Exam 6: Analytic Trigonometry

Write the word or phrase that best completes each statement or answers the question. Establish the identity. - $\frac { \sin x } { 1 - \cos x } + \frac { \sin x } { 1 + \cos x } = 2 \csc x$

Choose the one alternative that best completes the statement or answers the question. Express the product as a sum containing only sines or cosines. - $\sin \frac { \theta } { 2 } \cos \frac { 5 \theta } { 2 }$

Complete the identity. - $\frac { \sin \alpha + \sin \beta } { \cos \alpha + \cos \beta } = ?$

Solve the equation. Give a general formula for all the solutions. - $\cos \theta - 1 = 0$

Choose the one alternative that best completes the statement or answers the question. Complete the identity. - $\cos ( \alpha + \beta ) \cos ( \alpha - \beta ) =$ ?

Find the exact value of the expression. - $\tan ^ { - 1 } \frac { \sqrt { 3 } } { 3 }$

Choose the one alternative that best completes the statement or answers the question. Complete the identity. - $\tan ^ { 4 } \theta - \sec ^ { 4 } \theta =$ ?

Complete the identity. - $\frac { \sin ( 4 \theta ) + \sin ( 8 \theta ) } { \cos ( 4 \theta ) + \cos ( 8 \theta ) } = ?$

Find the exact value of the expression. - $\tan \left( \tan ^ { - 1 } \frac { 3 } { 4 } + \sin ^ { - 1 } \frac { 1 } { 2 } \right)$

Choose the one alternative that best completes the statement or answers the question. Express the product as a sum containing only sines or cosines. - $\sin ( 4 \theta ) \sin ( 6 \theta )$

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

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

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

Write the word or phrase that best completes each statement or answers the question. - $\tan ^ { 2 } u ( 1 + \cos 2 u ) = 1 - \cos 2 u$

Choose the one alternative that best completes the statement or answers the question. Complete the identity. - $\sin \theta [ \sin ( 2 \theta ) + \sin ( 4 \theta ) ] = ?$

Find the exact value of the expression. - $\tan \frac { 5 \pi } { 12 }$

Choose the one alternative that best completes the statement or answers the question. Complete the identity. - $\frac { 1 } { \cot ^ { 2 } \theta } + \sec \theta \cos \theta = \text { ? }$

Write the word or phrase that best completes each statement or answers the question. -The path of a projectile fired at an inclination θ (in degrees) to the horizontal with an initial velocity v0 is a parabola. The range R of the projectile, that is, the horizontal distance that the projectile travels, is found by using the formula $\mathrm { R } = \frac { \mathrm { v } _ { 0 } ^ { 2 } } { \mathrm {~g} } \sin ( 2 \theta )$ s where $g$ is the acceleration due to gravity. Suppose the projectile is fired with an initial velocity of 400 feet per se and $g = 32$ feet per second 2 . What angle $\theta , 0 ^ { \circ } \leq \theta < 90 ^ { \circ }$ , would you select for the range to be 2500 feet? (There should be two values of $\theta . )$

Write the word or phrase that best completes each statement or answers the question. Establish the identity. - $\csc ^ { 4 } x - \cot ^ { 4 } x = \csc ^ { 2 } x + \cot ^ { 2 } x$

Choose the one alternative that best completes the statement or answers the question. Find the exact value of the expression. - $\sec \left[ \sin ^ { - 1 } \left( - \frac { \sqrt { 3 } } { 2 } \right) \right]$