Trigonometric Applications and Models 5.7 Trigonometric Applications and Models

Trigonometric Functions on a Calculator Example 1: Calculate sin 40. Set the calculator in degree mode. Calculator keystrokes: sin 40 = Display: 0.6427876 Example 2: Calculate sec 40. Calculator keystrokes: 1  cos 40 = Display: 1.3054072 Copyright © by Houghton Mifflin Company, Inc. All rights reserved. Trigonometric Functions on a Calculator

Solving Right Triangles Solving a right triangle means to find the lengths of the sides and the measures of the angles of a right triangle. Some information is usually given. θ a an angle  and a side a, b a or two sides, a and b. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. Solving Right Triangles

Solving A Right Triangle Given an Angle and a Side Solve the right triangle. 30○ 5 The third angle is 60, the complement of 30. Use the values of the trigonometric functions of 30o. Since = sin 30 = = , it follows that hyp = 10. hyp opp 5 30○ 5 To get the last side, note that = cos 30 = ; therefore, adj = 10 60○ Copyright © by Houghton Mifflin Company, Inc. All rights reserved. Solving A Right Triangle Given an Angle and a Side

Example 1: A bridge is to be constructed across a small river from post A to post B. A surveyor walks 100 feet due south of post A. She sights on both posts from this location and finds that the angle between the posts is 73. Find the distance across the river from post A to post B. Post B Post A 100 ft. x 73○ Use a calculator to find tan 73o = 3.27. 3.27 = tan 73= = adj opp It follows that x = 327. The distance across the river from post A to post B is 327 feet. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. Example 1: Application

Inverse Trigonometric Functions on a Calculator Labels for sin1, cos1, and tan1 are usually written above the sin, cos, and tan keys. Inverse functions are often accessed by using a key that maybe be labeled SHIFT, INV, or 2nd. Check the manual for your calculator. Example: Find the acute angle  for which cos  = 0.25. Calculator keystrokes: (SHIFT) cos1 0.25 = Display: 75.522487 Copyright © by Houghton Mifflin Company, Inc. All rights reserved. Inverse Trigonometric Functions on a Calculator

Solving a Right Triangle Given Two Sides Solve the right triangle shown. θ 5 Solve for the hypotenuse: hyp2 = 62 + 52 = 61 6 hyp = = 7.8102496 6 5 50.2○ Solve for  : tan  = = and  = tan-1( ). adj opp 39.8○ Calculator Keystrokes: (SHIFT) tan1 ( 5  6 ) Display: 39.805571 Subtract to calculate the third angle: 90 39.805571 = 50.194428. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. Solving a Right Triangle Given Two Sides

Angle of Elevation and Angle of Depression When an observer is looking upward, the angle formed by a horizontal line and the line of sight is called the: angle of elevation. observer object line of sight horizontal angle of elevation When an observer is looking downward, the angle formed by a horizontal line and the line of sight is called the: observer object line of sight horizontal angle of depression. angle of depression Copyright © by Houghton Mifflin Company, Inc. All rights reserved. Angle of Elevation and Angle of Depression

Example 2: A ship at sea is sighted by an observer at the edge of a cliff 42 m high. The angle of depression to the ship is 16. What is the distance from the ship to the base of the cliff? observer horizontal 16○ angle of depression cliff 42 m line of sight 16○ ship d d = = 146.47. The ship is 146 m from the base of the cliff. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. Example 2: Application

Example 3: A house painter plans to use a 16 foot ladder to reach a spot 14 feet up on the side of a house. A warning sticker on the ladder says it cannot be used safely at more than a 60 angle of inclination. Does the painter’s plan satisfy the safety requirements for the use of the ladder? ladder house sin  = = 0.875 16 14 θ Next use the inverse sine function to find .  = sin1(0.875) = 61.044975 The angle formed by the ladder and the ground is about 61. The painter’s plan is unsafe! Copyright © by Houghton Mifflin Company, Inc. All rights reserved. Example 3: Application