Ken Ward's Mathematics
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Trigonometry Compound Angles
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Page Contents
- Compound Angles
- sin(α±β)
- cos(α±β)
- tan(α±β)
- Double Angles
- sin(2α)
- cos(2α)
- tan(2α)
Compound Angles (Addition Formulae)
Sine
We have
already shown that:
[1.1]
By substituting
−β for β in Equation 1.1, we get:
[1.2]
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Cosine
By noting that sin (π/2−θ)=cos θ, we can write Equation 1.1 (with θ=α+β) as:
sin (π/2-α-β)=
cos(α+β)=
sin(π/2-α)·cos(−β)+cos(π/2-α)·sin(−β)
=cosα·cosβ−sinα·sinβ
Hence:
[1.3]
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Similarly, by substituting −β for β in Equation 1.3:
[1.4]
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Tangent
[1.4]
To prove this we divide Equation 1.1 by Equation 1.3:
Which yields:
After dividing throughout by cosα·cosβ.
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By substituting
−β for β in 1.4, we get:
[1.5]
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Double Angles
The formulae for double angles follow from those for compound angles.
Sine
Using:
[1.1, repeated]
And setting β to α, we have:
sin(2·α)=sinα·cosα+sinα·cosα=
by [2.1]
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Cosine
As with sine, setting β to α in the following:
[1.3, repeated]
We have:
cos2α=cosα·cosα-sinα·sinα=
cos2α−sin2α=
2·cos2α−1
Because
sin2α=1−cos2α
So:
[2.2]
Or:
[ [2.3]
[2.4]
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Tangent
As before, setting β to α in the following:
[1.4, repeated]
After simplifying, we obtain the formula:
[2.4]
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