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Trigonometric Functions

1. Trigonometric Functions. Trigonometric Functions. 1. 1.1 Angles 1.2 Angle Relationships and Similar Triangles 1.3 Trigonometric Functions 1.4 Using the Definitions of the Trigonometric Functions. Angles. 1.1.

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Trigonometric Functions

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  1. 1 Trigonometric Functions

  2. Trigonometric Functions 1 • 1.1 Angles • 1.2 Angle Relationships and Similar Triangles • 1.3 Trigonometric Functions • 1.4 Using the Definitions of the Trigonometric Functions

  3. Angles 1.1 Basic Terminology ▪ Degree Measure▪ Standard Position▪ Coterminal Angles

  4. 1.1Example 1 Finding the Complement and the Supplement of an Angle (page 3) • For an angle measuring 55°, find the measure of its complement and its supplement. Complement: 90° − 55° = 35° Supplement: 180° − 55° = 125°

  5. 1.1Example 2(a) Finding Measures of Complementary and Supplementary Angles (page 3) and • Find the measure of each angle. The two angles form a right angle, so they are complements. The measures of the two angles are

  6. 1.1Example 2(b) Finding Measures of Complementary and Supplementary Angles (page 3) and • Find the measure of each angle. The two angles form a straight angle, so they are supplements. The measures of the two angles are

  7. 1.1Example 3 Calculating with Degrees, Minutes, and Seconds (page 4) • Perform each calculation. (a) (b)

  8. 1.1Example 4 Converting Between Decimal Degrees and Degrees, Minutes, and Seconds (page 5) (a) Convert 105°20′32″ to decimal degrees. (b) Convert 85.263° to degrees, minutes, and seconds.

  9. 1.1Example 5 Finding Measures of Coterminal Angles (page 6) • Find the angles of least possible positive measure coterminal with each angle. (a) 1106° Add or subtract 360° as many times as needed to obtain an angle with measure greater than 0° but less than 360°. An angle of 1106° is coterminal with an angle of 26°. (b) –150° An angle of –150° is coterminal with an angle of 210°.

  10. 1.1Example 5 Finding Measures of Coterminal Angles (cont.) (c) –603° An angle of –603° is coterminal with an angle of 117°.

  11. 1.1Example 6 Analyzing the Revolutions of a CD Player (page 7) The wheel makes 270 revolutions in one minute or revolutions per second. In five seconds, the wheel makesrevolutions. Each revolution is 360°, so a point on the edge of the wheel will move A wheel makes 270 revolutions per minute. Through how many degrees will a point on the edge of the wheel move in 5 sec?

  12. Angles 1.2 Geometric Properties ▪ Triangles

  13. 1.2Example 1 Finding Angle Measures (page 11) • Find the measures of angles 1, 2, 3, and 4 in the figure, given that lines m and n are parallel. Angles 2 and 3 are interior angles on the same side of the transversal, so they are supplements.

  14. 1.2Example 1 Finding Angle Measures (cont.) Angles 1 and 2 have equal measure because they are vertical angles, and angles 1 and 4 have equal measure because they are alternate exterior angles.

  15. 1.2Example 2 Finding Angle Measures (page 12) • The measures of two of the angles of a triangle are 33° and 26°. Find the measure of the third angle. The sum of the measures of the angles of a triangle is 360°. Let x = the measure of the third angle. The third angle measures 121°.

  16. 1.2Example 3 Finding Angle Measures in Similar Triangles (page 13) • In the figure, triangles DEF and GHI are similar. Find the measures of angles G and I. The triangles are similar, so the corresponding angles have the same measure.

  17. 1.2Example 4 Finding Side Lengths in Similar Triangles (page 14) • Given that triangle MNP and triangle QSR are similar, find the lengths of the unknown sides of triangle QSR. The triangles are similar, so the lengths of the corresponding sides are proportional. PM corresponds to RQ. PN corresponds to RS. MN corresponds to QS.

  18. 1.2Example 4 Finding Side Lengths in Similar Triangles (cont.)

  19. 1.2Example 5 Finding the Height of a Flagpole (page 14) • Joey wants to know the height of a tree in a park near his home. The tree casts a 38-ft shadow at the same time that Joey, who is 63 in. tall, casts a 42-in. shadow. Find the height of the tree. Let x = the height of the tree The tree is 57 feet tall.

  20. Trigonometric Functions 1.3 Trigonometric Functions ▪ Quadrantal Angles

  21. 1.3Example 1 Finding Function Values of an Angle (page 22) • The terminal side of an angle θ in standard position passes through the point (12, 5). Find the values of the six trigonometric functions of angle θ. x = 12 and y = 5. 13

  22. 1.3Example 2 Finding Function Values of an Angle (page 22) • The terminal side of an angle θ in standard position passes through the point (8, –6). Find the values of the six trigonometric functions of angle θ. x = 8 and y = –6. 6 10

  23. 1.3Example 2 Finding Function Values of an Angle (cont.)

  24. 1.3Example 3 Finding Function Values of an Angle (page 23) • Find the values of the six trigonometric functions of angle θ in standard position, if the terminal side of θ is defined by 3x – 2y = 0, x ≤ 0. Since x ≤ 0, the graph of the line 3x – 2y = 0 is shown to the left of the y-axis. Find a point on the line:Let x = –2. Then A point on the line is (–2, –3).

  25. 1.3Example 3 Finding Function Values of an Angle (cont.)

  26. 1.3Example 4(a) Finding Function Values of Quadrantal Angles (page 25) • Find the values of the six trigonometric functions of a 360° angle. The terminal side passes through (2, 0). So x = 2 and y = 0 and r = 2.

  27. 1.3Example 4(b) Finding Function Values of Quadrantal Angles (page 25) • Find the values of the six trigonometric functions of an angle θ in standard position with terminal side through (0, –5). x = 0 and y = –5 and r = 5.

  28. Using the Definitions of the Trigonometric Functions 1.4 Reciprocal Identities ▪ Signs and Ranges of Function Values▪ Pythagorean Identities▪ Quotient Identities

  29. 1.4Example 1 Using the Reciprocal Identities (page 29) • Find each function value. (a) tan θ, given that cot θ = 4. tan θ is the reciprocal of cot θ. (b) sec θ, given that sec θ is the reciprocal of cos θ.

  30. 1.4Example 2 Finding Function Values of an Angle (page 30) (b) A 260º angle in standard position lies in quadrant III, so its sine, cosine, secant, and cosecant are negative, while its tangent and cotangent are positive. (c) A –60º angle in standard position lies in quadrant IV, so cosine and secant are positive, while its sine, cosecant, tangent, and cotangent are negative. • Determine the signs of the trigonometric functions of an angle in standard position with the given measure. (a) 54° (b) 260° (c) –60° (a) A 54º angle in standard position lies in quadrant I, so all its trigonometric functions are positive.

  31. 1.4Example 3 Identifying the Quadrant of an Angle (page 31) • Identify the quadrant (or possible quadrants) of an angle θthat satisfies the given conditions. (a) tan θ > 0, csc θ < 0 tan θ > 0 in quadrants I and III, while csc θ < 0 in quadrants III and IV. Both conditions are met only in quadrant III. (b) sin θ > 0, csc θ > 0 sin θ > 0 in quadrants I and II, as is csc θ. Both conditions are met in quadrants I and II.

  32. 1.4Example 4 Deciding Whether a Value is in the Range of a Trigonometric Function (page 32) (a) cot θ = –0.999 is possible because the range of cot θ is (b) cos θ = –1.7 is impossible because the range of cos θ is [–1, 1]. (c) csc θ = 0 is impossible because the range of csc θ is • Decide whether each statement is possible or impossible. (a) cot θ = –0.999 (b)cos θ = –1.7 (c) csc θ = 0

  33. 1.4Example 5 Finding All Function Values Given One Value and the Quadrant (page 32) Since and θlies in quadrant III, then x = –5 and y = –8. • Angle θlies in quadrant III, and Find the values of the other five trigonometric functions.

  34. 1.4Example 5 Finding All Function Values Given One Value and the Quadrant (cont.)

  35. 1.4Example 6 Finding Other Function Values Given One Value and the Quadrant (page 34) • Find cos θ and tan θ given that sin θandcos θ > 0. Reject the negative root.

  36. 1.4Example 6 Finding Other Function Values Given One Value and the Quadrant (cont.)

  37. 1.4Example 7 Using Identities to Find Function Values (page 35) • Find sin θ and cos θ given that cot θandθ is in quadrant II. Since θ is in quadrant II, sin θ > 0 and cos θ < 0.

  38. 1.4Example 7 Finding Other Function Values Given One Value and the Quadrant (cont.)

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