804 CHAPTER 8 Applications of Trigonometry sin CAB 60 = sin 130° 98 Law of sines (alternative form) sin CAB ≈0.46900680 Multiply by 60 and use a calculator. CAB ≈28° Use the inverse sine function. Finally, a ≈180° - 28° = 152°. Incline Applications We can use vectors to solve incline problems. EXAMPLE 3 Finding a Required Force Find the force required to keep a 50-lb wagon from sliding down a ramp inclined at 20° to the horizontal. (Assume there is no friction.) SOLUTION In Figure 28, the vertical 50-lb force BA represents the force of gravity. It is the sum of vectors BC and -AC. The vector BC represents the force with which the weight pushes against the ramp. The vector BF represents the force that would pull the weight up the ramp. Because vectors BF and AC are equal, AC gives the magnitude of the required force. Vectors BF and AC are parallel, so angle EBD equals angle A by alternate interior angles. Because angle BDE and angle C are right angles, triangles CBA and DEB have two corresponding angles equal and, thus, are similar triangles. Therefore, angle ABC equals angle E, which is 20°. From right triangle ABC, we have the following. sin 20° = AC 50 sin B = side opposite B hypotenuse AC = 50 sin 20° Multiply by 50 and rewrite. AC ≈17 Use a calculator. A force of approximately 17 lb will keep the wagon from sliding down the ramp. S Now Try Exercise 39. Ramp 208 E D 208 A C B F 50 Figure 28 EXAMPLE 4 Finding an Incline Angle A force of 16.0 lb is required to hold a 40.0-lb lawn mower on an incline. What angle does the incline make with the horizontal? SOLUTION This situation is illustrated in Figure 29. Consider right triangle ABC. Angle B equals angle u, the magnitude of vector BA represents the weight of the mower, and vector AC equals vector BE, which represents the force required to hold the mower on the incline. sin B = 16.0 40.0 sin B = side opposite B hypotenuse sin B = 0.4 Simplify. B ≈23.6° Use the inverse sine function. The hill makes an angle of about 23.6° with the horizontal. S Now Try Exercise 41. A C E B u 16.0 40.0 Figure 29 S Now Try Exercise 31.
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