Exam 6: Applications of Trigonometry

Solve the problem. -A rectangle has its center at the origin. It has two sides of length a which are parallel to the horizontal axis and two sides of length 5 a parallel to the vertical axis. Find polar coordinates of the vertices.

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. - $\langle - 7 , - 6 \rangle$

With your calculator set to radian mode and polar graphics capability, graph the following function in the window specified. - $r ^ { 2 } = \sin 3 \theta , 0 \leq \theta \leq 2 \pi , [ - 2,2 ]$ by $[ - 2,2 ]$

Solve the problem. -What is the maximum $r$ -value for the polar equation $r = 5 \cos 2 \theta$ ? Note that a maximum $r$ -value is a maximum value of $| \mathrm { r } |$ , the maximum distance from the pole.

Find the product or quotient, as indicated. Leave your answer in polar form. -Find the quotient. $\frac { \cos ( \pi / 3 ) + i \sin ( \pi / 3 ) } { \cos ( \pi / 6 ) + i \sin ( \pi / 6 ) }$

Use your grapher to determine which of the graphs matches the given polar equation. - $\mathrm { r } = 3 \sin 2 \theta$

Find the product or quotient. Write the answer in standard form. - $\frac { 4 + 4 i } { 5 + 7 i }$

Use the given graph to find the values of t that produce the graph in the given quadrant. - $x=4-|t|, y=t+1,-6 \leq t \leq 6$ Quadrant III

Write the complex number in the form a + bi. - $3 \left( \cos 270 ^ { \circ } + i \sin 270 ^ { \circ } \right)$

Eliminate the parameter. -x = 3t, y = t + 7

The polar coordinates of point P are given. Find all of its polar coordinates. - $\mathrm { P } = ( 2 , - \pi / 5 )$

Write the vector u as a sum of two orthogonal vectors, one of which is the vector projection of u onto v, $\operatorname{proj}_{\mathbf{v}} \mathbf{u}$ . - $\mathbf { u } = \langle 5 , - 5 \rangle , \mathbf { v } = \langle 6,8 \rangle$

Find the product or quotient. Write the answer in standard form. - $( 5 - 8 i ) ( 7 + 4 i )$

Solve the problem. -A constant force $\mathbf { F } = 49 \mathbf { i } + 31 \mathbf { j }$ moves an object along a vector $\mathbf { D } = 20 \mathbf { i } - 14 \mathrm { j }$ , where units are in pounds and feet. Find the work done.

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

Determine whether each statement below is true or false. If either statement is false, provide a counterexample. - If two vectors are perpendicular, then their dot product must be zero. - If two vectors are orthogonal, then they must also be perpendicular.

With your calculator set to radian mode and polar graphics capability, graph the following function in the window specified. - $r = 1 + 2 \cos \theta , 0 \leq \theta \leq 2 \pi , [ - 4,4 ]$ by $[ - 4,4 ]$

Write the complex number in the form a + bi. - $\sqrt { 6 } \left( \cos 315 ^ { \circ } + i \sin 315 ^ { \circ } \right)$

Find the unit vector in the direction of the given vector. Write your answer in the indicated form. -Let $\mathbf { u } = \langle - 4,3 \rangle$ . Find the unit vector in the direction of $\mathbf { u }$ , and write your answer as a linear combination of the standard unit vectors $\mathrm { i }$ and $\mathbf { j }$ .

Use De Moivre's Theorem to find the indicated power of the complex number. Write your answer in standard form a + bi. - $( - \sqrt { 3 } + i ) ^ { 6 }$