1. Slide 1-1 Chapter 2 Acute Angles and Right Triangle Y. Ath

2. Slide 1-2 Section 2. 1 Trigonometric Functions of Acute Angle

3. Slide 1-3 Six Trigonometric Functions using Right Triangle SOH-CAH-TOA 1. Sine 2. Cosine 3. Tangent 4. Cosecant 5. Secant 6. Cotangent Note:

4. Slide 1-4 Find the sine, cosine, and tangent values for angles A and B. Example 1

5. Slide 1-5 Cofunction Identities

6. Slide 1-6 Find one solution for the equation. Assume all angles involved are acute angles. Example 2 SOLVING EQUATIONS USING COFUNCTION IDENTITIES (a) (b)

7. Slide 1-7 Special Triangles 45º-45º-90º (Isosceles Right Triangle) 30º-60º-90º (Equilateral Triangle)

8. Slide 1-8 Section 2.2 Trigonometric Functions of Non-Acute Angles

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10. Slide 1-10 Reference Angles Quad I Quad II Quad III Quad IV θ’ = θθ’ = 180° – θ θ’ = θ – 180°θ’ = 360° – θ

11. Slide 1-11 Example 3 Find the values of the six trigonometric functions for 210°.

12. Slide 1-12 Finding Trigonometric Function Values For Any Nonquadrantal Angle θ Step 1If θ > 360°, or if θ < 0°, 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 θ′. Step 3Find the trigonometric function values for reference angle θ′.

13. Slide 1-13 Finding Trigonometric Function Values For Any Nonquadrantal Angle θ (continued) Step 4Determine the correct signs for the values found in Step 3. This gives the values of the trigonometric functions for angle θ.

14. Slide 1-14 Find the exact value of sin (–150°). Example 4 Step 1. Find a coterminal angle of –150° Step 2. Find a reference angle.

15. Slide 1-15 Find the exact value of cot 780°. Example 5 Step 1. Find a coterminal angle of 780° Step 2. Find a reference angle.

16. Slide 1-16 Evaluate. Example 6

17. Slide 1-17 Example 7

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19. Slide 1-19 Section 2.3 Finding Trig Function Values Using a Calculator

20. Slide 1-20 Degree mode

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22. Slide 1-22 Example 8 FINDING GRADE RESISTANCE

23. Slide 1-23 Example 8 FINDING GRADE RESISTANCE (cont.) (a)Calculate F to the nearest 10 lb for a 2500-lb car traveling an uphill grade with θ = 2.5°. (b)Calculate F to the nearest 10 lb for a 5000-lb truck traveling a downhill grade with θ = –6.1°. F is negative because the truck is moving downhill.

24. Slide 1-24 Example 8 FINDING GRADE RESISTANCE (cont.) (c)Calculate F for θ = 0° and θ = 90°. Do these answers agree with your intuition? If θ = 0°, then there is level ground and gravity does not cause the vehicle to roll. If θ = 90°, then the road is vertical and the full weight of the vehicle would be pulled downward by gravity, so F = W.

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28. Slide 1-28 Significant Digits A significant digit is a digit obtained by actual measurement. The significant digits in the following numbers are identified in color. 40821.518.006.7000.0025 0.09810 7300

29. Slide 1-29 To determine the number of significant digits for answers in applications of angle measure, use the following table.

30. Slide 1-30 Example 9 SOLVING A RIGHT TRIANGLE GIVEN AN ANGLE AND A SIDE Solve right triangle ABC, if A = 34°30′ and c = 12.7 in.

31. Slide 1-31 Example 11 SOLVING A RIGHT TRIANGLE GIVEN TWO SIDES Solve right triangle ABC, if a = 29.43 cm and c = 53.58 cm. or 33º 19΄

32. Slide 1-32 Angles of Elevation or Depression

33. Slide 1-33 Example FINDING A LENGTH GIVEN THE ANGLE OF ELEVATION Pat Porterfield knows that when she stands 123 ft from the base of a flagpole, the angle of elevation to the top of the flagpole is 26°40′. If her eyes are 5.30 ft above the ground, find the height of the flagpole. Since Pat’s eyes are 5.30 ft above the ground, the height of the flagpole is 61.8 + 5.30 = 67.1 ft.

34. Slide 1-34 Example 12 FINDING AN ANGLE OF DEPRESSION From the top of a 210-ft cliff, David observes a lighthouse that is 430 ft offshore. Find the angle of depression from the top of the cliff to the base of the lighthouse.

35. Slide 1-35 Example 12 FINDING AN ANGLE OF DEPRESSION (continued)

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37. Slide 1-37 Bearing There are two methods for expressing bearing. When a single angle is given, such as 164°, it is understood that the bearing is measured in a clockwise direction from due north.

38. Slide 1-38 Example 13 SOLVING A PROBLEM INVOLVING BEARING (METHOD 1) Radar stations A and B are on an east-west line, 3.7 km apart. Station A detects a plane at C, on a bearing of 61°. Station B simultaneously detects the same plane, on a bearing of 331°. Find the distance from A to C.

39. Slide 1-39 Bearing The second method for expressing bearing starts with a north-south line and uses an acute angle to show the direction, either east or west, from this line.

40. Slide 1-40 Example 14 SOLVING A PROBLEM INVOLVING BEARING (METHOD 2) A ship leaves port and sails on a bearing of N 47º E for 3.5 hr. It then turns and sails on a bearing of S 43º E for 4.0 hr. If the ship’s rate of speed is 22 knots (nautical miles per hour), find the distance that the ship is from port.

41. Slide 1-41 Example 14 SOLVING A PROBLEM INVOLVING BEARING (METHOD 2) (cont.) Now find c, the distance from port at point A to the ship at point B.

42. Slide 1-42 Example 15 USING TRIGONOMETRY TO MEASURE A DISTANCE The subtense bar method is a method that surveyors use to determine a small distance d between two points P and Q. The subtense bar with length b is centered at Q and situated perpendicular to the line of sight between P and Q. Angle θ is measured, then the distance d can be determined. (a)Find d with θ = 1°23′12″ and b = 2.0000 cm. From the figure, we have

43. Slide 1-43 Let b = 2. Convert θ to decimal degrees: Example 15 USING TRIGONOMETRY TO MEASURE A DISTANCE (continued)

44. Slide 1-44 Since θ is 1″ larger, θ = 1°23′13″ ≈ 1.386944º. (b)How much change would there be in the value of d if θ were measured 1″ larger? Example 15 USING TRIGONOMETRY TO MEASURE A DISTANCE (continued) The difference is