Chapter 11

Write each complex number in trigonometric form, where \(r\) is exact and \(0 \leq \theta<2 \pi\) $$\sqrt{3}-i$$

Find all complex solutions for each equation. Leave your answers in trigonometric form. $$x^{5}+1=0$$

Write each vector in the form ai \(+\) bj. Round a and b to the nearest hundredth, if necessary. Direction angle \(208^{\circ},\) magnitude 0.9

The screen shown to the right is an example of a Lissajous figure. Lissajous figures occur in electronics and may be used to find the frequency of an unknown voltage. Graph each Lissajous figure for \(0 \leq t \leq 6.5\) in the window \([-6.6,6.6]\) by \([-4.1,4.1]\). (GRAPH CANNOT COPY) $$x=3 \sin 4 t, y=3 \cos 3 t$$

Answer each question.If a point lies on an axis in the rectangular plane, then what kind of angle must \(\theta\) be if \((r, \theta)\) represents the point in polar coordinates?

Solve triangle. There may be two, one, or no such triangle. $$B=32^{\circ} 50^{\prime}, a=7540 \text { centimeters, } b=5180 \text { centimeters }$$

Write each complex number in trigonometric form, where \(r\) is exact and \(0 \leq \theta<2 \pi\) $$-\sqrt{2}+i \sqrt{2}$$

Find the magnitude and direction angle (to the nearest tenth) for each vector. Give the measure of the direction angle as an angle in \(\left[0,360^{\circ}\right)\). $$\langle 1,1\rangle$$

Solve each problem. Points \(A\) and \(B\) are on opposite sides of Lake Yankee. From a third point, \(C\), the angle between the lines of sight to \(A\) and \(B\) is \(46.3^{\circ} .\) If \(A C\) is 350 meters long and \(B C\) is 286 meters long, find \(A B\).

The screen shown to the right is an example of a Lissajous figure. Lissajous figures occur in electronics and may be used to find the frequency of an unknown voltage. Graph each Lissajous figure for \(0 \leq t \leq 6.5\) in the window \([-6.6,6.6]\) by \([-4.1,4.1]\). (GRAPH CANNOT COPY) $$x=4 \sin 4 t, y=3 \sin 5 t$$

Solve triangle. There may be two, one, or no such triangle. $$C=22^{\circ} 50^{\prime}, b=159 \text { millimeters, } c=132 \text { millimeters }$$

Write each complex number in trigonometric form, where \(r\) is exact and \(0 \leq \theta<2 \pi\) $$-5-5 i$$

Find the magnitude and direction angle (to the nearest tenth) for each vector. Give the measure of the direction angle as an angle in \(\left[0,360^{\circ}\right)\). $$\langle- 4,4 \sqrt{3}\rangle$$

Solve each problem. The sides of a parallelogram are 4.0 centimeters and 6.0 centimeters. One angle is \(58^{\circ}\) and another is \(122^{\circ} .\) Find the lengths of the diagonals of the parallelogram.

Do the following. (a) Determine the parametric equations that model the path of the projectile. (b) Determine the rectangular equation that models the path of the projectile. (c) Determine the time the projectile is in flight and the horizontal distance covered. A model rocket is launched from the ground with a velocity of 48 feet per second at an angle of \(60^{\circ}\) with respect to the ground.

Write each complex number in trigonometric form, where \(r\) is exact and \(0 \leq \theta<2 \pi\) $$-4$$

We examine how the three complex cube roots of \(-8\) can be found in two different ways. All complex roots of the equation \(x^{3}+8=0\) are cube roots of \(-8 .\) Factor \(x^{3}+8\) as the sum of two cubes.

Find the magnitude and direction angle (to the nearest tenth) for each vector. Give the measure of the direction angle as an angle in \(\left[0,360^{\circ}\right)\). $$\langle 8 \sqrt{2},-8 \sqrt{2}\rangle$$

Do the following. (a) Determine the parametric equations that model the path of the projectile. (b) Determine the rectangular equation that models the path of the projectile. (c) Determine the time the projectile is in flight and the horizontal distance covered. A golfer hits a golf ball from the ground at an angle of \(60^{\circ}\) with respect to the ground at a velocity of 150 feet per second. (PICTURE CANNOT COPY)

A trigonometry student makes the statement "If we know any two angles and one side of a triangle, then the triangle is uniquely determined." Is this a valid statement? Explain, referring to the congruence axioms given in this section.

Write each complex number in trigonometric form, where \(r\) is exact and \(0 \leq \theta<2 \pi\) $$5 i$$

Find the magnitude and direction angle (to the nearest tenth) for each vector. Give the measure of the direction angle as an angle in \(\left[0,360^{\circ}\right)\). $$\langle\sqrt{3},-1\rangle$$

Solve each problem. Two Ships Two ships leave a harbor together, traveling on courses that have an angle of \(135^{\circ} 40^{\prime}\) between them. If they each travel 402 miles, how far apart are they?

Do the following. (a) Determine the parametric equations that model the path of the projectile. (b) Determine the rectangular equation that models the path of the projectile. (c) Determine the time the projectile is in flight and the horizontal distance covered. A batter hits a softball when it is 2 feet above the ground. The ball leaves her bat at an angle of \(20^{\circ}\) with respect to the ground at a velocity of 88 feet per second.

The graphs of rose curves have equations of the form \(r=a \sin n \theta\) or \(r=a \cos n \theta .\) What does the value of \(a\) determine? What does the value of \(n\) determine?

To find the distance \(A B\) across a river, a distance \(B C=354\) meters is laid off on one side of the river. It is found that \(B=112^{\circ} 10^{\prime}\) and \(C=15^{\circ} 20^{\prime} .\) Find \(A B\)

Find each product in rectangular form, using exact values. $$\left[3\left(\cos 60^{\circ}+i \sin 60^{\circ}\right)\right]\left[2\left(\cos 90^{\circ}+i \sin 90^{\circ}\right)\right]$$

Find the magnitude and direction angle (to the nearest tenth) for each vector. Give the measure of the direction angle as an angle in \(\left[0,360^{\circ}\right)\). $$\langle 15,-8\rangle$$

Do the following. (a) Determine the parametric equations that model the path of the projectile. (b) Determine the rectangular equation that models the path of the projectile. (c) Determine the time the projectile is in flight and the horizontal distance covered. A batter hits a baseball when it is 2.5 feet above the ground. The ball leaves his bat at an angle of \(29^{\circ}\) from the horizontal with a velocity of 136 feet per second. (PICTURE CANNOT COPY)

To determine the distance \(R S\) across a deep canyon, Joanna lays off a distance \(T R=582\) yards. She then finds that \(T=32^{\circ} 50^{\prime}\) and \(R=102^{\circ} 20^{\prime} .\) Find \(R S\).

Find each product in rectangular form, using exact values. $$\left[4\left(\cos 30^{\circ}+i \sin 30^{\circ}\right)\right]\left[5\left(\cos 120^{\circ}+i \sin 120^{\circ}\right)\right]$$

We examine how the three complex cube roots of \(-8\) can be found in two different ways. Use the method described in this section to find the three complex cube roots of \(-8 .\) Give them in trigonometric form.

Find the magnitude and direction angle (to the nearest tenth) for each vector. Give the measure of the direction angle as an angle in \(\left[0,360^{\circ}\right)\). $$\langle- 7,24\rangle$$

If an object is projected on the moon, then the parametric equations of flight are $$ x=(v \cos \theta) t \quad \text { and } \quad y=(v \sin \theta) t-2.66 t^{2}+h $$ Estimate the distance that a golf ball hit at 88 feet per second \((60 \mathrm{mph})\) at an angle of \(45^{\circ}\) with the horizontal travels on the moon if the moon's surface is level.

For each equation, find an equivalent equation in rectangular coordinates. Then graph the result. $$r=2 \sin \theta$$

Radio direction finders are at points \(A\) and \(B\), which are 3.46 miles apart on an east-west line, with \(A\) west of \(B\). From \(A,\) the bearing of a certain radio transmitter is \(47.7^{\circ}\); from \(B,\) the bearing is \(302.5^{\circ} .\) Find the distance of the transmitter from \(A\).

Find each product in rectangular form, using exact values. $$\left[2\left(\cos 45^{\circ}+i \sin 45^{\circ}\right)\right]\left[2\left(\cos 225^{\circ}+i \sin 225^{\circ}\right)\right]$$

Solve each problem. \(\quad\) A ship is sailing east. At one point, the bearing of a submerged rock is \(45^{\circ} 20^{\prime} .\) After the ship has sailed 15.2 miles, the bearing of the rock has become \(308^{\circ} 40^{\circ} .\) Find the distance of the ship from the rock at the latter point.

Find the magnitude and direction angle (to the nearest tenth) for each vector. Give the measure of the direction angle as an angle in \(\left[0,360^{\circ}\right)\). $$\langle- 6,0\rangle$$

A baseball is hit from a height of 3 feet at a \(60^{\circ}\) angle above the horizontal. Its initial velocity is 64 feet per second. (a) Write parametric equations that model the flight of the baseball. (b) Determine the horizontal distance traveled by the ball in the air. Assume that the ground is level. (c) What is the maximum height of the baseball? At that time, how far has the ball traveled horizontally? (d) Would the ball clear a 5 -foot-high fence that is 100 feet from the batter?

For each equation, find an equivalent equation in rectangular coordinates. Then graph the result. $$r=2 \cos \theta$$

A ship is sailing due north. At a certain point, the bearing of a lighthouse 12.5 kilometers away is \(N 38.8^{\circ}\) E. Later on, the captain notices that the bearing of the lighthouse has become \(S 44.2^{\circ} \mathrm{E} .\) How far did the ship travel between the two observations of the lighthouse?

Find each product in rectangular form, using exact values. $$\left[8\left(\cos 300^{\circ}+i \sin 300^{\circ}\right)\right]\left[5\left(\cos 120^{\circ}+i \sin 120^{\circ}\right)\right]$$

Solve each problem. Two boats leave a dock together. Each travels in a straight line. The angle between their courses measures \(54^{\circ} 10^{\prime} .\) One boat travels 36.2 kilometers per hour and the other 45.6 kilometers per hour. How far apart will they be after 3 hours?

Find the magnitude and direction angle (to the nearest tenth) for each vector. Give the measure of the direction angle as an angle in \(\left[0,360^{\circ}\right)\). $$\langle 0,-12\rangle$$

A projectile has been launched from the ground with an initial velocity of 88 feet per second. You are given parametric equations that model the path of the projectile. (a) Graph the parametric equations. (b) Approximate \(\theta\), the angle the projectile makes with the horizontal at launch, to the nearest tenth of a degree. (c) On the basis of your answer to part (b), write parametric equations for the projectile, using the cosine and sine functions. $$x=82.69265063 t, y=-16 t^{2}+30.09777261 t$$

For each equation, find an equivalent equation in rectangular coordinates. Then graph the result. $$r=\frac{2}{1-\cos \theta}$$

Find each product in rectangular form, using exact values. $$\left[5 \text { cis } \frac{\pi}{2}\right]\left[3 \text { cis } \frac{\pi}{4}\right]$$

Find the dot product of each pair of vectors. $$\langle 6,-1\rangle,\langle 2,5\rangle$$

A projectile has been launched from the ground with an initial velocity of 88 feet per second. You are given parametric equations that model the path of the projectile. (a) Graph the parametric equations. (b) Approximate \(\theta\), the angle the projectile makes with the horizontal at launch, to the nearest tenth of a degree. (c) On the basis of your answer to part (b), write parametric equations for the projectile, using the cosine and sine functions. $$x=56.56530965 t, y=-16 t^{2}+67.41191099 t$$