Exam 6: Applications of Trigonometry

Solve the problem using a graphing calculator. -Estimate the maximum height reached by a baseball during its flight if it is thrown with a velocity of 65 feet per second at an angle of $76 ^ { \circ }$ relative to level ground.

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

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

Find the rectangular coordinates of the point with the given polar coordinates. - $( \sqrt { 3 } , \pi / 6 )$

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 II

Find the angle between the given vectors to the nearest tenth of a degree. -

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

Solve the problem. -An airplane flies on a compass heading of $90.0 ^ { \circ }$ at $230 \mathrm { mph }$ . The wind affecting the plane is blowing from $338 ^ { \circ }$ at $23.0 \mathrm { mph }$ . What is the true course and ground speed of the airplane? Round results to an appropriate number of significant digits.

Find the product or quotient, as indicated. Leave your answer in polar form. -Find the quotient. $\frac { 9 ( \cos 7 \pi + i \sin 7 \pi ) } { 4 ( \cos 5 \pi + i \sin 5 \pi ) }$

Solve the problem using a graphing calculator. -Determine whether a baseball hit 133 feet per second at an angle of $30 ^ { \circ }$ relative to level ground will clear a 10-foot wall 400 feet away.

Write the vector u as a sum of two orthogonal vectors, one of which is the vector projection of u onto v, $\operatorname{proj}_{\mathbf{v}} \mathbf{u}$ . - $\mathbf { u } = \langle 9,8 \rangle , \mathbf { v } = \langle 1,2 \rangle$

Find an equivalent equation in polar coordinates. - $x y = 1$

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 $\mathrm { r } = 3 - 4 \sin 3 \theta$ ? Note that a maximum $\mathrm { r }$ -value occurs at a point that is the maximum distance from the pole.

Analyze the graph of the given polar curve. Include the following information: If possible, describe the shape of the graph (circle, rose curve, limacon, etc.), and state the domain, range, and maximum r-value of the graph. State whether the graph is continuous and whether it is bounded. Describe any symmetry that the graph has. Give the equations of any asymptotes or state that the graph has no asymptotes. - $\theta = \pi / 5$

Find the product or quotient. Write the answer in standard form. - $( 3 + 5 i ) ( 8 + 4 i )$

Find a· b. - $\mathbf { a } = 10 \mathbf { i } - 6 \mathbf { j } , \mathbf { b } = 5 \mathbf { i } + 2 \mathbf { j }$

Find the product or quotient, as indicated. Leave your answer in polar form. -Find the quotient. $\frac { 9 \left( \cos 125 ^ { \circ } + i \sin 125 ^ { \circ } \right) } { 5 \left( \cos 35 ^ { \circ } + i \sin 35 ^ { \circ } \right) }$

Write the complex number in the form a + bi. - $\left( \cos \frac { \pi } { 2 } + i \sin \frac { \pi } { 2 } \right)$

Solve the problem. -A constant force $F = 13 i + 26 j$ moves an object from the point $P = ( 5,2 )$ to the point $Q = ( 3,15 )$ , where units are in pounds and feet. Find the work done.

Let $\mathrm { z } _ { 1 } = \mathrm { r } _ { 1 } \left( \cos \theta _ { 1 } + \mathrm { i } \sin \theta _ { 1 } \right)$ and $\mathrm { z } _ { 2 } = \mathrm { r } _ { 2 } \left( \cos \theta _ { 2 } + \mathrm { i } \sin \theta _ { 2 } \right) , \mathrm { r } _ { 2 } \neq 0$ . Verify that $\mathrm { z } _ { 1 } / \mathrm { z } _ { 2 } = \mathrm { r } _ { 1 } / \mathrm { r } _ { 2 } \left[ \cos \left( \theta _ { 1 } - \theta _ { 2 } \right) + \mathrm { i } \sin \left( \theta _ { 1 } - \theta _ { 2 } \right) \right]$ .