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Chapter 2 Trigonometric Functions

Chapter 2 Trigonometric Functions

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Chapter 2 Trigonometric Functions

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  1. Chapter 2Trigonometric Functions 2.1 Degrees and Radians 2.2 Linear and Angular Velocity 2.3 Trigonometric Functions: Unit Circle Approach 2.4 Additional Applications 2.5 Exact Values and Properties of Trigonometric Functions

  2. 2.1 Degrees and Radians • Degree and radian measure of angles • Angles in standard position • Arc length and area of a sector of a circle

  3. Radian Measure of Central Angles • Example: • Find the radian measure of the central angle subtended by an arc of 32 cm in a circle of radius 8 cm. • Solution: q = 32cm/8cm = 4 rad

  4. Radian-Degree Conversion Formulas • Example: • Find the radian measure of -1.5 rad in terms of p and in decimal form to 4 decimal places. • Solution: qd = (qr)(180º/p rad) = • (-1.5)(180/p) = 270º/p = -85.9437º

  5. Angles in Standard Position

  6. Sketching angles in Standard Position • Sketch these angles in standard position: A. -60º B. 3p/2 rad C. -3p rad D. 405º

  7. Coterminal Angles • Angles that differ by an integer multiple of 2 p or 360º are coterminal. • Example: Are the angles –p/3 rad and 2p/3 rad coterminal? Solution: (-p/3) – (2p/3) = -3p/3 = -pNo • Example: Are the angles -135º and 225º coterminal? Solution: -135º - (-225º) = -1(360)º Yes

  8. Area of a Sector of a Circle • A = ½ r2q, r = radius and q = central angle • Example: In a circle of radius 3 m find the area of the sector with central angle 0.4732. Solution: A = ½ 3m2(0.4732) ≈ 2.13 m2

  9. 2.2 Linear and Angular Velocity

  10. Electrical Wind Generator • This wind generator has propeller blades 5 m long. If the blades are rotating at 8 p rad/sec, what is the angular velocity of a point on the tip of one blade? • Solution: V = 5 (8p) = 126 m/sec

  11. 2.3 Trigonometric Functions: The Unit Circle Approach • Definition of Trigonometric Functions • Calculator Evaluation • Application • Summary of Sign Properties

  12. Trigonometric Functions

  13. The Unit Circle If a point (a,b) lies on the unit circle, then the following are true for the angle x associated with that point: sin x = b cos x = a tan x = b/a (a ≠ 0) csc x = 1/b (b ≠ 0) sec x = 1/a (a ≠ 0) cot x = a/b (b ≠ 0)

  14. Evaluating Trigonometric Functions Example: Find the exact values of the 6 trigonometric functions for the point (-4, -3) The Pythagorean Theorem shows that the distance from the point to the origin is 5. sin x = -3/5 cos x = -4/5 tan x = 3/4 csc x = -5/3 sec x = -5/4 cot x = 4/3

  15. Using Given Information to Evaluate Trigonometric Functions • Example: • Given that the terminal side of an angle is in Quadrant IV and cos x = 3/5 find the remaining trigonometric functions. • b2 = 25 – 9 = 16, so b = 4 • Sin x = 4/5, tan x = -4/3, csc x = -5/4, • sec x = 5/3 and cot x = -3/4

  16. Reciprocal Relationships

  17. Calculator Evaluation • Set the calculator in the proper mode for each method of evaluating trigonometric functions. Use degree mode or radian mode. • Example: Find tan 3.472 rad Solution: tan 3.472 rad ≈ .3430 • Example: Find csc 192º 47’ 22” Solution: csc 192º 47’ 22” ≈ 1/ sin 192.7894… ≈ -4.517

  18. Additional Applications • Modeling light waves and refraction • Modeling bow waves • Modeling sonic booms • High-energy physics: Modeling particle energy • Psychology: Modeling perception

  19. Light Rays

  20. Reflected Light • Example: • What is the angle of incidence a that will cause a light beam to be totally reflected? • Solution: sin a = (sin 90º)1/1.33 • a = sin-1 (1/1.33)≈ 48.8º

  21. 2.5 Exact Values and Properties of Trigonometric Functions • Exact values of trigonometric functions at special angles • Reference triangles • Periodic functions • Fundamental identities

  22. Special Angles

  23. Using Special Angles for Points (a,b) • Example: Find sec 5p/4 Solution: (a, b) = (-1/√2, -1/√2) sec 5p/4 = 1/a = -√2 • Example: Find sin 135º Solution: (a, b) = (-1/√2, 1/√2) sin 135º = b = 1/√2

  24. Angles on the Unit Circle

  25. Using Special Angles for Points (a,b) • Example: • Find sin 7p/6 • Solution: (a, b) = (-√3/2, -1/2) • sin 7p/6 = b = -1/2

  26. Reference Triangle and Reference Angle

  27. Reference Triangles and Angles • Example: Sketch the reference triangle and find the reference angle a for q = -315º. Solution:

  28. Periodic Functions • Adding any integer multiple of 2p to x returns the same point on the circle. • sin x = sin (x + 2p) • cos x = cos (x + 2p) • If sin x = 0.7714 then sin(x + 2p) = 0.7714

  29. Fundamental Identities • csc x = 1/b = 1/sin x • sec x = 1/a = 1/cos x • cot x = a/b = 1/tan x • tan x = b/a = sin x / cos x • cot x = a/b = cos x / sin x • sin2x + cos2x = 1

  30. Use of Identities Claim: Proof: