5.3 Trigonometric ratios FOR angles greater than 90o

SPECIAL TRIANGLES

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

PRINCIPAL ANGLE & RELATED ACUTE ANGLE

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

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

ACUTE ANGLe = 30o

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

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

ACUTE ANGLe = 60o

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

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

ACUTE ANGLe = 45o

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

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

Quadrant 1 all trig ratios are positive

Quadrant 2

Quadrant 3

Quadrant 4

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)

What about negative angles??

LET’s repeat with negative angles

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

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

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

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

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

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

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

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

HOMEFUN  pg 291 Reflecting H, I, J pg 292 #1-4 Check out examples 1, 2, 3 on pg 283-286 (if needed)