Exam 6: Applications of Trigonometry

Find the unit vector in the direction of the given vector. Write your answer in the indicated form. -Let $\mathbf { u } = \langle 5,3 \rangle$ . Find the unit vector in the direction of $\mathbf { u }$ , and write your answer as a linear combination of the standard unit vectors $i$ and $\mathbf { j }$ .

Find the product or quotient, as indicated. Leave your answer in polar form. -Find the product of $\mathrm { z } _ { 1 }$ and $\mathrm { z } _ { 2 }$ . $\mathrm { z } _ { 1 } = 8 \left( \cos 35 ^ { \circ } + \mathrm { i } \sin 35 ^ { \circ } \right) , \mathrm { z } _ { 2 } = 4 \left( \cos 125 ^ { \circ } + \mathrm { i } \sin 125 ^ { \circ } \right)$

Solve the problem using a graphing calculator. -Determine which will travel farther: baseball x hit 95 feet per second at an angle of $35 ^ { \circ }$ relative to level ground or baseball y hit 125 feet per second at an angle of $30 ^ { \circ }$ .

Find the rectangular coordinates of the point with the given polar coordinates. - $\left( 9,0 ^ { \circ } \right)$

Choose the graph of the given parametric equations. - $x=5 \cos t, y=3 \sin t, 0 \leq t \leq \pi$

Find the component form and magnitude of the indicated vector. -Given that $\mathrm { P } = ( - 1,7 )$ and $\mathrm { Q } = ( - 2,10 )$ , find the component form and magnitude of the vector $\overrightarrow { \mathrm { PQ } }$ .

Solve the problem. -For what values of $\theta ( 0 \leq \theta < 2 \pi )$ do maximum r-values occur on the graph of the polar equation $r = 1 + 5 \cos 2 \theta$ ? Note that a maximum $r$ -value occurs at a point that is the maximum distance from the pole.

Determine whether the vectors u and v are parallel, orthogonal, or neither. - $\mathbf { u } = \langle 8,2 \rangle , \mathbf { v } = \langle 12 , - 48 \rangle$

Use De Moivre's Theorem to find the indicated power of the complex number. Write your answer in standard form a + bi. - $( 2 ( \cos \pi / 2 + i \sin \pi / 2 ) ) ^ { 5 }$

Use the dot product to find $|\mathbf{v}|$ - $\mathbf { v } = \langle - 2 , - 1 \rangle$

The graph of a limacon curve is given. Without using your graphing calculator, determine which equation is correct for the graph. - $[ - 5,5 ]$ by $[ - 5,5 ]$

Plot the point with the given polar coordinates. - $\left( 2 , - \frac { \pi } { 4 } \right)$

Find the component form of the vector v. -

Express the complex number in polar form. - $- 4 + 4 \sqrt { 3 } i$

Find the component form of the vector v. -

Find the product or quotient, as indicated. Leave your answer in polar form. -Find the product of $\mathrm { z } _ { 1 }$ and $\mathrm { z } _ { 2 }$ . $z _ { 1 } = 7 \left( \cos \frac { 5 \pi } { 4 } + i \sin \frac { 5 \pi } { 4 } \right) , z _ { 2 } = 5 \left( \cos \frac { 2 \pi } { 3 } + i \sin \frac { 2 \pi } { 3 } \right)$

Find the magnitude and direction angle for the following vector. Give the direction angle as an angle in [0°, 360°) rounded to the nearest tenth. - $\langle 14 \sqrt { 3 } , - 14 \rangle$

Find the product or quotient, as indicated. Leave your answer in polar form. -Find the product of $z _ { 1 }$ and $z _ { 2 }$ . $z _ { 1 } = 5 \left[ \cos \left( 75 ^ { \circ } \right) + i \sin \left( 75 ^ { \circ } \right) \right] , z _ { 2 } = \sqrt { 3 } \left( \cos - 135 ^ { \circ } + i \sin - 135 ^ { \circ } \right)$

Find the magnitude and direction angle for the following vector. Give the direction angle as an angle in [0°, 360°) rounded to the nearest tenth. - $3 \left( \cos 102 ^ { \circ } \mathbf { i } + \sin 102 ^ { \circ } \mathbf { j } \right)$

Find the indicated roots. Write the answer in polar form. -Fourth roots of $256 \left( \cos 280 ^ { \circ } + i \sin 280 ^ { \circ } \right)$