Exam 6: Analytic Trigonometry

Solve the problem. -A weight suspended from a spring is vibrating vertically with up being the positive direction. The function $f ( t ) = 10 \sin \left( \frac { 3 \pi t } { 4 } - \frac { \pi } { 4 } \right)$ represents the distance in centimeters of the weight from its rest position as a function of time t, where t is measured in seconds. Find the smallest positive value of t for which the displacement of the weight above its rest position is 5 cm. Round answer to three decimal places, if necessary.

Choose the one alternative that best completes the statement or answers the question. Complete the identity. - $\tan ( \pi - \theta ) =$ ?

Write the word or phrase that best completes each statement or answers the question. Establish the identity. - $\frac { \cot x } { 1 + \csc x } = \frac { \csc x - 1 } { \cot x }$

Write the word or phrase that best completes each statement or answers the question. Establish the identity. - $\sec \left( \frac { \pi } { 2 } + u \right) = - \csc u$

Find the exact value under the given conditions. - $\sin \alpha = - \frac { 5 } { 13 } , \frac { 3 \pi } { 2 } < \alpha < 2 \pi ; \quad \tan \beta = - \frac { 7 } { 24 } , \frac { \pi } { 2 } < \beta < \pi \quad$ Find $\cos ( \alpha + \beta )$ .

Choose the one alternative that best completes the statement or answers the question. Solve the problem. -Find $\tan \frac { \theta } { 2 }$ , given that $\tan \theta = 3$ and $\theta$ terminates in $\pi < \theta < 3 \pi / 2$ .

Write the word or phrase that best completes each statement or answers the question. Establish the identity. - $\frac { 1 - \sin t } { \cos t } = \frac { \cos t } { 1 + \sin t }$

Write the word or phrase that best completes each statement or answers the question. Establish the identity. - $\frac { 1 - 2 \sec x - 3 \sec ^ { 2 } x } { - \tan ^ { 2 } x } = \frac { 1 - 3 \sec x } { 1 - \sec x }$

Write the trigonometric expression as an algebraic expression containing u and v. - $\cos \left( \sin ^ { - 1 } u + \cos ^ { - 1 } v \right)$

Find the exact solution of the equation. - $2 \cos ^ { - 1 } x = \pi$

Express the sum or difference as a product of sines and/or cosines. - $\sin ( 8 \theta ) - \sin ( 4 \theta )$

Write the trigonometric expression as an algebraic expression containing u and v. - $\sin \left( \tan ^ { - 1 } u + \tan ^ { - 1 } v \right)$

Choose the one alternative that best completes the statement or answers the question. Solve the problem. -Find $\sin \frac { \theta } { 2 }$ , given that $\cos \theta = \frac { 1 } { 4 }$ and $\theta$ terminates in $0 < \theta < 90 ^ { \circ }$ .

Write the word or phrase that best completes each statement or answers the question. Use a calculator to solve the equation on the interval 0 . Round the answer to one decimal place if necessary. $0 \leq x < 2 \pi$ - $x + 3 \sin x = 1$

Find the exact value of the expression. - $\cos \left[ 2 \sin ^ { - 1 } \left( - \frac { 5 } { 13 } \right) \right]$

Solve the problem. -If $\cos \theta = - \frac { 5 } { 13 }$ , and $\theta$ terminates in quadrant II, then find $\cos 2 \theta$ .

Use the Half-angle Formulas to find the exact value of the trigonometric function. - $\sin 165 ^ { \circ }$

Solve the problem. -If $\tan \theta = \frac { 7 } { 24 }$ , and $\pi < \theta < \frac { 3 \pi } { 2 }$ , then find $\tan 2 \theta$ .

Choose the one alternative that best completes the statement or answers the question. Solve the problem. -Find $\sin \frac { \theta } { 2 }$ , given that $\sin \theta = - \frac { 3 } { 5 }$ and $\theta$ terminates in $270 ^ { \circ } < \theta < 360 ^ { \circ }$ .

Solve the problem using Snell's Law: $\frac { \sin \theta _ { 1 } } { \sin \theta _ { 2 } } = \frac { v _ { 1 } } { v _ { 2 } }$ -A light beam in air travels at $2.99 \times 10 ^ { 8 }$ meters per second. If its angle of incidence to a second medium is $42 ^ { \circ }$ and its angle of refraction in the second medium is $32 ^ { \circ }$ , what is its speed in the second medium (to two decimal places)?