Chapter 7

Show that the Law of Cosines applied to a right triangle yields the Pythagorean Theorem.

Round your answers to two decimal places. A glider traveling at 90 miles per hour in the direction \(\mathrm{N} 20^{\circ} \mathrm{W}\) encounters a mild wind with speed 15 miles per hour. If the wind is traveling from east to west, find the resulting speed of the glider and its direction.

Sweepstakes Patrons of a nationwide fast-food chain are given a ticket that gives them a chance of winning a million dollars. The ticket shows a triangle \(A B C\) with the lengths of two sides marked as \(a=6.1 \mathrm{cm}\) and \(b=5.4 \mathrm{cm},\) and the measure of angle \(A\) marked as \(72.5^{\circ} .\) The winning ticket will be chosen from all the entries that correctly state the value of \(c\) rounded to the nearest tenth of a centimeter and the measures of angles \(B\) and \(C\) rounded to the nearest tenth of a degree. To be eligible for the prize, what should you submit as the values of \(c, B,\) and \(C ?\)

This set of exercises will draw on the ideas presented in this section and your general math background. Prove the following for any vector \(\mathbf{u}\) and any real number \(k\) : \((k \mathbf{u}) \cdot(\mathbf{v})=k(\mathbf{u} \cdot \mathbf{v})=\mathbf{u} \cdot(k \mathbf{v})\)

Convert each of the given polar equations to rectangular form. $$r=4$$

Explain why you cannot use the Law of Cosines directly to solve an oblique triangle if you are given only the measures of two angles and one side of the triangle (either AAS or ASA) and no two of the angles of the triangle are of equal measure.

Round your answers to two decimal places. The net velocity of a ship is the vector sum of the velocity imparted to the ship by its engine and the velocity of the wind. The engine propels the ship at a velocity of 20 miles per hour in the direction \(S 35^{\circ} \mathrm{E}\). (a) What are the components of the velocity imparted to the ship by its engine? (b) If the wind is blowing from north to south at 12 miles per hour, find the magnitude and direction of the net velocity of the ship. (c) Rework part (b) for the case where the wind is blowing from north to south at 15 miles per hour.

This set of exercises will draw on the ideas presented in this section and your general math background. Derive the Law of Sines for right triangles.

Convert each of the given polar equations to rectangular form. $$\theta=\frac{\pi}{4}$$

Can you use the Law of Cosines directly to solve an oblique triangle if you are given only two of the sides and the angle opposite one of them (SSA) and the two given sides are not of equal length? Explain.

Round your answers to two decimal places. Lucas pulls a 40 -pound box along a level surface from left to right by attaching a piece of rope to the box and pulling on it with a force \(\mathbf{F}_{1}\) of 20 pounds in the direction \(25^{\circ}\) above the horizontal. A friction force \(\mathbf{F}_{2}\) of 5 pounds is acting on the box as it is being pulled. (A friction force acts in the direction opposite to the direction of motion.) (a) Find the \(x\) and \(y\) components of \(\mathbf{F}_{1}\) (b) Find the \(x\) and \(y\) components of \(\mathbf{F}_{2}\) (c) Use your answers to parts (a) and (b) to express the vector sum \(\mathbf{F}_{1}+\mathbf{F}_{2}\) in terms of its \(x\) and \(y\) components. (d) Give the magnitude and direction of each of the other forces acting on the box. (e) Find the magnitude and direction of the net force acting on the box.

This set of exercises will draw on the ideas presented in this section and your general math background. Explain how you can use the Law of Sines to solve a right triangle. Is this the best way to solve a right triangle? Explain.

Convert each of the given polar equations to rectangular form. $$\theta=\pi$$

Is it possible for a triangle to have sides \(a=3, b=2,\) and \(c=5 ?\) (Hint: What happens if you apply the Law of Cosines to this triangle?)

This set of exercises will draw on the ideas presented in this section and your general math background. Determine the set of positive values of \(a\) for which there is exactly one triangle \(A B C\) with \(A=60^{\circ}\) and \(b=10,\) where \(a\) and \(b\) are the sides opposite angles \(A\) and \(B\), respectively. Then find the set of positive values of \(a\) for which exactly two such triangles \(A B C\) exist, and the set of positive values of \(a\) for which no such triangle exists.

Write the polar equation \(r=2-2 \cos \left(\theta+\frac{\pi}{2}\right)\) in terms of just the sine function.

Convert each of the given polar equations to rectangular form. $$r \cos \theta=4$$

If you are given all three sides of a triangle (SSS), how can you tell whether it has a right angle?

This set of exercises will draw on the ideas presented in this section and your general math background. Explain why you cannot use the Law of Sines to solve an oblique triangle if you are given only the three sides of the triangle (SSS) and no two of them are of equal length.

Write the polar equation \(r=4 \sin \left(\theta-\frac{3 \pi}{2}\right)\) in terms of just the cosine function.

Convert each of the given polar equations to rectangular form. $$r \sin \theta=3$$

If you are given all three sides of a triangle (SSS), how can you tell whether it has an obtuse angle?

Show that if \(\|\mathbf{v}\|=0,\) then \(\mathbf{v}=\langle 0,0\rangle\)

This set of exercises will draw on the ideas presented in this section and your general math background. Can you use the Law of Sines to solve an oblique triangle if you are given only two of the sides and the included angle (SAS) and the two given sides are not of equal length? Explain.

If you are given two sides of a triangle and the included angle (SAS), how can you tell whether the triangle has a right angle if the included angle is acute?

Use a graphing utility to graph \(r_{1}=2 \sin (3 \theta)\) and \(r_{2}=2 \sin \left(3\left(\theta+\frac{\pi}{3}\right)\right) .\) Explain the relationship between the two graphs in terms of rotations.

Convert each of the given polar equations to rectangular form. $$2 r \cos \theta+r \sin \theta=4$$

Show that if \(\mathbf{u}\) is a nonzero vector, then the vector \(\frac{\mathbf{u}}{\|\mathbf{u}\|}\) has magnitude \(1 .\)

If you are given two sides of a triangle and the included angle (SAS), how can you tell whether the triangle has an obtuse angle if the included angle is acute?

Use a graphing utility to graph \(r_{1}=1+\cos \theta\) and \(r_{2}=1+\cos \left(\theta-\frac{\pi}{2}\right) .\) Explain the relationship between the two graphs in terms of rotations.

Convert each of the given polar equations to rectangular form. $$r \cos \theta-3 r \sin \theta=5$$

If \(u\) is a nonzero vector, for what values of \(k\) does the equation \(\|k \mathbf{u}\|=k\|\mathbf{u}\|\) hold? Explain.

Show that the formula Area \((A B C)=\frac{1}{2} a b \sin C\) holds if \(A B C\) is a right triangle.

Convert each of the given polar equations to rectangular form. $$r=2 \cos \theta$$

Convert each of the given polar equations to rectangular form. $$r=4 \sin \theta$$

Convert each of the given polar equations to rectangular form. $$r^{2} \cos 2 \theta=4$$

Convert each of the given polar equations to rectangular form. $$r^{2} \cos 2 \theta=1$$

A patented device converts a radar signal given in polar coordinates to a format in rectangular coordinates so that it is better suited to display in a televisiontype display device. If a radar signal is at the point \(\left(3,-\frac{2 \pi}{3}\right)\) find the exact values of the corresponding rectangular coordinates in the television display.

A boat departs its starting point and travels 4 miles north and 3 miles west. Determine its current location in polar coordinates, using the starting point as the origin. Use a scientific calculator to approximate \(\theta,\) in radians, to three decimal places.

Explain why \((r, \theta)\) and \((r, \theta+2 \pi)\) represent the same point in the polar coordinate system.

Explain why \((r, \theta)\) and \((-r, \theta+\pi)\) represent the same point in the polar coordinate system.

List at least two features of the polar coordinate system that are different from those of the rectangular coordinate system.