839 8.7 Polar Equations and Graphs 1. Given vectors a = 8-1, 49 and b = 85, 29, find each of the following. (a) 3a (b) 4a - 2b (c) a (d) a # b (e) the angle between a and b 2. Walking Dogs on Leashes While Michael is walking his two dogs, Gus and Dotty, they reach a corner and must wait for a WALK sign. Michael is holding the two leashes in the same hand, and the dogs are pulling on their leashes at the angles and forces shown in the figure. Find the magnitude of the equilibrant force (to the nearest tenth of a pound) that Michael must apply to restrain the dogs. 3. For the following complex numbers, w = 3 + 5i and z = -4 + i, find w + z and give a geometric representation of the sum. 4. Express 11 - i23 in rectangular form. 5. Write each complex number in trigonometric (polar) form, where 0° … u 6360°. (a) -4i (b) 1 - i23 (c) -3 - i 6. Write each complex number in rectangular form. (a) 41cos 60° + i sin 60°2 (b) 5 cis 130° (c) 71cos 270° + i sin 270°2 7. Write each of the following in the form specified for the complex numbers w = 121cos 80° + i sin 80°2 and z = 31cos 50° + i sin 50°2. (a) wz (trigonometric form) (b) w z (rectangular form) (c) z3 (rectangular form) 8. Find the four complex fourth roots of -16. Write them in both trigonometric and rectangular forms. Chapter 8 Quiz (Sections 8.3 – 8.6) 8.7 Polar Equations and Graphs ■ Polar Coordinate System ■ Graphs of Polar Equations ■ Conversion from Polar to Rectangular Equations ■ Classification of Polar Equations Polar Coordinate System Previously we have used the rectangular coordinate system to graph points and equations. In the rectangular coordinate system, each point in the plane is specified by giving two numbers 1x, y2. These represent the directed distances from a pair of perpendicular axes, the x-axis and the y-axis. Now we consider the polar coordinate system which is based on a point, called the pole, and a ray, called the polar axis. The polar axis is usually drawn in the direction of the positive x-axis, as shown in Figure 56. Pole Polar axis Figure 56 258 358 15 lb 20 lb
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