Exam 6: Additional Topics in Trigonometry

Which of the following is a fifth root of $2 ( - 1 + i ) ? \quad 0 \quad 2$ I. $\sqrt [ 10 ] { 8 } \left( \cos \left( \frac { 3 \pi } { 20 } \right) + i \sin \left( \frac { 3 \pi } { 20 } \right) \right)$ II. $\sqrt [ 10 ] { 8 } \left( \cos \left( \frac { 19 \pi } { 20 } \right) + i \sin \left( \frac { 19 \pi } { 20 } \right) \right)$ III. $\sqrt [ 5 ] { 2 } \left( \cos \left( \frac { 3 \pi } { 20 } \right) + i \sin \left( \frac { 3 \pi } { 20 } \right) \right)$

Perform the indicated operation using trigonometric form. Leave answer in trigonometric form. $( 3 + 3 i ) ( 5 + 5 i )$

Given $a = 6 , b = 9$ , and $c = 7$ , use the Law of Cosines to solve the triangle for the value of $A$ . Round answer to two decimal places.

If $\| \mathbf { u } \| = 5$ and $\| \mathbf { v } \| = 3$ , and the vectors make angles of $150 ^ { \circ }$ and $110 ^ { \circ }$ with the $x$ -axis respectively, find the component form of the sum of $\mathbf { u }$ and $\mathbf { v }$ . Round answers to two decimal places.

In the figure below, $( a , b ) = ( 4,6 )$ and $( c , d ) = ( 6,5 )$ . Find $( x , y )$ so that $\mathbf { u } = \mathbf { v }$ .

Find the fourth roots of $- \frac { 1 } { 2 } + \frac { \sqrt { 3 } } { 2 } i$ . Write the roots in trigonometric form.

Find the standard form of the complex number shown below. Round answers to two decimal places. $2 \left( \cos 340 ^ { \circ } + i \sin 340 ^ { \circ } \right)$

Find the projection of $\mathbf { u }$ onto $\mathbf { v }$ if $\mathbf { u } = \langle 5 , - 4 \rangle , \mathbf { v } = \langle - 2,3 \rangle$ .

A straight road makes an angle, $A$ , of $16 ^ { \circ }$ with the horizontal. When the angle of elevation, $B$ , of the sun is $58 ^ { \circ }$ , a vertical pole beside the road casts a shadow 6 feet long parallel to the road. Approximate the length of the pole. Round answer to two decimal places.

Given $a = 5 , b = 9$ , and $c = 12$ , use Heron's Area Formula to find the area of triangle $A B C$ . Round answer to two decimal places.

Determine whether $\mathbf { u }$ are $\mathbf { v }$ and orthogonal, parallel, or neither. $\mathbf { u } = \langle 6,4 \rangle , \mathbf { v } = \langle - 20,30 \rangle$

After a severe storm, three sisters, April, May, and June, stood on their front porch and noticed that the tree in their front yard was leaning $5 ^ { \circ }$ from vertical toward the house. From the porch, which is 100 feet away from the base of the tree, they noticed that the angle of elevation to the top of the tree was $29 ^ { \circ }$ . Approximate the height of the tree. Round answer to two decimal places.

Perform the indicated operation using trigonometric form. Leave answer in trigonometric form. $( 8 + 8 i ) ( 2 + 2 i )$

Given $A = 54 ^ { \circ } , B = 69 ^ { \circ }$ , and $a = 7.1$ , use the Law of Sines to solve the triangle for the value of $b$ . Round answer to two decimal places.

In the following triangle, $A = 58 ^ { \circ } , b = 105$ , and $a = 126$ . Use the Law of Sines to find the measure of angle $C$ , in degrees. Round your answer to two decimals.

Given $\mathbf { u } = - 3 \mathbf { i } - 2 \mathbf { j }$ and $\mathbf { v } = - \mathbf { i } - 5 \mathbf { j }$ , determine $- 9 \mathbf { u } + 6 \mathbf { v }$ .

Find the trigonometric form of the complex number shown below. $- 6 i$

Given vectors $\mathbf { u } = \left\langle u _ { 1 } , u _ { 2 } \right\rangle , \mathbf { v } = \left\langle v _ { 1 } , v _ { 2 } \right\rangle$ , and $\mathbf { w } = \left\langle w _ { 1 } , w _ { 2 } \right\rangle$ , determine whether the result of the following expression is a vector or a scalar. $( w \cdot w ) w$

Let $\mathbf { u } = \langle 6,5 \rangle$ and $\mathbf { v } = \langle 12,7 \rangle$ . Find $\mathbf { u } - \mathbf { v }$ .

Given $C = 126 ^ { \circ } , a = 10.10$ , and $c = 18$ , use the Law of Sines to solve the triangle (if possible) for the value of $b$ . If two solutions exist, find both. Round answer to two decimal places.