Exam 8: Trigonometric Identities and Equations

Graph the expression on each side of the equals symbol to determine whether the equation might be an identity. - $\frac { \sin x + \csc x } { \sin x - \csc x } = - 2 \tan ^ { 2 } x$

Use the fundamental identities to simplify the expression. - $\frac { \cos ^ { 2 } x } { \sin ^ { 2 } x } + \cos x \sec x$

Verify that the equation is an identity. - $\cot ( \pi - \theta ) = - \cot \theta$

Verify that the equation is an identity. -Two low-frequency sound waves with similar frequencies, producing the pressures $P _ { 1 } = 0.003 \sin ( 14 \pi t )$ and $P _ { 2 } = 0.002 \sin ( 24 \pi t )$ , combine to form a pressure $P = P _ { 1 } + P _ { 2 }$ . The variable $\mathrm { t }$ is time measured in seconds. Graph $\mathrm { P }$ in $[ 0,1.2 , .05 ]$ by $[ - 0.01,0.01,0.001 ]$ and use the graph to determine the frequency of the displayed beats.

Write the product as a sum or difference of trigonometric functions. - $2 \sin 3 x \sin 18 x$

Use identities to find the indicated value for each angle measure. - $\cos 2 \theta = \frac { 2 } { 3 }$ and $\theta$ terminates in quadrant $\mathrm { I } \quad$ Find $\sin \theta$

Solve the equation (x in radians and $\boldsymbol { \theta }$ in degrees) for all exact solutions where appropriate. Round approximate answers in radians to four decimal places and approximate answers in degrees to the nearest tenth. - $4 \csc ^ { 2 } \theta = 5 \csc \theta$

Verify that each equation is an identity. - $\frac { \sin x + \cos x } { \sin x - \cos x } = \frac { 1 + 2 \sin x \cos x } { 2 \sin ^ { 2 } x - 1 }$

Use identities to fill in the blank with the appropriate trigonometric function name. - $14 ^ { \circ } = \csc 76 ^ { \circ }$

Verify that each equation is an identity. - $\sec ( 2 \theta ) = \frac { \csc ^ { 2 } \theta } { \csc ^ { 2 } \theta - 2 }$

The statement tan(tan $\tan \left( \tan ^ { - 1 } x \right) = x$ x for all real numbers in the interval - $- \frac { \pi } { 2 } < x < \frac { \pi } { 2 }$

Provide an appropriate response. -Graph the expression $\frac { 1 - \sin ^ { 2 } x } { \sin x - \csc x }$ on your calculator. Determine what constant or single circular function is equivalent to the given expression.

Solve the problem. -Suppose that a household appliance draws a current represented by the relation $\mathrm { I } ( \mathrm { t } ) = 5 \cos ( 120 \pi \mathrm { t } )$ , where $t$ is time measured in seconds. The power consumed by the appliance is $\mathrm { P } = \mathrm { I } ^ { 2 } \mathrm { R }$ , where $\mathrm { R }$ is a constant. Take R to be 12 and graph the power in [0, 0.04, 0.01] by $[ - 200,2000,200 ]$ and use an identity to write the expression for the power in the form $\mathrm { P } = \mathrm { a } \cos ( \mathrm { k } \pi \mathrm { t } ) + \mathrm { d }$ , where $a , k$ , and $d$ are constants.

Use an identity to write the expression as a single trigonometric function or as a single number. - $\sin 22.5 ^ { \circ } \cos 22.5 ^ { \circ }$

Solve the equation in the interval $\left[ 0 ^ { \circ } , 360 ^ { \circ } \right)$ ). Give solutions to the nearest tenth, if necessary. - $\sin 2 \theta = - \sin \theta$

Write the following as an algebraic expression in u, u > 0. - $\tan \left( \operatorname { arcsec } \frac { \sqrt { \mathrm { u } ^ { 2 } + 9 } } { \mathrm { u } } \right)$

Rewrite the following as a product of trigonometric functions. - $\sin \frac { \pi } { 11 } + \sin \frac { \pi } { 4 }$

Solve the equation for solutions in the interval [0 $\left[ 0 ^ { \circ } , 360 ^ { \circ } \right)$ ). Round to the nearest degree. - $\sin 2 \theta = - \frac { 1 } { 2 }$

Use the fundamental identities to simplify the expression. - $\tan ^ { 2 } \theta \csc ^ { 2 } \theta$

Solve the equation (x in radians and $\boldsymbol { \theta }$ in degrees) for all exact solutions where appropriate. Round approximate answers in radians to four decimal places and approximate answers in degrees to the nearest tenth. - $\frac { 4 \tan \theta } { 12 - \tan ^ { 2 } \theta } = 1$