Exam 6: Additional Topics in Trigonometry

Given vectors $\mathbf { u } = \langle - 2 , - 3 \rangle$ and $\mathbf { v } = \langle 4 , - 4 \rangle$ , determine the quantity indicated below. $- 4 \mathbf { u } \cdot 3 \mathbf { v }$

Use DeMoivre's Theorem to find the indicated power of the following complex number. $( 3 + 3 \sqrt { 3 } i ) ^ { 6 }$

Find the component form of $\mathbf { v }$ if $\| \mathbf { v } \| = 8$ and the angle it makes with the $x$ -axis is $120 ^ { \circ }$ .

In the figure below, $a = 8 , b = 12$ , and $d = 13$ . Use this information to solve the parallelogram for $\beta$ . The diagonals of the parallelogram are represented by $c$ and $d$ . Round answer to two decimal places.

Find the trigonometric form of the complex number shown below. $8 i$

Find the component form of vector $v$ .

Write the complex number shown below in trigonometric form.

Which of the following is a fourth root of $1 ?$ I. $\cos \left( \frac { 3 \pi } { 2 } \right) + i \sin \left( \frac { 3 \pi } { 2 } \right)$ II. $\frac { 1 } { 4 } ( \cos ( 0 ) + i \sin ( 0 ) )$ III. $\cos \left( \frac { \pi } { 4 } \right) + i \sin \left( \frac { \pi } { 4 } \right)$

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

Determine whether $\mathbf { u }$ and $\mathbf { v }$ are orthogonal, parallel, or neither. $\mathbf { u } = \langle - 7,3 \rangle , \mathbf { v } = \langle 12,35 \rangle$

Determine whether $\mathbf { u }$ and $\mathbf { v }$ are orthogonal, parallel, or neither. $\mathbf { u } = \langle 1,5 \rangle , \mathbf { v } = \langle - 25,2 \rangle$

A park ranger at point $A$ observes a fire in the direction $\mathrm { N } 27 ^ { \circ } 27 ^ { \prime } \mathrm { E }$ . Another ranger at point $B , 9$ miles due east of $A$ , sites the same fire at $\mathrm { N } _ { 5 } 4 ^ { \circ } 28 ^ { \prime } \mathrm { W }$ . Determine the distance from point $B$ to the fire. Round answer to two decimal places.

Perform the operation below and leave the result in trigonometric form. $\frac { 17 \left( \cos 59 ^ { \circ } + i \sin 59 ^ { \circ } \right) } { 21 \left( \cos 12 ^ { \circ } + i \sin 12 ^ { \circ } \right) }$

If $\mathbf { u } = - \mathbf { i } - 8 \mathbf { j }$ and $\mathbf { v } = - 4 \mathbf { i } + \mathbf { j }$ , find $\mathbf { w } = - 3 \mathbf { u } - 4 \mathbf { v }$ .

Find the absolute value of the complex number $- 2 - 6 i$ .

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. $\mathbf { u } \cdot 4 \mathbf { v }$

Using the figure below, sketch a graph of the given vector. [The graphs in the answer choices are drawn to the same scale as the graph below.] $2 \mathbf { u } + \mathbf { v }$

Use Heron's area formula to find the area of the triangle pictured below, if $a = 10$ inches, $\mathrm { b } = 12$ inches, and $\mathrm { c } = 4$ inches.

Determine the area of a triangle having the following measurements. Round your answer to two decimal places. $B = 76 ^ { \circ } 30 ^ { \prime } , a = 8$ , and $c = 13$

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 $6 ^ { \circ }$ from vertical toward the house. From the porch, which is 110 feet away from the base of the tree, they noticed that the angle of elevation to the top of the tree was $27 ^ { \circ }$ . Approximate the height of the tree. Round answer to two decimal places.