Exam 7: Trigonometric Identities and Equations

Solve the problem. -Two low-frequency sound waves with similar frequencies, producing the pressures $\mathrm { P } _ { 1 } = 0.005 \sin ( 14 \pi \mathrm { t } )$ and $\mathrm { P } _ { 2 } = 0.002 \sin ( 22 \pi \mathrm { t } )$ , combine to form a pressure $\mathrm { P } = \mathrm { P } _ { 1 } + \mathrm { P } _ { 2 }$ . The variable $\mathrm { t }$ is time measured in seconds. Graph $P$ in $[ 0,1.2 , .05 ]$ by $[ - 0.01,0.01,0.001 ]$ and use the graph to determine the frequency of the displayed beats.

Complete the sentence so the result is an identity. Let x be any real number. - $ـــــــــ= \frac { 1 } { \sin x }$

Use the fundamental identities to find the value of the trigonometric function. -Find $\sin \theta$ if $\sec \theta = - \frac { 8 } { 5 }$ and $\tan \theta < 0$ .

Verify that the equation is an identity. - $\tan \left( \frac { \pi } { 2 } + x \right) = - \cot x$

Use a sum or difference identity to find the exact value. - $\tan 75 ^ { \circ }$

Solve the problem. -An airplane flying faster than the speed of sound creates sound waves that form a cone. If $\theta$ is the vertex angle of the cone and $m$ is the Mach number for the speed of the plane, then $\sin \frac { \theta } { 2 } = \frac { 1 } { m }$ $( m > 1 )$ . Find the Mach number to the nearest tenth if $\theta = 30 ^ { \circ }$ .

Find the exact value by using a half-angle identity. - $\cos 75 ^ { \circ }$

Write the expression in terms of sine and cosine, and simplify so that no quotients appear in the final expression. - $\frac { \sin ^ { 2 } x - 1 } { \cos ( - x ) }$

Solve the equation (x in radians and θ 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. - $\frac { 4 \tan \theta } { 21 - \tan ^ { 2 } \theta } = 1$

Use the fundamental identities to find the value of the trigonometric function. -Find $\cos \theta$ if $\csc \theta = - \frac { 5 } { 3 }$ and $\theta$ is in quadrant IV.

Use the fundamental identities to find the value of the trigonometric function. -Find $\sec \theta$ if $\cos \theta = \frac { 1 } { 4 }$ and $\sin \theta > 0$ .

Solve the equation in the interval [0°, 360°). Give solutions to the nearest tenth, if necessary. - $2 \cos ^ { 3 } \theta = \cos \theta$

Verify that each equation is an identity. - $\sec ^ { 2 } \frac { \mathrm { u } } { 2 } = \frac { 2 \sec \mathrm { u } } { \sec u + 1 }$

Use a calculator to give the real number value. Round the answer to 7 decimal places. - $\mathrm { y } = \operatorname { arcsec } ( 2.8842912 )$

Use the fundamental identities to simplify the expression. - $\cot \theta \sec \theta \sin \theta$

Find the exact value by using a half-angle identity. - $\sin 22.5 ^ { \circ }$

Write the product as a sum or difference of trigonometric functions. - $2 \sin 4 x \sin 14 x$

Use identities to find the indicated value for each angle measure. - $\sin \theta = - \frac { 4 } { 5 } , \frac { 3 \pi } { 2 } < \theta < 2 \pi \quad$ Find $\cos ( 2 \theta )$

Find the exact value of the real number y. - $\mathrm { y } = \tan ^ { - 1 } ( 1 )$

Provide an appropriate response. -Explain what is wrong with the following solution for the equation $\tan 3 \theta = \sqrt { 3 }$ in the interval $[ 0,2 \pi )$ . $\tan 3 \theta = \sqrt { 3 }$ $\tan \theta = \frac { \sqrt { 3 } } { 3 }$ $\theta = \frac { \pi } { 6 }$ or $\theta = \frac { 7 \pi } { 6 }$