6.5 Double Angle & Half Angle Formulas
Objectives: • Use the double angle formula to find exact values and establish identities.
• Use the half angle formula to find exact values.
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Remember the sum formula, we can use this to create a new formula.
If α = β = θ then it will look much different.
sin(α + β) = sin( α ) cos( β ) + cos( α ) sin( β )
sin(2θ) = sin( θ ) cos( θ ) + cos( θ ) sin( θ )
sin(2θ) = 2sin( θ ) cos( θ )
The same can be done for cosine.
cos(α + β) = cos( α ) cos( β ) ­ sin( α ) sin( β )
cos(2θ) = cos( θ ) cos( θ ) ­ sin( θ ) sin( θ )
cos(2θ) = cos2( θ ) ­ sin2( θ )
With our identity sin 2( θ ) + cos2( θ ) = 1 this can take many forms
Since sin2( θ ) = 1 ­ cos 2( θ )
cos(2θ) = cos2( θ ) ­ (1 ­ cos 2( θ ))
cos(2θ) = 2cos2( θ ) ­ 1
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Also cos(2θ) = 1 ­ 2sin 2( θ )
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We can derive some more equations, but we will give them to you to save time and confusion. These are for halfing an angle.
P.S. Please let me know if you want to see the derivation.
1 ­ cos( α )
sin2( α/2 ) = 2
1 + cos( α )
cos2( α/2 ) = 2
1 ­ cos( α )
tan2( α/2 ) = 1 + cos( α )
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Formula Page!
sin(2θ) = 2sin( θ ) cos( θ )
±√
1 ­ cos( α )
2
±√
1 + cos( α )
2
±√
1 ­ cos( α )
1 + cos( α )
sin( α/2 ) = cos(2θ) = cos2( θ ) ­ sin2( θ )
cos(2θ) = 2cos2( θ ) ­ 1
cos(2θ) = 1 ­ 2sin 2( θ )
tan(2θ) = 2tan( θ )
1 ­ tan2( θ )
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cos( α/2 ) = tan( α/2 ) = VOTE
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Example
Given: sin(θ) = ­5/13 for π < θ < 3π/2
Find the exact value for:
a) sin (2θ) b) cos(2θ) c) sin(θ/2) d) cos(θ/2)
Step 1: π < θ < 3π/2 means quadrant III ⇒ x and y are negative
Step 2: Draw a right triangle and find all 3 sides
13
­5
θ
­12
a) sin (2θ) = 2sinθcosθ = b) cos(2θ) = cos2θ ­ sin2θ = c) sin(θ/2) = ± √1 ­ cosθ
/2 = √1 ­ cosθ
/2 = d) cos(θ/2) = ± √1 + cosθ
/2 = ­ √1 + cosθ
/2 = Title: Apr 25­9:13 AM (9 of 11)
Note: For half angle formulas you need to determine a new quadrant!!! π/2 < θ/2 < 3π/4 means quad II ⇒ only x is negative
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Homework
page 490
(9 ­ 15)
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