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...
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Trigonometric Func.ons of Any Angle
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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
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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
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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
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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