Exam 6: Additional Topics in Trigonometry

A park ranger at point $A$ observes a fire in the direction $\mathrm { N } \mathrm {} 29 ^ { \circ } 15 ^ { \prime } \mathrm { E }$ . Another ranger at point $B , 9$ miles due east of $A$ , sites the same fire at $\mathrm { N } 50 ^ { \circ } 39 ^ { \prime } \mathrm { W }$ . Determine the distance from point $B$ to the fire. Round answer to two decimal places.

Let $\mathbf { u } = \langle 6,8 \rangle$ and $\mathbf { v } = \langle 14,10 \rangle$ . Find $\mathbf { u } - \mathbf { v }$

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

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.

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

Find the absolute value of the complex number $- 4 + 6 i$ .

Given vectors $\mathbf { u } = \langle - 3,1 \rangle$ and $\mathbf { v } = \langle 2 , - 2 \rangle$ , determine the quantity indicated below. $\mathbf { u } \cdot - 4 \mathbf { v }$

Perform the operation shown below and leave the result in trigonometric form. $\frac { 36 ( \cos 2.5 + i \sin 2.5 ) } { 6 ( \cos 1.7 + i \sin 1.7 ) }$

Given $C = 130 ^ { \circ } , a = 19.9$ , and $c = 15.3$ , 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 vector $\mathbf { v }$ that has a magnitude of 4 and is in the same direction as $\mathbf { u }$ , where $\mathbf { u } = \langle - 5,4 \rangle$ .

Determine whether $\mathbf { u }$ and $\mathbf { v }$ are orthogonal, parallel, or neither. $\mathbf { u } = \left\langle \frac { 7 } { 3 } , \frac { - 1 } { 2 } \right\rangle , \mathbf { v } = \langle 14 , - 3 \rangle$

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. $\mathbf { u } \cdot \mathbf { w } + \mathbf { v } \cdot \mathbf { w }$

In the figure below, $a = 7 , c = 20$ , and $d = 14$ . Use this information to solve the parallelogram for $b$ . The diagonals of the parallelogram are represented by $c$ and $d$ . Round answer to two decimal places.

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.

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.

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

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

A plane flies 440 kilometers with a bearing of $320 ^ { \circ }$ 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.

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

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.