Exam 6: Applications of Trigonometry

Give examples of three real-world quantities that could be represented by a vector and three quantities that could be represented by a single real number. Explain why some quantities can be represented by a number and others must be represented by a vector.

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. - $\langle 5 , - 9 \rangle$

Find an equivalent equation in rectangular coordinates. - $r \sin \theta = 10$

Eliminate the parameter. - $x = t - 4 , y = \frac { 3 } { t + 7 }$

Solve the problem using a graphing calculator. -Determine whether a baseball hit 123 feet per second at an angle of $40 ^ { \circ }$ relative to level ground will clear a 10-foot wall 400 feet away.

Find the indicated roots. Write the answer in polar form. -Fifth roots of $7 \left( \cos \frac { \pi } { 3 } + i \sin \frac { \pi } { 3 } \right)$

Find the unit vector in the direction of the given vector. Write your answer in the indicated form. -Let $\mathbf { u } = \langle 2,1 \rangle$ . Find the unit vector in the direction of $\mathbf { u }$ , and write your answer in component form.

Use the dot product to find $|\mathbf{v}|$ - $v = 2 i$

Use De Moivre's Theorem to find the indicated power of the complex number. Write your answer in standard form a + bi. - $\left( \cos 11 ^ { \circ } + i \sin 11 ^ { \circ } \right) ^ { 3 }$

Eliminate the parameter. - $x = t ^ { 2 } + 1 , y = t ^ { 2 } - 1$

Find the indicated roots. Write the answer in a + bi form. -Fourth roots of $- 16$

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

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

Express the complex number in polar form. -

With your calculator set to radian mode and polar graphics capability, graph the following function in the window specified. - $\mathrm { r } = 3 \theta , 0 \leq \theta \leq 2 \pi , [ - 20,20 ]$ by $[ - 20,20 ]$

Solve the problem. -A constant force $\mathbf { F } = \langle 23,27 \rangle$ moves an object along a vector $\mathbf { D } = \langle 16,8 \rangle$ , where units are in pounds and feet. Finv the work done.

Determine whether the vectors u and v are parallel, orthogonal, or neither. - $\mathbf { u } = \langle - 6,9 \rangle , \mathbf { v } = \langle - 2,3 \rangle$

Find the component form of the indicated vector. -Let $\mathbf { u } = \langle - 5,1 \rangle , \mathbf { v } = \langle - 7 , - 1 \rangle$ . Find $8 \mathbf { u } + 2 \mathbf { v }$ .

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 = 5

Solve the problem. -The angle between a 150 -pound force $\mathbf { F }$ and $\overrightarrow { \mathrm { AB } } = 4 \mathbf { i } + 3 \mathbf { j }$ is $30 ^ { \circ }$ . Find the work done by $\mathbf { F }$ in moving an object from $A$ to $B$ .