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Introduction to Trigonometric Functions: Key Concepts and Examples

Understand trigonometric functions, trig ratios in triangles, angles on the coordinate plane, DMS conversions, reference angles, and how to find trig ratios in quadrants 2, 3, and 4 with practical examples.

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Introduction to Trigonometric Functions: Key Concepts and Examples

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  1. Introduction to the trigonometric functions

  2. What is a Trigonometric Function? • There are 6 trig functions that represent the ratios of the sides in any right triangle. • They are functions of an ANGLE so you must always specify the angle in the abbreviation. • Sine (sin α) • Cosine (cosθ) • Tangent (tan β) • Cosecant (cscA) • Secant (sec Q) • Cotangent (cot γ)

  3. The Sine and Cosine Functions • The sine function of an acute angle θin a right triangleis the ratio of the opposite side from angle θdivided by the hypotenuse of the triangle. • The cosine function of an acute angle θin a right triangleis the ratio of the adjacent side to an angle θdivided by the hypotenuse of the triangle.

  4. Quick Examples: • Find the value of the sine function in each of the following. α θ

  5. THE RECIPROCAL IDENTITIES

  6. Examples: • Find the value of each trigonometric function below. α θ

  7. Examples: • Find the value of the 6 trigonometric functions for angle  shown below.  25 24

  8. Examples: • Use the triangle below to fill in the blank in each expression. 63 2√5 2 27 4

  9. Examples: • Find the value of the other 5 trigonometric functions if you know that 

  10. THINK ABOUT IT???? • Is it possible for Explain why or why not. • Is it possible for Explain why or why not.

  11. Special Triangles Trig ratios for the 45°- 45° - 90° Triangle and the 30° – 60° – 90° Triangle.

  12. The 45°- 45° - 90° Triangle Let there be an isosceles triangle such that one leg has a length of 1. Then…. 1 1

  13. The 30° – 60° – 90° Triangle Let there be an equilateral triangle such that one side has a length of 2. Then…. 2 1

  14. Positive angles are always measured • counterclockwise. • Vertex is always at the origin • Initial side is always the positive x-axis  ANGLES ON THE COORDINATE PLANE & THE UNIT CIRCLE

  15. Positions of Angles on the Coordinate Plane 1st quadrant: 0 - 90 2nd quadrant: 90 - 180 3rd quadrant: 180 - 270 4th quadrant: 270 - 360

  16. Examples: • Name the quadrant in which each angle lies. a) 80 b) 120 c) 230 d) 145 a) 1st b) 2nd c) 3rd d) 2nd e) 200 f) 340 g) 90 e) 3rd f) 4th g) positive y-axis

  17. Negative Angles • Negative angles are measured clockwise on the Coordinate Plane.

  18. Examples: • Name the quadrant in which each angle lies. a) -280 b) -190 c) -130 d) -245 a) 1st b) 2nd c) 3rd d) 2nd e) -100 f) -80 g) -90 e) 3rd f) 4th g) negative y-axis

  19. Coterminal Angles • Angles whose terminal sides are in the same position on the coordinate plane. •  + 360(n),where ‘n’ is any integer. • Least positive coterminal angle: coterminal angle between 0 - 360

  20. Examples: Find the least positive coterminal angle for the following angles on the coordinate plane: a) 380 b) 420 c) -130 d) -245 a) 20 b) 60 c) 230 d) 115

  21. Degrees – Minutes - Seconds What is DMS? • 52 32’ 22” • Conversions: • 1 degree = 60 minutes • 1 minute = 60 seconds • 1 degree = 3600 seconds

  22. CONVERTING DMS TO DECIMAL • 13 13’ 22” • 46 34’ 51”

  23. CONVERTING AN ANGLE TO DMS

  24. Converting a Negative Angle • When working with a negative angle, disregard the negative until the final answer, then reinsert. • Examples: • -57.62 • -57 37’ 12” • -39 30’ 13” • -39.50

  25. REFERENCE ANGLES Right triangle equivalents in Q2, 3, 4 Will always be an angle between 0 - 90˚ Always create your triangle with the x-axis

  26. Reference angles in quadrant 2 • Quadrant 2:

  27. Reference angles in quadrant 3 • Quadrant 3:

  28. Reference angles in quadrant 4 • Quadrant 4:

  29. Reference Angles - Summary • If  is a nonquadrantal angle in standard position, then the reference angle for  is the positive acute angle ’ formed by the terminal side of  and the x-axis 3rd quad: ’ =  - 180 1st quad: ’ =  2st quad: ’ = 180 -  4th quad: ’ = 360 - 

  30. Finding reference angles • Find the reference angle for each of the following angles. a) 190 b) 120 c) -30 d) -210 a) 10 b) 60 c) 30 d) 30 e) 90 f) 490 g) 260 h) -160 e) none f) 50 g) 80 h) 20

  31. Reference Triangles for Specials • What angles would have the following reference angles in the given quadrants? • Quadrant 2: • Reference: 30 - • Reference: 45 - • Reference: 60 - • Quadrant 3: • Reference: 30 - • Reference: 45 - • Reference: 60 - • Quadrant 4: • Reference: 30 - • Reference: 45 - • Reference: 60 -

  32. HOW DO YOU FIND A TRIG RATIO FOR AN ANGLE IN QUADRANTS 2, 3 OR 4? The trigonometric functions of an angle in Q 2, 3, or 4 will have the same numerical values as the trig function of their reference angles, Q’ Depending on the quadrant you may need to adjust the sign (±)

  33. Trig Ratio for angles in quadrant 2 • Quadrant 2:

  34. Trig Ratio for angles in quadrant 3 • Quadrant 3:

  35. Trig Ratio for angles in quadrant 4 • Quadrant 4:

  36. Summary

  37. Unit Circle Centered at origin with radius of 1 • Let θbe an angle in standard position on the coordinate plane. • Let (x, y) be a point on the Unit Circle and on the terminal side of θ. • Drop a perpendicular from (x,y) to the x-axis forming a right triangle.

  38. Summary • Any point on the Unit Circle (x, y) is the same as (cosθ , sin θ) where θ is an angle in standard position. θ θ θ

  39. Unit Circle & Reference Triangles

  40. Using the Unit Circle to Find the Trig function value for the Quadrantals • 0 • Sin 0= • Cos 0 = • 90 • Sin 90 = • Cos 90 = • 180 • Sin 180 = • Cos 180 = • 270 • Sin 270 = • Cos 270 =

  41. RADIAN MEASURE OF AN ANGLE EQ: What is radian measure and why do we use it to measure angles? Definition: measure of an angle based off of the length of the intercepted arc on the Unit Circle.

  42. Positions of Angles on the Coordinate Plane 1st quadrant: 0 - /2 2nd quadrant: /2 -  3rd quadrant:  - 3/2 4th quadrant: 3/2 - 2

  43. Examples: • Name the quadrant in which each angle lies. a) /3 b) 5/6 c) -3/4 a) 1st b) 2nd c) 3rd d) 4/3 e) 3/2 f) 8/3 d) 3rd e) negative y-axis f) 2nd

  44. Converting Between Degrees and Radians • To convert an angle from degrees to radians or radians to degrees, use the following conversion factor: • 210

  45. Converting from degrees to radians • Convert each angle into radians. Express answers in terms of . a) 90 b) 120 c) -130 d) -225 a) /2 b) 2/3 c) -13/18 d) -5/4

  46. Converting from radians to degrees • Convert each angle into degree measure. a) 3/2 b) /3 c) -/8 d) -7/4 a) 270 b) 60 c) -22.5 d) -315

  47. The 45°- 45° - 90° Triangle The /4 - /4 - /2 triangle

  48. The 30° – 60° – 90° Triangle The /6 - /3 - /2 triangle

  49. Reference Angles in Radians • Quadrant 2: • 5/6 - Reference: /6 • 3/4 - Reference: /4 • 2/3 - Reference: /3 • Quadrant 3: • 7/6 - Reference: /6 • 5/4 - Reference: /4 • 4/3 - Reference: /3 • Quadrant 4: • 11/6 - Reference: /6 • 7/4 - Reference: /4 • 5/3 - Reference: /3

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