1. 5.3 Trigonometric ratios FOR angles greater than 90o

2. SPECIAL TRIANGLES

5. Angle in standard position
has its vertex at the origin
is measured from the initial (fixed) arm on the positive x- axis to the terminal arm that rotates about the origin

6. PRINCIPAL ANGLE & RELATED ACUTE ANGLE

7. POSTIVE ANGLES vs negative angles
all positive angles are measured counterclockwise
all negative angles are measured clockwise

8. INVESTIGATION
How are the primary trigonometric ratios for the acute angle related to the corresponding ratios for the principal angle?

11. ACUTE ANGLe = 30o

12. Acute Angle 30o
Quadrant
sin
cos
tan
Principal Angle 30o 1
Principal Angle ____o 2 3 4

13. Acute Angle 30o
Quadrant
sin
cos
tan
Principal Angle 30o 1 2 √3
1 x √3= √3
√3 x√
Principal Angle 150o
-√3 3
Principal Angle 210o -1
Principal Angle 330o 4

14. ACUTE ANGLe = 60o

15. Acute Angle 60o
Quadrant
sin
cos
tan
Principal Angle 60o 1
Principal Angle ____o 2 3 4

16. Acute Angle 60o
Quadrant
sin
cos
tan
Principal Angle 60o 1 √3 2
√3 = √3
Principal Angle 120o -1
-√3
Principal Angle 240o 3
Principal Angle 300o 4

17. ACUTE ANGLe = 45o

18. Acute Angle 45o
Quadrant
sin
cos
tan
Principal Angle 45o 1
Principal Angle ____o 2 3 4

19. Acute Angle 45o
Quadrant
sin
cos
tan
Principal Angle 45o 1
1 x √2= √2
√2 x√ √2 2
1 = 1
Principal Angle 135o
-√2 -1
Principal Angle 225o 3
Principal Angle 315o 4

20. Quadrant 1
all trig ratios are positive

21. Quadrant 2

22. Quadrant 3

23. Quadrant 4

24. CAST rule quadrant 1, All ratios are +ve (x & y are +ve )
quadrant 2, only Sine is +ve
(x is -ve, y is +ve )
quadrant 3, only Tangent is +ve
(x & y are -ve)
quadrant 4, only Cosine is +ve
(x is +ve, y is -ve)

25. What about negative angles??

26. LET’s repeat with negative angles

27. negative angles
What are the equivalent positive angles to these negative angles? -30o = -150o = -210o = -330o =

28. negative angles
What are the equivalent positive angles to these negative angles? -30o = 330o -150o = 210o -210o = 150o -330o = 30o

29. negative angles
What are the equivalent positive angles to these negative angles? -60o = -120o = -240o = -300o =

30. negative angles
What are the equivalent positive angles to these negative angles? -60o = 300o -120o = 240o -240o = 120o -300o = 60o

31. negative angles
What are the equivalent positive angles to these negative angles? -45o = -135o = -225o = -315o =

32. negative angles
What are the equivalent positive angles to these negative angles? -45o = 315o -135o = 225o -225o = 135o -315o = 45o

33. Example 1 Determine the exact value. sin(240o) cos(135o) cot(210o)
csc(510o )

34. Example 2
Determine angles if 0o    360o . Round to the nearest tenth if necessary.

35. HOMEFUN 
pg 291 Reflecting H, I, J pg 292 #1-4 Check out examples 1, 2, 3 on pg (if needed)