Exam 8: Trigonometric Identities and Equations

Solve the equation in the interval $\left[ 0 ^ { \circ } , 360 ^ { \circ } \right)$ ). Give solutions to the nearest tenth, if necessary. - $2 \cos 2 \theta + 7 \sin \theta = 5$

Verify that the equation is an identity. -In a particular situation, the pressure, $\mathrm { P }$ , exerted on a person's eardrum at a distance of 6 feet fron the source is given by the function: $\mathrm { P } ( \mathrm { t } ) = \frac { \mathrm { a } } { 6 } \cos ( 2 \mathrm { n } \pi - \mathrm { ct } )$ , where $\mathrm { n }$ is a positive integer. Use the difference identity for cosine to simplify $\mathrm { P }$ in this situation.

Solve the equation for solutions over the interval [0, $[ 0,2 \pi )$ ). Write solutions as exact values or to four decimal places, as appropriate. -The weekly sales in thousands of items of a product has a seasonal sales record approximated by $\mathrm { n } = 84.77 + 17.9 \sin \frac { \pi \mathrm { t } } { 24 }$ ( $\mathrm { t }$ is time in weeks with $\mathrm { t } = 1$ referring to the first week in the year). During which week(s) will the sales equal 93,720 items?

Use a sum or difference identity to find the exact value. - $\tan \frac { \pi } { 12 }$

Use a sum or difference identity to find the exact value. - $\frac { \tan 165 ^ { \circ } - \tan 45 ^ { \circ } } { 1 + \tan 165 ^ { \circ } \tan 45 ^ { \circ } }$

Use a sum or difference identity to find the exact value. - $\sin \frac { 7 \pi } { 24 } \cos \frac { \pi } { 8 } - \cos \frac { 7 \pi } { 24 } \sin \frac { \pi } { 8 }$

Solve the equation for exact solutions. - $\sin ^ { - 1 } x + \tan ^ { - 1 } x _ { x } = 0$

Find the exact value by using a half-angle identity. - $\sin 165 ^ { \circ }$

Use a sum or difference identity to find the exact value. - $\sin 10 ^ { \circ } \cos 50 ^ { \circ } + \cos 10 ^ { \circ } \sin 50 ^ { \circ }$

Use an identity to write the expression as a single trigonometric function or as a single number. - $\frac { \sin 40 ^ { \circ } } { 1 - \cos 40 ^ { \circ } }$

Solve the equation for solutions in the interval [0 $\left[ 0 ^ { \circ } , 360 ^ { \circ } \right)$ ). Round to the nearest degree. - $\tan 4 \theta = 0$

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

Use Identities to find the exact value. - $\cos \frac { 7 \pi } { 12 } \cos \frac { 5 \pi } { 12 } + \sin \frac { 7 \pi } { 12 } \sin \frac { 5 \pi } { 12 }$

Solve the equation for solutions over the interval [0, $[ 0,2 \pi )$ ). Write solutions as exact values or to four decimal places, as appropriate. - $\sin \frac { x } { 2 } + \cos \frac { x } { 2 } = 0$

Graph the inverse circular function. - $y = \operatorname { arccsc } \frac { 1 } { 2 } x$

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

Verify that each equation is an identity. - $\sec \beta + \tan \beta = \frac { \cos \beta } { 1 - \sin \beta }$

Solve the equation for x, where x is restricted to the given interval. - $y = - 5 \sin 8 x$ , for $x$ in $\left[ - \frac { \pi } { 16 } , \frac { \pi } { 16 } \right]$

Use a calculator to give the real number value. Round the answer to 7 decimal places. - $y = \cos ^ { - 1 } ( - 0.3907 )$

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. - $\cos ^ { 2 } x - \cos x = 0$