Exam 6: Additional Topics in Trigonometry

Find the angle between the vectors $\mathbf { u }$ and $\mathbf { v }$ . $\mathbf { u } = \cos \left( \frac { \pi } { 3 } \right) \mathbf { i } + \sin \left( \frac { \pi } { 3 } \right) \mathbf { j } , \quad \mathbf { v } = \cos \left( \frac { 5 \pi } { 4 } \right) \mathbf { i } + \sin \left( \frac { 5 \pi } { 4 } \right) \mathbf { j }$

Determine the area of a triangle having the following measurements. Round your answer to two decimal places. $A = 128 ^ { \circ } , b = 7 \text {, and } c = 14$

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.

Find the component form of vector $\mathbf { v }$ with initial point $( - 5,1 )$ and terminal point $( - 3 , - 2 )$ .

Use the figure to determine the tension in the cable CA supporting the load. Round your answer to two decimals.

Find the component form of vector $\mathbf { v }$ .

Given $a = 12 , b = 8$ , and $c = 11$ , use the Law of Cosines to solve the triangle for the value of $B$ . Round answer to two decimal places.

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

Find the component form of $\mathbf { v }$ if $\| \mathbf { v } \| = 4$ and the angle it makes with the $x$ -axis is $120 ^ { \circ }$ .

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

Determine the area of a triangle having the following measurements. Round your answer to two decimal places. $B = 75 ^ { \circ } 22 ^ { \prime } , a = 11 , \text { and } c = 14$

The vector $\mathbf { u } = \langle 3700,4600 \rangle$ gives the number of units of two models of laptops produced by a company. The vector $\mathbf { v } = \langle 1550,1150 \rangle$ gives the prices (in dollars) of the two models of laptops, respectively. Identify the vector operation used to increase revenue by $3 \%$ .

Determine whether $\mathbf { u }$ and $\mathbf { v }$ are orthogonal, parallel, or neither. $\mathbf { u } = \langle 1,5 \rangle , \mathbf { v } = \langle - 30 , - 5 \rangle$

Find the component form of vector $\mathbf { v }$ .

Given $a = 12 , b = 5$ , and $c = 15$ , use the Law of Cosines to solve the triangle for the value of $C$ . Round answer to two decimal places.

A plane flies 440 kilometers with a bearing of 320° from City A to City B (see figure below). The plane then flies 800 kilometers from City B to City C, which is directly West of City A. Find the bearing of the flight from City B to City C. Round your Answer to two decimals.

Plot the following complex number. $3 i$

Given $a = 3 , b = 8$ , and $c = 6$ , use the Law of Cosines to solve the triangle for the value of $C$ . Round answer to two decimal places.

Perform the operation shown below and leave the result in trigonometric form. $\left[ 6 \left( \cos 90 ^ { \circ } + i \sin 90 ^ { \circ } \right) \right] \left[ 2 \left( \cos 280 ^ { \circ } + i \sin 280 ^ { \circ } \right) \right]$

Find the projection of $\mathbf { u }$ onto $\mathbf { v }$ if $\mathbf { u } = \langle 2 , - 1 \rangle , \mathbf { v } = \langle 5,2 \rangle$ .