Exam 5: Analytic Trigonometry

Solve. -A ranger in fire tower A spots a fire at a direction of $40 ^ { \circ }$ . A ranger in fire tower $\mathrm { B }$ , which is 28 miles directly east of tower $\mathrm { A }$ , spots the same fire at a direction of $116 ^ { \circ }$ . How far from tower $\mathrm { A }$ is the fire?

Simplify the expression. - $\frac { \cos ^ { 2 } x + \sin ^ { 2 } x } { \cot ^ { 2 } x - \csc ^ { 2 } x }$

Solve. -To find the distance $\mathrm { AB }$ across a river, a distance $\mathrm { BC }$ of $786 \mathrm {~m}$ is laid off on one side of the river. It is found that $\mathrm { B }$ $= 110.2 ^ { \circ }$ and $\mathrm { C } = 13.5 ^ { \circ }$ . Find $\mathrm { AB }$ .

Prove the identity. - $1 + \sec ^ { 2 } x \sin ^ { 2 } x = \sec ^ { 2 } x$

Given the ASA parts of a triangle, is it better to use the law of sines or the law of cosines as the first step in solving the triangle? Explain.

Find all solutions in the interval [0, 2π). - $\sin ^ { 2 } \left( \frac { x } { 2 } \right) = \sin ^ { 2 } x$

State whether the given measurements determine zero, one, or two triangles. - $\mathrm { C } = 41 ^ { \circ } , \mathrm { a } = 16 , \mathrm { c } = 12$

Simplify the expression to either 1 or -1. - $- \sin ^ { 2 } x - \cos ^ { 2 } x$

Solve the problem. -Two boats leave a dock together, each traveling in a straight line. One boat travels at 17 mph and the other at 31 mph. If the angle between their courses measures $64.9 ^ { \circ } ,$ how far apart are they after 33 minutes? Give your Answer in miles and round your answer to the nearest tenth.

Solve the triangle. - $\mathrm { A } = 53 ^ { \circ } , \mathrm { b } = 10 , \mathrm { c } = 8$

Solve the problem. -To find the distance between two small towns, an electronic distance measuring (EDM) instrument is placed on a hill from which both towns are visible. If the distance from the EDM to the towns is 4.7 miles and 2.4 miles and The angle between the two lines of sight is $41 ^ { \circ }$ what is the distance between the towns? Round your answer to The nearest tenth of a mile.

Solve the problem. -At a certain baseball diamond, the distance from home plate through the pitcher's rubber to the far wall (i.e., to dead center) is 363 ft. Given that the infield forms a square 90 ft on each side, how far is it from third base to Dead center. Round your answer to the nearest tenth.

Determine if the following is an identity. - $\tan ^ { 2 } x = \sec ^ { 2 } x - \sin ^ { 2 } x - \cos ^ { 2 } x$

Simplify the expression. - $\frac { \cot x } { \cos x } - \csc x$

Two triangles can be formed using the given measurements. Solve both triangles. - $\mathrm { C } = 69 ^ { \circ } , \mathrm { a } = 19 , \mathrm { c } = 18$

Find all solutions in the interval [0, 2π). - $\sin \left( \frac { x } { 2 } \right) = 2 \cos ^ { 2 } x - 1$

Determine if the following is an identity. - $1 + \sec ^ { 2 } x \sin ^ { 2 } x = \sec ^ { 2 } x$

Write the expression as the sine, cosine, or tangent of an angle. - $\frac { \tan ( \pi / 5 ) + \tan ( \pi / 3 ) } { 1 - \tan ( \pi / 5 ) \tan ( \pi / 3 ) }$

Use basic identities to simplify the expression. - $\sin ^ { 2 } \theta + \tan ^ { 2 } \theta + \cos ^ { 2 } \theta$

Prove the identity. - $\sin 4 x = ( 4 \sin x \cos x ) \left( 1 - 2 \sin ^ { 2 } x \right)$