Exam 6: Applications of Trigonometry

Solve the problem. -A plane is heading due south with an airspeed of $223 \mathrm { mph }$ . A wind from a direction of $60.0 ^ { \circ }$ is blowing at $18.0$ mph. Find the bearing of the plane. (Note that bearings are measured from north, clockwise.) Round results to an appropriate number of significant digits.

Find the angle between the given vectors to the nearest tenth of a degree. - $\mathbf { u } = \left( 3 \cos \frac { \pi } { 6 } \right) \mathbf { i } + \left( 3 \sin \frac { \pi } { 6 } \right) \mathbf { j } , \mathbf { v } = \left( \cos \frac { 4 \pi } { 3 } \right) \mathbf { i } + \left( \sin \frac { 4 \pi } { 3 } \right) \mathbf { j }$

Solve the problem using a graphing calculator. -Given that the parametric equations of flight on the moon are $x = ( v \cos \theta ) t$ and $y = ( v \sin \theta ) t - 2.66 t ^ { 2 }$ , determine the approximate distance that a baseball travels if it is thrown on the moon with a velocity of 91 feet per second at an angle of $23 ^ { \circ }$ relative to level ground.

Write the vector u as a sum of two orthogonal vectors, one of which is the vector projection of u onto v, $\operatorname{proj}_{\mathbf{v}} \mathbf{u}$ . - $\mathbf { u } = \langle - 9 , - 4 \rangle , \mathbf { v } = \langle 7,1 \rangle$

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. - $r = 5 \cos 5 \theta$

Find the indicated roots. Write the answer in a + bi form. -Square roots of $- 1 + \sqrt { 3 } i$

Write the complex number in the form a + bi. - $6 \left( \cos 330 ^ { \circ } + i \sin 330 ^ { \circ } \right)$

The polar coordinates of point P are given. Find all of its polar coordinates. - $\mathrm { P } = ( 9 , \pi / 3 )$

Use the given graph to find the values of t that produce the graph in the given quadrant. - $x = 4 - | t | , y = t + 3 , - 6 \leq t \leq 6$ Quadrant II

Eliminate the parameter. - $x = \sin t , y = 3 \cos t$

Express the complex number in polar form. - $- 6 + 6 i$

Find the angle between the given vectors to the nearest tenth of a degree. - $\mathbf { u } = \langle - 7,5 \rangle , \mathbf { v } = \langle - 6,3 \rangle$

Find a· b. - $\mathbf { a } = \langle 1 , - 2 \rangle , \quad \mathbf { b } = \langle - 3,4 \rangle$

Find the indicated roots. Write the answer in a + bi form. -Fourth roots of $- 1 - \sqrt { 3 } i$

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. - $\langle - 3,0 \rangle$

Draw a graph of the rose curve. - $r=4 \cos 2 \theta, \quad 0 \leq \theta \leq 2 \pi$

Choose the graph of the given parametric equations. - $x=3 t, y=t+1,-2 \leq t \leq 3$

Determine whether the vectors u and v are parallel, orthogonal, or neither. - $\mathbf { u } = \langle 9,6 \rangle , \mathbf { v } = \langle 45,30 \rangle$

Determine whether the vectors u and v are parallel, orthogonal, or neither. - $\mathbf { u } = \langle 7,4 \rangle , \mathbf { v } = \langle 6,3 \rangle$

Find the product or quotient. Write the answer in standard form. - $( 9 - 7 i ) ( 3 + 9 i )$