Exam 6: Applications of Trigonometry

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 $\mathrm { r } = - 2 + 2 \cos \theta$ ? Note that a maximum $\mathrm { r }$ -value occurs at a point that is the maximum distance from the pole.

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

Determine two pairs of polar coordinates for the point with 0° ≤θ < 360°. - $( - 5 \sqrt { 2 } , - 5 \sqrt { 2 } )$

Determine two pairs of polar coordinates for the point with 0° ≤θ < 360°. - $( - 5 \sqrt { 3 } , 5 \sqrt { 3 } )$

Graph the equation. - $\mathrm { r } \sin \theta = 2$

Find the indicated roots. Write the answer in a + bi form. -Square roots of $4 + 4 \sqrt { 3 } i$

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. - $- 7 i + 2 j$

Use the dot product to find $|\mathbf{v}|$ - $\mathbf { v } = \langle - 8,3 \rangle$

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. - $r = 4 + 5 \sin \theta$

Find an equivalent equation in polar coordinates. - $x ^ { 2 } + 4 y ^ { 2 } = 4$

Find the unit vector in the direction of the given vector. Write your answer in the indicated form. -Let $\mathbf { u } = \langle - 3 , - 4 \rangle$ . Find the unit vector in the direction of $\mathbf { u }$ , and write your answer in component form.

Prove that $\overrightarrow{R S}$ and $\overrightarrow{O P}$ are equivalent by showing that they represent the same vector. -R = (-6, 3) , S = (-1, 6), O = (0, 0), and P = (5, 3)

Determine whether the vectors u and v are parallel, orthogonal, or neither. - $\mathbf { u } = \langle 10,0 \rangle , \mathbf { v } = \langle 0 , - 10 \rangle$

Write the vector u as a sum of two orthogonal vectors, one of which is the vector projection of u onto v, $\operatorname{proj}_{\mathbf{v}} \mathbf{u}$ . - $\mathbf { u } = \langle 6 , - 9 \rangle , \mathbf { v } = \langle - 9 , - 3 \rangle$

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. - $\mathrm { r } = 0.25 \theta$

Plot the point with the given polar coordinates. - $\left( 2,45 ^ { \circ } \right)$

Find the rectangular coordinates of the point with the given polar coordinates. - $\left( 2 , \frac { 3 } { 2 } \pi \right)$

Express the complex number in polar form. - $6 \mathrm { i }$

Find a· b. - $\mathbf { a } = 10 \mathbf { i } + 6 \mathbf { j } , \quad \mathbf { b } = 5 \mathbf { i } + 3 \mathbf { j }$

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. - $r = 3 \cos 3 \theta - \sin 3 \theta$