Trigonometric  Func.ons  of  Any  Angle  

Example:  

x  =  -­‐3    y  =  4        

x2 + y2 = r2

r2 = −3( )2 + 42

r2 = 25r = 5

sinθ = yr

sinθ = 45

cosθ = xr

cosθ = −35

tanθ = yx

tanθ = 4−3

Example:  

tanθ < 0cosθ > 0

tanθ = − 54=yx

y = −5x = 4

x2 + y2 = r2

42 + −5( )2 = r2

r2 = 41

r = ± 41 Radius  is  always  posi2ve,  so  r  =  √41  

sinθ = yr=

secθ = rx=

−541

= −5 4141

414

Example:  

a. 300°→ quadrant IV θ ' = 360°−300° θ ' = 60°

b. 2.3→ quadrant II θ ' = 3.14− 2.3° θ ' = 0.84

c. −135°

−135°+360° = 225°θ ' = 225°−180° θ ' = 45°225°→ quadrant III

Example:  

a. 4π3→ quadrant III θ ' = 4π

3−π θ ' = π

3→

12, 32

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cos π3

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−12, because cosθ is negative in QIII

b. − 210°+360° =150°→ quadrant II θ ' =180°−150° θ ' = 30°

tan 30°( ) =1232

=33

−33, because tanθ is negative in QII

c. 11π4

− 2π = 3π4→ quadrant II θ ' = π − 3π

4θ ' = π

4→

22, 22

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csc π4

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22= 2 2, because cosθ is positive in QII