4.4 Trigonometric Functions of any angle...Evaluating Trigonometric Functions of Any Angle To find...
Transcript of 4.4 Trigonometric Functions of any angle...Evaluating Trigonometric Functions of Any Angle To find...
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