Exam 8: Trigonometric Identities and Equations

Verify that the equation is an identity. -When a person bends at the waist with a straight back, the force, F, exerted by the person's lower back muscles can be estimated with the formula $F = 2.89 \mathrm { w } \sin \left( \theta + 90 ^ { \circ } \right)$ , where $w$ is the person's weight, and $\theta$ is the angle made by the person's torso relative to the horizontal. Suppose that a 220 -pound person bends at an angle of $30 ^ { \circ }$ . Determine the force exerted by the person's back muscles, and estimate the angle that requires the back muscles to exert a force of 381 pounds. Round the force to the nearest pound, and the angle to the nearest tenth, if necessary.

Find the exact value of the expression using the provided information. -Find $\cos ( \mathrm { s } + \mathrm { t } )$ given that $\sin \mathrm { s } = - \frac { 1 } { 2 }$ , with $\mathrm { s }$ in quadrant IV, and $\sin \mathrm { t } = \frac { 1 } { 4 }$ , with $\mathrm { t }$ in quadrant II.

Use the cofunction identities to find an angle $\theta$ that makes the statement true. - $\tan \theta = \cot \left( 30 ^ { \circ } + 5 \theta \right)$

Graph the expression on each side of the equals symbol to determine whether the equation might be an identity. - $( \cos \theta + \cot \theta ) ( 1 - \csc \theta ) = \cot ^ { 2 } \theta$

Use identities to fill in the blank with the appropriate trigonometric function name. - $\tan \frac { \pi } { 4 } =....... \quad \frac { \pi } { 4 }$

Find the exact value of the expression using the provided information. -Find $\cos ( \mathrm { s } - \mathrm { t } )$ given that $\cos \mathrm { s } = - \frac { 1 } { 6 }$ , with $\mathrm { s }$ in quadrant III, and $\cos \mathrm { t } = - \frac { 3 } { 5 }$ , with $\mathrm { t }$ in quadrant III.

Use an identity to write the expression as a single trigonometric function or as a single number. - $\sqrt { \frac { 1 + \cos 22 ^ { \circ } } { 2 } }$

Solve the equation for x, where x is restricted to the given interval. - $y = \sin x - \pi$ , for $x$ in $\left[ - \frac { \pi } { 2 } , \frac { \pi } { 2 } \right]$

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

Find the exact value of the real number y. - $y = \csc ^ { - 1 } ( - 2 )$

Use identities to find the indicated value for each angle measure. - $\tan \theta = \frac { 7 } { 24 } , \quad \pi < \theta < \frac { 3 \pi } { 2 } \quad$ Find $\sin ( 2 \theta )$

Provide an appropriate response. -The statement cos $\cos ^ { - 1 } ( \cos x ) = x$ r all real numbers in the interval $- 1 \leq x \leq 1$

Verify that each equation is an identity. - $\left( 1 + \tan ^ { 2 } s \right) \left( 1 - \sin ^ { 2 } s \right) = 1$

Solve the equation for exact solutions. - $\arcsin 2 x + 2 \arccos x = \pi$

Write the product as a sum or difference of trigonometric functions. - $4 \cos 18 ^ { \circ } \cos 9 ^ { \circ }$

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

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. - $\tan x ( \tan x - 5 ) = 6$

Use a sum or difference identity to find the exact value. - $\sin \frac { 7 \pi } { 12 }$

Verify that the equation is an identity. - $\sin \left( \frac { 3 \pi } { 2 } - \theta \right) = - \cos \theta$

Graph the inverse circular function. - $y=\tan ^{-1} x$