1. Copyright © 2005 Pearson Education, Inc.

2. Chapter 2 Acute Angles and Right Triangles

3. Copyright © 2005 Pearson Education, Inc. 2.1 Trigonometric Functions of Acute Angles

4. Copyright © 2005 Pearson Education, Inc. Slide 2-4 Right-triangle Based Definitions of Trigonometric Functions For any acute angle A in standard position.

5. Copyright © 2005 Pearson Education, Inc. Slide 2-5 Example: Finding Trig Functions of Acute Angles Find the values of sin A, cos A, and tan A in the right triangle shown. A C B 52 48 20

6. Copyright © 2005 Pearson Education, Inc. Slide 2-6 Cofunction Identities For any acute angle A, sin A = cos(90   A)csc A = sec(90   A) tan A = cot(90   A)cos A = sin(90   A) sec A = csc(90   A)cot A = tan(90   A)

7. Copyright © 2005 Pearson Education, Inc. Slide 2-7 Example: Write Functions in Terms of Cofunctions Write each function in terms of its cofunction. a) cos 38  b) sec 78 

8. Copyright © 2005 Pearson Education, Inc. Slide 2-8 Example: Solving Equations Find one solution for the equation Assume all angles are acute angles.

9. Copyright © 2005 Pearson Education, Inc. Slide 2-9 Example: Comparing Function Values Tell whether the statement is true or false. sin 31  > sin 29  In the interval from 0  to 90 , as the angle increases, so does the sine of the angle, which makes sin 31  > sin 29  a true statement.

10. Copyright © 2005 Pearson Education, Inc. Slide 2-10 Special Triangles 30-60-90 Triangle 45-45-90 Triangle

11. Copyright © 2005 Pearson Education, Inc. Slide 2-11 14, 40, 54, 56, 62

12. Copyright © 2005 Pearson Education, Inc. Slide 2-12 Function Values of Special Angles 2 60  11 45  2 30  csc  sec  cot  tan  cos  sin 

13. Copyright © 2005 Pearson Education, Inc. 2.2 Trigonometric Functions of Non-Acute Angles

14. Copyright © 2005 Pearson Education, Inc. Slide 2-14 Reference Angles A reference angle for an angle  is the positive acute angle made by the terminal side of angle  and the x-axis.

15. Copyright © 2005 Pearson Education, Inc. Slide 2-15 Example: Find the reference angle for each angle. a) 218  Positive acute angle made by the terminal side of the angle and the x- axis is 218   180  = 38 . 1387 

16. Copyright © 2005 Pearson Education, Inc. Slide 2-16 Example: Finding Trigonometric Function Values of a Quadrant Angle Find the values of the trigonometric functions for 210 .

17. Copyright © 2005 Pearson Education, Inc. Slide 2-17 Finding Trigonometric Function Values for Any Nonquadrantal Angle  Step 1If  > 360 , or if  < 0 , then find a coterminal angle by adding or subtracting 360  as many times as needed to get an angle greater than 0  but less than 360 . Step 2Find the reference angle  '. Step 3Find the trigonometric function values for reference angle  '. Step 4Determine the correct signs for the values found in Step 3. (Use the table of signs in section 5.2, if necessary.) This gives the values of the trigonometric functions for angle .

18. Copyright © 2005 Pearson Education, Inc. Slide 2-18 Example: Finding Trig Function Values Using Reference Angles Find the exact value of each expression. cos (  240  )

19. Copyright © 2005 Pearson Education, Inc. Slide 2-19 Example: Evaluating an Expression with Function Values of Special Angles Evaluate cos 120  + 2 sin 2 60   tan 2 30 .

20. Copyright © 2005 Pearson Education, Inc. Slide 2-20 Example: Using Coterminal Angles Evaluate each function by first expressing the function in terms of an angle between 0  and 360 . cos 780 

21. Copyright © 2005 Pearson Education, Inc. Slide 2-21 45, 48, 49, 8, 10, 12, 14, 36, 38

22. Copyright © 2005 Pearson Education, Inc. 2.3 Finding Trigonometric Function Values Using a Calculator

23. Copyright © 2005 Pearson Education, Inc. Slide 2-23 Function Values Using a Calculator Calculators are capable of finding trigonometric function values. When evaluating trigonometric functions of angles given in degrees, remember that the calculator must be set in degree mode. To check if your calculator is in degree mode enter sin 90. The answer should be 1. Remember that most calculator values of trigonometric functions are approximations.

24. Copyright © 2005 Pearson Education, Inc. Slide 2-24 Example: Finding Function Values with a Calculator a) b) cot 68.4832 

25. Copyright © 2005 Pearson Education, Inc. Slide 2-25 Angle Measures Using a Calculator Graphing calculators have three inverse functions. If x is an appropriate number, then gives the measure of an angle whose sine, cosine, or tangent is x.

26. Copyright © 2005 Pearson Education, Inc. Slide 2-26 Example: Using Inverse Trigonometric Functions to Find Angles Use a calculator to find an angle in the interval that satisfies each condition.

27. Copyright © 2005 Pearson Education, Inc. Slide 2-27 Example: Using Inverse Trigonometric Functions to Find Angles continued

28. Copyright © 2005 Pearson Education, Inc. Slide 2-28 50, 58

29. Copyright © 2005 Pearson Education, Inc. 2.4 Solving Right Triangles

30. Copyright © 2005 Pearson Education, Inc. Slide 2-30 Significant Digits for Angles A significant digit is a digit obtained by actual measurement. Your answer is no more accurate then the least accurate number in your calculation. Tenth of a minute, or nearest thousandth of a degree 5 Minute, or nearest hundredth of a degree4 Ten minutes, or nearest tenth of a degree3 Degree2 Angle Measure to Nearest:Number of Significant Digits Example 2

31. Copyright © 2005 Pearson Education, Inc. Slide 2-31 Example: Solving a Right Triangle, Given an Angle and a Side Solve right triangle ABC, if A = 42  30' and c = 18.4.

32. Copyright © 2005 Pearson Education, Inc. Slide 2-32 Example: Solving a Right Triangle Given Two Sides Solve right triangle ABC if a = 11.47 cm and c = 27.82 cm.

33. Copyright © 2005 Pearson Education, Inc. Slide 2-33 Definitions Angle of Elevation: from point X to point Y (above X) is the acute angle formed by ray XY and a horizontal ray with endpoint X. Angle of Depression: from point X to point Y (below) is the acute angle formed by ray XY and a horizontal ray with endpoint X.

34. Copyright © 2005 Pearson Education, Inc. Slide 2-34 Solving an Applied Trigonometry Problem Step 1Draw a sketch, and label it with the given information. Label the quantity to be found with a variable. Step 2Use the sketch to write an equation relating the given quantities to the variable. Step 3Solve the equation, and check that your answer makes sense.

35. Copyright © 2005 Pearson Education, Inc. Slide 2-35 46

36. Copyright © 2005 Pearson Education, Inc. Slide 2-36 Example: Application The length of the shadow of a tree 22.02 m tall is 28.34 m. Find the angle of elevation of the sun. Draw a sketch. 22.02 m 28.34 m B

37. Copyright © 2005 Pearson Education, Inc. 2.5 Further Applications of Right Triangles

38. Copyright © 2005 Pearson Education, Inc. Slide 2-38 Bearing Other applications of right triangles involve bearing, an important idea in navigation.

39. Copyright © 2005 Pearson Education, Inc. Slide 2-39 Example An airplane leave the airport flying at a bearing of N 32  E for 200 miles and lands. How far east of its starting point is the plane?

40. Copyright © 2005 Pearson Education, Inc. Slide 2-40 16, 22

41. Copyright © 2005 Pearson Education, Inc. Slide 2-41 Example: Solving a Problem Involving Angles of Elevation Sean wants to know the height of a Ferris wheel. From a given point on the ground, he finds the angle of elevation to the top of the Ferris wheel is 42.3. He then moves back 75 ft. From the second point, the angle of elevation to the top of the Ferris wheel is 25.4. Find the height of the Ferris wheel.

42. Copyright © 2005 Pearson Education, Inc. Slide 2-42 33