Exam 6: Analytic Trigonometry

Solve the problem. -The altitude of a projectile in feet (neglecting air resistance) is given by $y = ( \tan \theta ) x - \frac { 16 } { v ^ { 2 } \cos ^ { 2 } \theta } x ^ { 2 }$ where $x$ is the horizontal distance covered in feet and $v$ is the initial velocity of the projectile at an angle $\theta$ from $t$ ] horizontal. Find the firing angle (in degrees) of a projectile fired at an initial velocity of 100 feet per second so that it strikes the ground $312.5$ feet from the firing point.

Complete the identity. - $\tan \alpha + \cot \beta = ?$

Find the exact value of the expression. - $\sin 15 ^ { \circ }$

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

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

Solve the problem. -If $\cos \theta = - \frac { 5 } { 13 }$ , and $\frac { \pi } { 2 } < \theta < \pi$ , then find $\tan 2 \theta$ .

Find the exact value of the composition. - $\sin \left( \tan ^ { - 1 } 2 \right)$

Write the word or phrase that best completes each statement or answers the question. -The average daily temperature $\mathrm { T }$ of a city in the United States is approximated by $\mathrm { T } = 55 - 23 \cos \frac { 2 \pi } { 365 } ( \mathrm { t } - 30 )$ where $t$ is in days, $1 \leq t \leq 365$ , and $t = 1$ corresponds to January 1 . For what range of values of $t$ is the average dail temperature above $70 ^ { \circ } \mathrm { F }$ ? Use a calculator and round answers to the nearest whole number.

Simplify the trigonometric expression by following the indicated direction. -Factor and simplify: $\frac { 8 \sin ^ { 2 } \theta + 9 \sin \theta + 1 } { \sin ^ { 2 } \theta - 1 }$

Find the exact value of the expression. Do not use a calculator. - $\tan \left[ \tan ^ { - 1 } ( - 0.4 ) \right]$

Find the exact value of the composition. - $\sin \left( \tan ^ { - 1 } ( 2 ) \right)$

Choose the one alternative that best completes the statement or answers the question. - $\sin ^ { 2 } x - 8 \sin x + 16 = 0$

Express the sum or difference as a product of sines and/or cosines. - $\sin ( 2 \theta ) - \sin ( 4 \theta )$

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

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

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

Write the trigonometric expression as an algebraic expression in u. - $\cos \left( \sin ^ { - 1 } \mathrm { u } \right)$

Find the exact value under the given conditions. - $\sin \alpha = \frac { 7 } { 25 } , 0 < \alpha < \frac { \pi } { 2 } ; \quad \cos \beta = \frac { 20 } { 29 } , 0 < \beta < \frac { \pi } { 2 } \quad$ Find $\cos ( \alpha + \beta )$ .

Write the word or phrase that best completes each statement or answers the question. Establish the identity. - $\csc ^ { 3 } x \tan ^ { 2 } x = \csc x \left( 1 + \tan ^ { 2 } x \right)$

Solve the equation. Give a general formula for all the solutions. - $\sin \theta = \frac { \sqrt { 3 } } { 2 }$