Exam 6: Applications of Trigonometry

With your calculator set to radian mode and polar graphics capability, graph the following function in the window specified. - $\mathrm { r } = \theta + 3 \cos \theta , 0 \leq \theta \leq 3 \pi , [ - 4,4 ]$ by $[ - 4,4 ]$

Express the indicated roots of unity in standard form a + bi. -Cube roots of unity

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 $r = 2 + 3 \sin \theta$ ? Note that a maximum r-value occurs at a point that is the maximum distance from the pole.

Eliminate the parameter. - $x = \frac { 1 } { 4 } t , y = 2 t ^ { 3 } - 4$

Use De Moivre's Theorem to find the indicated power of the complex number. Write your answer in standard form a + bi. - $\left( 5 \left( \cos 70 ^ { \circ } + i \sin 70 ^ { \circ } \right) \right) ^ { 3 }$

Solve the problem using a graphing calculator. -A Ferris wheel with a radius of 36 feet turns clockwise at the rate of one revolution every 12 sec. The lowest point of the Ferris wheel is 13 feet above ground level at the point $( 0,13 )$ on a rectangular coordinate system. Find parametric equations for the position of a person on the Ferris wheel as a function of time (in seconds) if the Ferris wheel starts $( t = 0 )$ with the person at the point $( 36,49 )$ .

Eliminate the parameter. - $\mathrm { x } = \mathrm { t } + 4 , \mathrm { y } = \mathrm { t } ^ { 2 }$

Find the parametrization for the curve. -The portion of the circle $x ^ { 2 } + y ^ { 2 } = 6$ that lies in the second quadrant

A plane is flying at $500 \mathrm { mph }$ on a bearing of $x ^ { \circ }$ where $x$ is greater than 90 . Give an expression for the component form of the velocity of the plane.

Find the component form of the indicated vector. -Let $\mathbf { u } = \langle - 4 , - 6 \rangle , \mathbf { v } = \langle 4,2 \rangle$ . Find $- \mathbf { u } - \mathbf { v }$ .

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. - $r = 1 - 4 \sin 4$

Find the component form and magnitude of the indicated vector. -Given that $P = ( 6,5 )$ and $Q = ( - 2,6 )$ , find the component form and magnitude of the vector $\sqrt { 3 } \overrightarrow { Q P }$ .

Determine if the graph is symmetric about the x-axis, the y-axis, or the origin. - $r = \frac { 2 } { 1 + 3 \cos \theta }$

Use De Moivre's Theorem to find the indicated power of the complex number. Write your answer in standard form a + bi. - $\left( - \frac { 1 } { 2 } - \frac { \sqrt { 3 } } { 2 } \mathrm { i } \right) 10$

Find an equivalent equation in rectangular coordinates. - $\mathrm { r } = 1 + 2 \sin \theta$

Find the component form of the vector v. -

Solve the problem. -A force of $90 \mathrm { lb }$ acts on an object at an angle of $55 ^ { \circ }$ . A second force of $100 \mathrm { lb }$ acts on the object at an angle of $- 60 ^ { \circ }$ . Find the direction and magnitude of the resultant force.

Write the complex number in the form a + bi. - $3 \left( \cos 60 ^ { \circ } + i \sin 60 ^ { \circ } \right)$

Find the unit vector in the direction of the given vector. Write your answer in the indicated form. -Let $\mathbf { u } = - 3 \mathbf { i } - 2 \mathbf { j }$ . Find the unit vector in the direction of $\mathbf { u }$ , and write your answer as a linear combination of the standard unit vectors $\mathbf { i }$ and $\mathbf { j }$ .

Find the rectangular coordinates of the point with the given polar coordinates. -