $\pi$ ). $\sin ^ { 3 } x = \sin x$

If x = 4 cot $\theta$ , use trigonometric substitution to write $\sqrt { 16 + x ^ { 2 } }$ as a trigonometric function of $\theta$ , where 0 < $\theta$ < $\pi$ .