Exam 8: Applications of Trigonometry

The graph of r = aθ in polar coordinates is an example of the spiral of Archimedes. With your calculator set to radian mode, use the given value of a and interval of θ to graph the spiral in the window specified. - $a = 2 , - 3 \pi \leq \theta \leq 3 \pi , [ - 20,20 ]$ by $[ - 20,20 ]$

The graph of a polar equation is given. Select the polar equation for the graph. -

Write the complex number in rectangular form. - $4 \operatorname { cis } 135 ^ { \circ }$

Sketch the vectors u and w with angle θ between them and sketch the resultant. - $| \mathbf { u } | = 5 , | \mathbf { w } | = 5 , \theta = 45 ^ { \circ }$

Graph the complex number. - $6 i$

Find all solutions of the equation. Leave answers in trigonometric form. - $x ^ { 4 } + 16 i = 0$

Solve the triangle. Round to the nearest tenth when necessary or to the nearest minute as appropriate. -

Find the quotient and write in rectangular form. First convert the numerator and denominator to trigonometric form. - $\frac { 3 i } { - 1 - i \sqrt { 3 } }$

Vector v has the given magnitude and direction. Find the magnitude of the indicated component of v rounded to the nearest tenth when necessary. - $\alpha = 25.8 ^ { \circ } , | \mathbf { v } | = 64.2$ ; Find the horizontal component of $\mathbf { v }$ .

Find the given power. Write answer in rectangular form. - $\left[ 2 \operatorname { cis } 15 ^ { \circ } \right] ^ { 4 }$

Vector v has the given magnitude and direction. Find the magnitude of the indicated component of v rounded to the nearest tenth when necessary. - $\alpha = 35.7 ^ { \circ } , | \mathbf { v } | = 368$ ; Find the vertical component of $\mathbf { v }$ .

Find the polar coordinates of the point(s) of intersection of the given curves for 0 ≤ θ < 2π. - $\mathrm { r } = 7 + \cos \theta , \mathrm { r } = 7 - \sin \theta$

Find the missing parts of the triangle. Round to the nearest tenth when necessary or to the nearest minute as - C=111. a=5.2 b=10.5

Solve the problem. -Suppose you would like to cross a 216 -foot wide river in a boat. Assume that the boat can travel $38 \mathrm { mph }$ relative to the water and that the current is flowing west at the rate of $6 \mathrm { mph }$ . If the bearing is chosen so that the boat will land at a point exactly across from its starting point, how long will it take for the boat to make the crossing? Give your answer to the nearest second.

Solve the problem. -What is the minimum force required to prevent a ball weighing $26.6 \mathrm { lb }$ from rolling down a ramp inclined $12.4 ^ { \circ }$ with the horizontal?

Find the indicated angle or side. Give an exact answer. -Find the exact length of side $a$ .

Find sum of the pair of complex numbers. - $- 5 i , 3$

Write the complex number in rectangular form. - $\sqrt { 7 } \operatorname { cis } 270 ^ { \circ }$

Find the magnitude and direction angle (to the nearest tenth) for each vector. Give the measure of the direction angle as an angle in [0,360°]. - $\langle 6,6 \rangle$

Solve the problem. -A rocket tracking station has two telescopes A and B placed $2.7$ miles apart. The telescopes lock onto a rocket and transmit their angles of elevation to a computer after a rocket launch. What is the distance to the rocket from telescope B at the moment when both tracking stations are directly east of the rocket telescope A reports an angle of elevation of $24 ^ { \circ }$ and telescope B reports an angle of elevation of $51 ^ { \circ }$ ? Round to the nearest hundredth of a mile.