Circular Trigonometric Functions

1. Circular Trigonometric Functions

2. Circular Trigonometric Functions Y • circle…center at (0,0) • radius r…vector with • length/direction r θ X • angle θ… determines • direction

3. Y-axis 90º Quadrant II Quadrant I Terminalside r r θ X-axis 0º 180º Initial side 360º Quadrant III Quadrant IV 270º

4. Y-axis -270º Quadrant II Quadrant I -360º X-axis -180º Terminal side Initial side 0º r θ Quadrant III Quadrant IV -90º

5. angle θ…measured from positive x-axis, • orinitial side, to terminal side • counterclockwise: positive direction • clockwise: negative direction • four quadrants…numbered I, II, III, IV counterclockwise

6. six trigonometric functions for angle θ whose terminal side passes thru point (x, y) on circle of radius r sin θ = y / r csc θ = r / y cos θ = x / r sec θ = r / x tan θ = y / x cot θ = x / y These apply to any angle in any quadrant.

7. For any angle in any quadrant x2 + y2 = r2 … So, r is positive by Pythagorean theorem. (x,y) r y θ x

8. NOTE: right-triangle definitions are special case of circular functions when θ is in quadrant I Y (x,y) r y θ X x

9. *Reciprocal Identities sin θ = y / r and csc θ = r / y cos θ = x / r and sec θ = r / x tan θ = y / x and cot θ = x / y

10. *Ratio Identities *Both sets of identities are useful to determine trigonometric functions of any angle.

11. Positive trig values in each quadrant: Y All all six positive Students sin positive (csc) (-, +) (+, +) II I X III IV Take Classes (-, -) (+, -) tan positive (cot) cos positive (sec)

12. REMEMBER: In the ordered pair (x, y), x represents cosine and y represents sine. Y (-, +) (+, +) II I X III IV (-, -) (+, -)

13. Examples

14. #1 Draw each angle whose terminal side passes through the given point, and find all trigonometric functions of each angle. θ1: (4, 3) θ2: (- 4, 3) θ3: (- 4, -3) θ4: (4, -3)

15. x = y = r = I (4,3) sin θ = cos θ = tan θ = csc θ = sec θ = cot θ = θ1

16. x = y = r = II (-4,3) θ2 sin θ = cos θ = tan θ = csc θ = sec θ = cot θ =

17. x = y = r = sin θ = cos θ = tan θ = csc θ = sec θ = cot θ = θ3 (-4,-3) III

18. x = y = r = sin θ = cos θ = tan θ = csc θ = sec θ = cot θ = θ4 (4,-3) IV

19. Y Perpendicular line from point on circle alwaysdrawn to the x-axis forming a reference triangle II I ref θ2 θ1 X ref θ3 refθ4 IV III

20. Value of trig function of angle in any quadrant is equal to trig function of its reference angle, or it differs only in sign. Y II I ref θ2 θ1 X ref θ3 refθ4 IV III

21. #2 Given: tan θ = -1 and cos θ is positive: • Draw θ. Show the values for x, y, and r.

22. Given: tan θ = -1 and cos θ is positive: • Find the six trigonometric functions of θ.

23. Calculator Exercise

24. # 1 Find the value of sin 110º. (First determine the reference angle.)

25. #2 Find the value of tan 315º. (First determine the reference angle.)

26. #3 Find the value of cos 230º. (First determine the reference angle.)

27. Practice

28. #1 Draw the angle whose terminal side passes through the given point.

29. Find all trigonometric functions for angle whose terminal side passes thru .

30. #2 Draw angle: sin θ = 0.6, cos θ is negative.

31. Find all six trigonometric functions: sin θ = 0.6, cos θ is negative.

32. #3Find remaining trigonometric functions: sin θ = - 0.7071, tan θ= 1.000

33. Find remaining trigonometric functions: sin θ = - 0.7071, tan θ= 1.000

34. Calculator Practice

35. #1 Express as a function of a reference angle and find the value: cot 306º .

36. #2 Express as a function of a reference angle and find the value: sec (-153º) .

37. #3 Find each value on your calculator. (Key in exact angle measure.) sin 260.5º tan 150º 10’

38. cot (-240º) csc 450º

39. cos 5.41 sec (7/4)

40. π/2 = 1.57 0 2π = 6.28 π = 3.14 3π/2 = 4.71

41. Application

42. # 1 The refraction of a certain prism is Calculate the value of n.

43. #2 A force vector F has components Fx= - 4.5 lb and Fy = 8.5 lb. Find sin θ and cos θ. Fy = 8.5 lb θ Fx=-4.5 lb

44. Fy = 8.5 lb θ Fx=-4.5 lb