1. Trigonometric Applications and Models Digital Lesson

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

3. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 3 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. an angle  and a side a, or two sides, a and b. Solving Right Triangles b a b a b a θ θ a a a θ

4. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 4 Solving A Right Triangle Given an Angle and a Side Solve the right triangle. The third angle is 60 , the complement of 30 . Use the values of the trigonometric functions of 30 o. 10 60 ○ 30 ○ 5 5 Since = sin 30  = =, it follows that hyp = 10. hyp opp hyp 5 Solving A Right Triangle Given an Angle and a Side To get the last side, note that = cos 30  = ; therefore, adj =

5. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 5 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. It follows that x = 327. The distance across the river from post A to post B is 327 feet. Use a calculator to find tan 73 o = 3.27. Post B Post A 100 ft. x 73 ○ Example 1: Application 3.27 = tan 73  = = adj opp

6. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 6 Inverse Trigonometric Functions on a Calculator Example: Find the acute angle  for which cos  = 0.25. Calculator keystrokes: ( SHIFT ) cos  1 0.25 = Inverse functions are often accessed by using a key that maybe be labeled SHIFT, INV, or 2 nd. Check the manual for your calculator. Labels for sin  1, cos  1, and tan  1 are usually written above the sin, cos, and tan keys. Inverse Trigonometric Functions on a Calculator Display: 75.22487

7. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 7 Solving a Right Triangle Given Two Sides hyp 2 = 6 2 + 5 2 = 61 Subtract to calculate the third angle: 90   39.805571  = 50.194428 . Solve for the hypotenuse: Solve for  : Calculator Keystrokes: (SHIFT) tan  1 ( 5  6 ) Display: 39.805571  hyp = = 7.8102496 39.8 ○ 6 5 50.2 ○ Solve the right triangle shown. θ 5 Solving a Right Triangle Given Two Sides tan  = = and  = tan -1 ( ). adj opp 6

8. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 8 angle of elevation When an observer is looking downward, the angle formed by a horizontal line and the line of sight is called the: Angle of Elevation and Angle of Depression When an observer is looking upward, angle of elevation. the angle formed by a horizontal line and the line of sight is called the: observer object line of sight horizontal observer object line of sight horizontal angle of depression angle of depression. Angle of Elevation and Angle of Depression

9. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 9 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? The ship is 146 m from the base of the cliff. line of sight angle of depression horizontal observer ship cliff 42 m 16 ○ d Example 2: Application d = = 146.47.

10. Copyright © by Houghton Mifflin Company, Inc. All rights reserved. 10 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? Next use the inverse sine function to find .  = sin  1 (0.875) = 61.044975 The painter’s plan is unsafe! ladder house 16 14 The angle formed by the ladder and the ground is about 61 . θ Example 3: Application sin  = = 0.875