Exam 7: Applications of Trigonometry

Divide the complex numbers below and leave the result in trigonometric form. $\frac { 2 \left( \cos 160 ^ { \circ } + i \sin 160 ^ { \circ } \right) } { 9 \left( \cos 290 ^ { \circ } + i \sin 290 ^ { \circ } \right) }$ ​

Two ocean liners leave from the same port in Puerto Rico at 10:00 a.m. One travels at a bearing of N 50°W at 12 miles per hour, and the other travels at a bearing of S 53°W at 14 miles per hour. Approximate the distance between them at noon the same day. Round your answer to two decimal places. ​

Use vectors to find the measure of the angle at vertex B of triangle ABC, when $A = ( 5,2 )$ , $B = ( - 4,1 )$ , and $C = ( - 1 , - 3 )$ . Round answer to two decimal places. ​

Find the fifth roots of the following complex number. Express the root(s) in trigonometric form. ​ $- \frac { 3125 } { 2 } + \frac { 3125 } { 2 } \sqrt { 3 } i$ ​

A 800-pound trailer is sitting on an exit ramp inclined at 33° on Highway 35. How much force is required to keep the trailer from rolling back down the exit ramp? Round your answer to two decimal places. ​

A straight road makes an angle, A, of 16° with the horizontal. When the angle of elevation, B, of the sun is 59°, a vertical pole beside the road casts a shadow 7 feet long parallel to the road, see figure. Approximate the length of the pole. Round answer to two decimal places.

Given a = 12, b = 11, and c = 10, use the Law of Cosines to solve the triangle for the value of B. Round your answer to two decimal places. ​

Use the vectors $\mathbf { u } = \langle 4,3 \rangle$ , $\mathbf { v } = \langle - 3,2 \rangle$ to find the indicated quantity. State whether the result is a vector or a scalar. ​ $( \mathbf { u } \cdot \mathbf { v } ) \mathbf { v }$ ​

Given A = 10 $\circ$ , b = 10 and c = 8, use the Law of Sines to solve the triangle (if possible) for the value of c. If two solutions exist, find both. Round answer to two decimal places.

Given A = 33°, b = 13, and c = 7, use the Law of Cosines to solve the triangle for the value of a. Round your answer to two decimal places. Figure not drawn to scale

Find the magnitude of vector v. ​

Find u + v. ​ $\mathbf { u } = \langle 5,1 \rangle , \mathbf { v } = \langle 1,6 \rangle$ ​

Use DeMoivre's theorem to find the indicated roots of the complex number. ​ Square roots of $11 \left( \cos 120 ^ { \circ } + i \sin 120 ^ { \circ } \right)$ . ​

Find the angle between the vectors u and v. $\mathbf { u } = \cos \left( \frac { \pi } { 3 } \right) \mathbf { i } + \sin \left( \frac { \pi } { 3 } \right) \mathbf { j }$ , $v = \cos \left( \frac { 3 \pi } { 4 } \right) i + \sin \left( \frac { 3 \pi } { 4 } \right) j$ ​

Use the law of Cosines to solve the given triangle. Round your answer to two decimal places. ​ a = 10, b = 5, C = 113° ​