Exam 9: Applications of Trigonometry

Given that the polar equation r models the orbits of the planets about the sun, -Suppose that a radio transmission pattern can be modeled by $\mathrm { r } ^ { 2 } = 28,900 \cos 2 \theta , \text { for } 0 ^ { \circ } \leq \theta \leq 360 ^ { \circ }$ where the units are miles. Determine how far the signal reaches to the east, assuming that east is represented by the direction of the positive x-axis.

Use the parallelogram rule to find the magnitude of the resultant force for the two forces shown in the figure. Round to one decimal place. -One rope pulls a barge directly east with a force of 61 newtons, and another rope pulls the barge directly north with a force of 58 newtons. Find the magnitude of the resultant force acting on the barge.

Write the complex number in trigonometric form r(c $r ( \cos \theta + i \sin \theta )$ ), with $\theta$ in the interval [0 $\left[ 0 ^ { \circ } , 360 ^ { \circ } \right)$ - $15 + 20 \mathrm { i }$

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

Find a rectangular equation for the plane curve defined by the parametric equations. - $x = t - 3 , y = t ^ { 2 } + 5$

Find the dot product for the pair of vectors. -4i + 4j, 10i - 5j

Solve the problem. -An airplane is sighted at the same time by two ground observers who are 3 miles apart and both directly west of the airplane. They report the angles of elevation as $15 ^ { \circ }$ and $23 ^ { \circ }$ . How high is the airplane? Round to the nearest hundredth of a mile.

Find all cube roots of the complex number. Leave answers in trigonometric form. - $2 + 2 \mathrm { i } \sqrt { 3 }$

Use the parallelogram rule to find the magnitude of the resultant force for the two forces shown in the figure. Round to one decimal place. -Suppose you would like to cross a 212 -foot wide river in a boat. Assume that the boat can travel 39 $\mathrm { mph }$ relative to the water and that the current is flowing west at the rate of $6 \mathrm { mph }$ . What bearing should be chosen so that the boat will land at a point exactly across from its starting point? Give your answer to the nearest tenth of a degree.

Find the missing parts of the triangle. - B=43 b=10.66 a=31.07 If necessary, round side lengths to the nearest hundredth.

Find the given power. Write answer in rectangular form. - $( 2 - 2 \mathrm { i } ) ^ { 5 }$

Find all specified roots. -Ninth roots of 1.

Find all cube roots of the complex number. Leave answers in trigonometric form. - $- 125$

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

Sketch the vectors u and w with angle $\theta$ between them and sketch the resultant. - $| \mathbf { u } | = 8 , | \mathbf { w } | = 3 , \theta = 80 ^ { \circ }$

Perform the indicated operation. Give answers in rectangular form expressing real and imaginary parts to four decimal places. - $\left( 12 \operatorname { cis } 18.5 ^ { \circ } \right) \left( 5 \operatorname { cis } 6.5 ^ { \circ } \right)$

Write the complex number in trigonometric form r(c $r ( \cos \theta + i \sin \theta )$ ), with $\theta$ in the interval [0 $\left[ 0 ^ { \circ } , 360 ^ { \circ } \right)$ - $5 - 5 \sqrt { 3 } i$

Find the polar coordinates of the point(s) of intersection of the given curves for 0 $0 \leq \theta < 2 \pi$ - $r = \frac { a \left( 1 - e ^ { 2 } \right) } { 1 + e \cos \theta }$ Given that the polar equation r models the orbits of the planets, compare Earth's orbit relative to the sun with that of Pluto and describe the difference. For Earth, a = 1.00 and e = 0.017. For Pluto, a = 39.4 and e = 0.249.

Find the missing parts of the triangle. Round to the nearest tenth when necessary or to the nearest minute as appropriate. - =324 =180 =169

Write the complex number in trigonometric form r(c $r ( \cos \theta + i \sin \theta )$ ), with $\theta$ in the interval [0 $\left[ 0 ^ { \circ } , 360 ^ { \circ } \right)$ -3