Exam 5: Analytic Trigonometry

Use the figure below to find the exact value of the given trigonometric expression. $\cot \frac { x } { 2 }$

Which of the following is a solution to the given equation? $\sec x - 2 = 0$

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

Use the half-angle formula to simplify the given expression. $\sqrt { \frac { 1 + \cos 16 x } { 2 } }$

Use fundamental identities to simplify the expression below and then determine which of the following is not equivalent. $\sin \alpha ( \csc \alpha - \sin \alpha )$

Determine which of the following are trigonometric identities. I. $\sin ( \theta ) + \cot ( \theta ) \cos ( \theta ) = \csc ( \theta )$ II. $\cot ( \theta ) - \sin ( \theta ) \cos ( \theta ) = 0$ III. $\sin ( \theta ) + \sin ( \theta ) \cos ( \theta ) = \csc ( \theta )$

Which of the following is a solution to the given equation? $2 \sin x - 1 = 0$

If $x = 8 \cos \theta$ , use trigonometric substitution to write $\sqrt { 64 - x ^ { 2 } }$ as a trigonometric function of $\theta$ , where $0 < \theta < \pi$ .

Solve the following equation. $\tan ^ { 2 } x + \tan x = 0$

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

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

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

If $\csc x = \frac { 4 \sqrt { 3 } } { 3 }$ and $\cos x < 0$ , evaluate the function below. $\sec x$

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

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. $\frac { \sin ( y ) + \cos ( x ) } { \sin ( y ) \cos ( x ) } = \sin ( x ) + \cos ( y )$

Use the graph below of the function to approximate the solutions to $3 \cos ( 2 x ) - \cos ( x ) = 0$ in the interval $[ 0,2 \pi )$ . Round your answers to one decimal.

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

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

Verify the identity shown below. $\left( 1 + \cot ^ { 2 } \theta \right) \tan ^ { 2 } \theta = \sec ^ { 2 } \theta$

Which of the following is a solution to the given equation? $2 \cos x + \sqrt { 3 } = 0$