Exam 6: Additional Topics in Trigonometry

Given $A = 52 ^ { \circ } , B = 79 ^ { \circ }$ , and $a = 5.1$ , use the Law of Sines to solve the triangle for the value of $b$ . Round answer to two decimal places.

Find the standard form of the complex number shown below. Round answers to two decimal places. $6 \left( \cos 190 ^ { \circ } + i \sin 190 ^ { \circ } \right)$

Use DeMoivre's Theorem to find $( 8 + 8 i ) ^ { 3 }$ . Write the result in standard form.

Determine the area of a triangle having the following measurements. Round your answer to two decimal places. $C = 59 ^ { \circ } 24 ^ { \prime } , a = 6 , \text { and } b = 14.5$

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

A force of 45 pounds is exerted along a rope attached to a crate at an angle of 30° above the horizontal. The crate is moved 50 feet. How much work has been Accomplished? Round answer to one decimal place.

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° 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°. Approximate the height of the tree. Round answer to two decimal places.

Given $C = 130 ^ { \circ } , a = 16.9$ , and $c = 12.3$ , 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.

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

Given $a = 7 , b = 5$ , and $c = 8$ , use the Law of Cosines to solve the triangle for the value of $B$ . Round answer to two decimal places. Figure not drawn to scale

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

A triangular parcel of land has sides of lengths 100,580 , and 600 feet. Approximate the area of the land. Round answer to nearest foot.

Use DeMoivre's Theorem to find the indicated power of the following complex number. $( 5 + 5 \sqrt { 3 } i ) ^ { 6 }$

Find the vector $\mathbf { v }$ that has a magnitude of 6 and is in the same direction as $\mathbf { u }$ , where $\mathbf { u } = \langle - 6 , - 5 \rangle$

Perform the operation below and leave the result in trigonometric form. $\frac { 4 \left( \cos 58 ^ { \circ } + i \sin 58 ^ { \circ } \right) } { 12 \left( \cos 13 ^ { \circ } + i \sin 13 ^ { \circ } \right) }$

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

Given vectors $\mathbf { u } = \langle - 2,5 \rangle$ and $\mathbf { v } = \langle 2 , - 4 \rangle$ , determine the quantity indicated below. $\mathbf { u } \cdot - \mathbf { v }$

Determine the area of a triangle having the following measurements. Round your answer to two decimal places. 128Ab=°, =8, and c=15

Use vectors to find the measure of the angle at vertex $B$ of triangle $A B C$ , when $A = ( 6,1 ) , B = ( - 1,3 )$ , and $C = ( - 4 , - 5 )$ . Round answer to two decimal places.

In the figure below, $( a , b ) = ( 2,7 )$ and $( c , d ) = ( 7,5 ) .$ Find $( x , y )$ so that $\mathbf { u } = \mathbf { v }$