Exam 7: Trigonometric Identities and Equations

Solve the problem. -The figure shows a stationary spy satellite positioned 19,000 miles above the equator. What percent, to the nearest tenth, of the equator can be seen from the satellite? The diameter of Earth is 7927 miles at the equator.

Solve the problem. -When a person bends at the waist with a straight back, the force, $\mathrm { F }$ , exerted by the person's lower back muscles can be estimated with the formula $F = 2.89 w \sin \left( \theta + 90 ^ { \circ } \right)$ , where $w$ is the person weight, and $\theta$ is the angle made by the person's torso relative to the horizontal. Suppose that a 200 -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 346 pounds. Round the force to the nearest pound, and the angle to the nearest tenth, if necessary.

Use a calculator to give the value to the nearest degree. - $\theta = \cos ^ { - 1 } ( - 0.3907 )$

Determine all solutions of the equation in radians. -Find $\sin \frac { \theta } { 2 }$ , given that $\sin \theta = - \frac { 3 } { 5 }$ and $\theta$ terminates in $270 ^ { \circ } < \theta < 360 ^ { \circ }$ .

Find the exact value of the expression using the provided information. -Find $\cos ( \mathrm { s } + \mathrm { t } )$ given that $\cos s = \frac { 1 } { 3 }$ , with $\mathrm { s }$ in quadrant $\mathrm { I }$ , and $\sin t = \frac { 1 } { 4 }$ , with $\mathrm { t }$ in quadrant II.

Use the fundamental identities to find the value of the trigonometric function. -Find $\cos \theta$ if $\sin \theta = - \frac { 5 } { 13 }$ and $\theta$ is in quadrant III.

Write the following as an algebraic expression in u, u > 0. - $\tan \left( \cos ^ { - 1 } \frac { \mathrm { u } } { 3 } \right)$

Give the degree measure of θ. - $\theta = \arcsin \left( - \frac { 1 } { 2 } \right)$

Use a calculator to find the value. Give answers as real numbers and round to 4 decimal places, if necessary. - $\sin \left( \cos ^ { - 1 } 0.8324 \right)$

Find the exact value of the expression using the provided information. -Find $\sin ( s - t )$ given that $\cos s = \frac { 1 } { 3 }$ , with $s$ in quadrant $\mathrm { I }$ , and $\sin t = \frac { 1 } { 4 }$ , with $\mathrm { t }$ in quadrant II.

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

Write the expression in terms of sine and cosine, and simplify so that no quotients appear in the final expression. - $\tan x ( \cot x - \cos x )$

Use identities to write the expression as a single function of x or θ. - $\sin \left( x - \frac { \pi } { 2 } \right)$

Write the product as a sum or difference of trigonometric functions. - $2 \cos 7 x \cos 4 x$

Write the following as an algebraic expression in u, u > 0. - $\cos ( \arcsin u )$

Perform the indicated operations and simplify the result so there are no quotients. - $\frac { ( \sin \theta + \cos \theta ) ^ { 2 } } { 1 + 2 \sin \theta \cos \theta }$

Use an identity to write the expression as a single trigonometric function or as a single number. - $\frac { \sin 76 ^ { \circ } } { 1 - \cos 76 ^ { \circ } }$

Verify that each equation is an identity. - $\csc ^ { 2 } t - \cos t \sec t = \cot ^ { 2 } t$

Use an identity to write the expression as a single trigonometric function or as a single number. - $2 \cos ^ { 2 } 22.5 ^ { \circ } - 1$

Solve the problem. -In an electric circuit, the electromotive force is defined as $\mathrm { V } = \cos 2 \pi \mathrm { t }$ , where $\mathrm { t }$ is time in seconds. Solve the equation for $\mathrm { t }$ .