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Chapter 2

Chapter 2. Acute Angles and Right Triangles. 2.1. Trigonometric Functions of Acute Angles. Right-triangle Based Definitions of Trigonometric Functions. For any acute angle A in standard position. 48. C. A. 20. 52. B. Example: Finding Trig Functions of Acute Angles.

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Chapter 2

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  1. Chapter 2 Acute Angles and Right Triangles

  2. 2.1 Trigonometric Functions of Acute Angles

  3. Right-triangle Based Definitions of Trigonometric Functions • For any acute angle A in standard position.

  4. 48 C A 20 52 B Example: Finding Trig Functions of Acute Angles • Find the values of sin A, cos A, and tan A in the right triangle shown.

  5. 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)

  6. Write each function in terms of its cofunction. a) cos 38 b) sec 78 Example: Write Functions in Terms of Cofunctions

  7. Example: Solving Equations • Find one solution for the equation Assume all angles are acute angles.

  8. 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.

  9. 30-60-90 Triangle 45-45-90 Triangle Special Triangles

  10. 14, 40, 54, 56, 62

  11. sin  cos  tan  cot  sec  csc  30 2 45 1 1 60 2 Function Values of Special Angles

  12. 2.2 Trigonometric Functions of Non-Acute Angles

  13. Reference Angles • A reference angle for an angle  is the positive acute angle made by the terminal side of angle  and the x-axis.

  14. a) 218 Positive acute angle made by the terminal side of the angle and the x-axis is 218  180 = 38. 1387 Example: Find the reference angle for each angle.

  15. Find the values of the trigonometric functions for 210. Example: Finding Trigonometric Function Values of a Quadrant Angle

  16. Finding Trigonometric Function Values for Any Nonquadrantal Angle  • Step 1 If  > 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 2 Find the reference angle '. • Step 3 Find the trigonometric function values for reference angle '. • Step 4 Determine 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 .

  17. Find the exact value of each expression. cos (240) Example: Finding Trig Function Values Using Reference Angles

  18. Example: Evaluating an Expression with Function Values of Special Angles • Evaluate cos 120 + 2 sin2 60  tan2 30.

  19. Example: Using Coterminal Angles • Evaluate each function by first expressing the function in terms of an angle between 0 and 360. • cos 780

  20. 45, 48, 49, 8, 10, 12, 14, 36, 38

  21. 2.3 Finding Trigonometric Function Values Using a Calculator

  22. 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.

  23. a) b) cot 68.4832  Example: Finding Function Values with a Calculator

  24. 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.

  25. Example: Using Inverse Trigonometric Functions to Find Angles • Use a calculator to find an angle in the interval that satisfies each condition.

  26. Example: Using Inverse Trigonometric Functions to Find Angles continued

  27. 50, 58

  28. 2.4 Solving Right Triangles

  29. Number of Significant Digits Angle Measure to Nearest: 2 Degree 3 Ten minutes, or nearest tenth of a degree 4 Minute, or nearest hundredth of a degree 5 Tenth of a minute, or nearest thousandth of a degree 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. • Example 2

  30. Example: Solving a Right Triangle, Given an Angle and a Side • Solve right triangle ABC, if A = 42 30' and c = 18.4.

  31. Example: Solving a Right Triangle Given Two Sides • Solve right triangle ABC if a = 11.47 cm and c = 27.82 cm.

  32. 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. Definitions

  33. Solving an Applied Trigonometry Problem • Step 1 Draw a sketch, and label it with the given information. Label the quantity to be found with a variable. • Step 2 Use the sketch to write an equation relating the given quantities to the variable. • Step 3 Solve the equation, and check that your answer makes sense.

  34. 46

  35. 22.02 m B 28.34 m 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.

  36. 2.5 Further Applications of Right Triangles

  37. Bearing • Other applications of right triangles involve bearing, an important idea in navigation.

  38. 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?

  39. 16, 22

  40. 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.

  41. 33

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