Exam 5: Analytic Trigonometry

Determine which of the following are trigonometric identities. I. $\frac { \sin ( y ) - \sin ( x ) } { \cos ( y ) + \cos ( x ) } + \frac { \cos ( y ) - \cos ( x ) } { \sin ( y ) + \sin ( x ) } = 0$ II. $\frac { \sin ( y ) + \sin ( x ) } { \cos ( y ) + \cos ( x ) } + \frac { \cos ( y ) + \cos ( x ) } { \sin ( y ) + \sin ( x ) } = 1$ III. $\frac { \sin ( y ) + \cos ( x ) } { \sin ( y ) \cos ( x ) } = \sin ( x ) + \cos ( y )$

Determine which of the following are trigonometric identities. I. $\frac { \sin ( \mathrm { t } ) + \sin ( \mathrm { s } ) } { \cos ( \mathrm { t } ) - \cos ( \mathrm { s } ) } + \frac { \cos ( \mathrm { t } ) + \cos ( \mathrm { s } ) } { \sin ( \mathrm { t } ) - \sin ( \mathrm { s } ) } = 0$ II. $\frac { \sin ( \mathrm { t } ) + \sin ( \mathrm { s } ) } { \cos ( \mathrm { t } ) + \cos ( \mathrm { s } ) } + \frac { \cos ( \mathrm { t } ) + \cos ( \mathrm { s } ) } { \sin ( \mathrm { t } ) + \sin ( \mathrm { s } ) } = 1$ III. $\frac { \sin ( \mathrm { t } ) + \cos ( \mathrm { s } ) } { \sin ( \mathrm { t } ) \cos ( \mathrm { s } ) } = \sin ( \mathrm { s } ) + \cos ( \mathrm { t } )$

Solve the multiple-angle equation in the interval $[ 0,2 \pi )$ $\tan 2 x = - 1$

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

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.

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.)

Verify the identity shown below. $\csc \theta - \cos \theta \cot \theta = \sin \theta$

Verify the given identity. $\cos ( x + y ) \cos ( x - y ) = \cos ^ { 2 } x - \sin ^ { 2 } y$

Use the graph below to approximate the solutions of the equation $- 2 \cos ( x ) - \sin ( x ) = 0$ on the interval $[ 0,2 \pi )$ . Round your answer to one decimal.

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

Solve the multiple-angle equation in the interval $[ 0,2 \pi )$ . $\sec 2 x = 2$

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

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

Determine which of the following are trigonometric identities. I. $\cot ( \theta ) \csc ( \theta ) = \sec ( \theta )$ II. $\cot ( \theta ) \sec ( \theta ) = \csc ( \theta )$ III. $\sec ( \theta ) \csc ( \theta ) = \cot ( \theta )$ IV. $\cot ( \theta ) \sin ( \theta ) = 1$

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

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

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.)

Find all solutions of the following equation in the interval $[ 0,2 \pi )$ . $2 \cos x - \sec x = 0$

Use the trigonometric substitution $x = 8 \sec ( \theta )$ to write the expression $\sqrt { x ^ { 2 } - 64 }$ as a trigonometric function of $\theta$ , where $0 < \theta < \frac { \pi } { 2 }$ .

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