Exam 6: Additional Topics in Trigonometry

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.

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

Given A =°16 , b =12 , and a = 10 , use the Law of Sines to solve the triangle (if possible) for the value of c. If two solutions exist, find both. Round answer to two Decimal places.

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 magnitude of vector $\mathbf { v }$ .

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. $( w \cdot w ) w$

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

Use DeMoivre's Theorem to find the indicated power of the folllowing complex number. $( - 6 + 6 i ) ^ { 4 }$

Perform the operation shown below and leave the result in trigonometric form. $\frac { 8 ( \cos 2.3 + i \sin 2.3 ) } { 4 ( \cos 1.7 + i \sin 1.7 ) }$

Find the trigonometric form of the complex number shown below. $- 3 i$

Use DeMoivre's Theorem to find the indicated power of the following complex number. $\left[ 2 \left( \cos \frac { \pi } { 3 } + i \sin \frac { \pi } { 3 } \right) \right] ^ { 12 }$

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

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.

Determine whether $\mathbf { u }$ and $\mathbf { v }$ are orthogonal, parallel, or neither. $\mathbf { u } = \langle 1 , - 5 \rangle , \mathbf { v } = \langle 20,3 \rangle$

Find the vector $\mathbf { v }$ that has a magnitude of 4 and is in the same direction as $\mathbf { u }$ , where $\mathbf { u } = \langle - 5,4 \rangle$ .

Use Heron's area formula to find the area of the triangle pictured below, if $a = 8$ inches, $\mathrm { b } = 10$ inches, and $\mathrm { c } = 4$ inches.

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

Find the vector v that has a magnitude of 4 and is in the same direction as u, where $\mathbf { u } = \langle - 5 , - 3 \rangle \text {. }$

A straight road makes an angle, $A$ , of $20 ^ { \circ }$ with the horizontal. When the angle of elevation, $B$ , of the sun is $60 ^ { \circ }$ , a vertical pole beside the road casts a shadow 8 feet long parallel to the road. Approximate the length of the pole. Round answer to two decimal places.

Given $\mathbf { u } = \langle - 3,7 \rangle$ and $\mathbf { v } = \langle 1 , - 5 \rangle$ , find $\mathbf { u } \cdot \mathbf { v }$ .