Chapter 11

A woman finds that the bearing of a tree on the opposite bank of a river flowing north is \(115.45^{\circ} .\) A man is on the same bank as the woman but 428.3 meters away. He finds that the bearing of the tree is \(45.47^{\circ} .\) The two banks are parallel. What is the distance across the river?

Find each product in rectangular form, using exact values. $$\left[6 \operatorname{cis} \frac{2 \pi}{3}\right]\left[5 \operatorname{cis}\left(-\frac{\pi}{6}\right)\right]$$

Solve each problem. A baseball diamond is a square 90 feet on a side, with home plate and the three bases as vertices. The pitcher's rubber is located 60.5 feet from home plate. Find the distance from the pitcher's rubber to each of the bases.

Find the dot product of each pair of vectors. $$\langle- 3,8\rangle,\langle 7,-5\rangle$$

Answer each of the following. The spiral of Archimedes has polar equation \(r=a \theta\) where \(r^{2}=x^{2}+y^{2}\). Show that a parametric representation of the spiral of Archimedes is \(x=a \theta \cos \theta, \quad y=a \theta \sin \theta, \quad\) for \(\theta\) in \((-\infty, \infty)\)

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

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

Solve each problem. A parallelogram has sides of lengths 25.9 centimeters and 32.5 centimeters. The longer diagonal has length 57.8 centimeters. Find the angle opposite the longer diagonal.

Find the dot product of each pair of vectors. $$\langle 2,-3\rangle,\langle 6,5\rangle$$

Answer each of the following. Show that the hyperbolic spiral given by \(r \theta=a,\) where \(r^{2}=x^{2}+y^{2},\) is given parametrically by \(x=\frac{a \cos \theta}{\theta}, \quad y=\frac{a \sin \theta}{\theta}, \quad\) for \(\theta\) in \((-\infty, 0) \cup(0, \infty)\)

For each equation, find an equivalent equation in rectangular coordinates. Then graph the result. $$r=\frac{3}{4 \cos \theta-\sin \theta}$$

Find each product in rectangular form, using exact values. $$\left[\sqrt{2} \operatorname{cis} \frac{5 \pi}{6}\right]\left[\sqrt{2} \operatorname{cis} \frac{3 \pi}{2}\right]$$f

Solve each problem. Distance between an Airplane and a Mountain A person in a plane flying straight north observes a mountain at a bearing of \(24.1^{\circ} .\) At the time, the plane is 7.92 kilometers from the mountain. A short time later, the bearing to the mountain becomes \(32.7^{\circ} .\) How far is the airplane from the mountain when the second bearing is taken?

Find the dot product of each pair of vectors. $$\langle 1,2\rangle,\langle 3,-1\rangle$$

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

The bearing of a lighthouse from a ship was found to be \(N 37^{\circ} \mathrm{E}\) After the ship sailed 2.5 miles due south, the new bearing was N \(25^{\circ} \mathrm{E}\). Find the distance between the ship and the lighthouse at each location.

Find each quotient in rectangular form, using exact values. $$\frac{4\left(\cos 120^{\circ}+i \sin 120^{\circ}\right)}{2\left(\cos 150^{\circ}+i \sin 150^{\circ}\right)}$$

Solve each problem. Distance between a Satellite and a Tracking Station \(\mathrm{A}\) satellite traveling in a circular orbit 1600 kilometers above Earth is due to pass directly over a tracking station at noon. Assume that the satellite takes 2 hours to make an orbit and that the radius of Earth is 6400 kilometers. Find the distance between the satellite and the tracking station at 12: 03 P.M. (Source: NASA.) (figure cannot copy)

Find the dot product of each pair of vectors. $$4 \mathbf{i}, 5 \mathbf{i}-9 \mathbf{j}$$

For each equation, find an equivalent equation in rectangular coordinates. Then graph the result. $$r=-5 \csc \theta$$

A balloonist is directly above a straight road 1.5 miles long that joins two towns. She finds that the town closer to her is at an angle of depression of \(35^{\circ}\) and the farther town is at an angle of depression of \(31^{\circ},\) How high above the ground is the balloon?

Find each product in rectangular form, using exact values. $$\frac{16\left(\cos 300^{\circ}+i \sin 300^{\circ}\right)}{8\left(\cos 60^{\circ}+i \sin 60^{\circ}\right)}$$

Find the dot product of each pair of vectors. $$2 \mathbf{i}+4 \mathbf{j},-\mathbf{j}$$

From shore station A, a ship \(C\) is observed in the direction \(N 22.4^{\circ}\) E. The same ship is observed to be in the direction \(\mathrm{N} 10.6^{\circ} \mathrm{W}\) from shore station \(\mathrm{B}\), located a distance of 25.5 kilometers exactly southeast of A. Find the distance of the ship from station A.

Find each product in rectangular form, using exact values. $$\frac{10\left(\cos \frac{5 \pi}{4}+i \sin \frac{5 \pi}{4}\right)}{5\left(\cos \frac{\pi}{4}+i \sin \frac{\pi}{4}\right)}$$

Solve each problem. Distance between Two Factories Two factories blow their whistles at exactly \(5: 00 .\) A man hears the two blasts at 3 seconds and 6 seconds after \(5: 00,\) respectively. The angle between his lines of sight to the two factories is \(42.20^{\circ} .\) If sound travels 344 meters per second, how far apart are the factories to the nearest meter?

Find the angle between each pair of vectors. $$\langle 2,1\rangle,\langle- 3,1\rangle$$

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

Find each product in rectangular form, using exact values. $$\frac{24\left(\cos \frac{5 \pi}{6}+i \sin \frac{5 \pi}{6}\right)}{2\left(\cos \frac{\pi}{6}+i \sin \frac{\pi}{6}\right)}$$

Solve each problem. Starting at point \(A,\) a ship sails 18.5 kilometers on a bearing of \(189^{\circ},\) then turns and sails 47.8 kilometers on a bearing of \(317^{\circ} .\) Find the distance of the ship from point \(A\)

Find the angle between each pair of vectors. $$\langle 4,0\rangle,\langle 2,2\rangle$$

The graph of \(r=a \theta\) is an example of the spiral of Archimedes. With a calculator set to radian mode. use the given value of a and interval of \(\theta\) to graph the spiral in the window specified. $$a=1,0 \leq \theta \leq 4 \pi,[-15,15] \text { by }[-15,10]$$

Find each product in rectangular form, using exact values. $$\frac{3 \operatorname{cis}\left(\frac{61 \pi}{36}\right)}{9 \operatorname{cis}\left(\frac{13 \pi}{36}\right)}$$

Find the angle between each pair of vectors. $$\langle 1,2\rangle,\langle- 6,3\rangle$$

The graph of \(r=a \theta\) is an example of the spiral of Archimedes. With a calculator set to radian mode. use the given value of a and interval of \(\theta\) to graph the spiral in the window specified. $$a=2,-4 \pi \leq \theta \leq 4 \pi,[-30,30] \text { by }[-30,30]$$

A surveyor standing 48.0 meters from the base of a building measures the angle to the top of the building to be \(37.4^{\circ} .\) The surveyor then measures the angle to the top of a clock tower on the building to be \(45.6^{\circ} .\) Find the height of the clock tower.

Find each product in rectangular form, using exact values. $$\frac{12 \text { cis } 293^{\circ}}{6 \text { cis } 23^{\circ}}$$

Solve each problem. Distance between Ends of the Vietnam Memorial The Vietnam Veterans Memorial in Washington, DC, is V-shaped with equal sides of length 246.75 feet, and the angle between these sides measures \(125^{\circ} 12^{\prime} .\) Find the distance between the ends of the two sides. (Source: Pamphlet obtained at Vietnam Veterans Memorial.) (picture cannot copy)

Find the angle between each pair of vectors. $$\langle 6,8\rangle,\langle- 4,3\rangle$$

The graph of \(r=a \theta\) is an example of the spiral of Archimedes. With a calculator set to radian mode. use the given value of a and interval of \(\theta\) to graph the spiral in the window specified. $$a=1.5,-4 \pi \leq \theta \leq 4 \pi,[-20,20] \text { by }[-20,20]$$

Find each quotient and express it in rectangular form by first converting the numerator and the denominator to trigonometric form. $$\frac{8}{\sqrt{3}+i}$$

Find the angle between each pair of vectors. $$\mathbf{i}+7 \mathbf{j}, \mathbf{i}+\mathbf{j}$$

The graph of \(r=a \theta\) is an example of the spiral of Archimedes. With a calculator set to radian mode. use the given value of a and interval of \(\theta\) to graph the spiral in the window specified. $$a=-1,0 \leq \theta \leq 12 \pi,[-40,40] \text { by }[-40,40]$$

Find each quotient and express it in rectangular form by first converting the numerator and the denominator to trigonometric form. $$\frac{2 i}{-1-i \sqrt{3}}$$

Find the angle between each pair of vectors. $$3 \mathfrak{i}+4 \mathfrak{j}, \mathfrak{j}$$

Find the polar coordinates of the points of intersection of the given curves for the specified interval of \(\theta\). $$r=4 \sin \theta, r=1+2 \sin \theta ; 0 \leq \theta<2 \pi$$

Find each quotient and express it in rectangular form by first converting the numerator and the denominator to trigonometric form. $$\frac{-i}{1+i}$$

Find the angle between each pair of vectors. $$\mathbf{i}+\mathbf{j}, 3 \mathbf{i}+4 \mathbf{j}$$

Find the polar coordinates of the points of intersection of the given curves for the specified interval of \(\theta\). $$r=3, r=2+2 \cos \theta ; 0^{\circ} \leq \theta<360^{\circ}$$

Find each quotient and express it in rectangular form by first converting the numerator and the denominator to trigonometric form. $$\frac{1}{2-2 i}$$