Ken Ward's Mathematics Pages

Trigonometry Compound Angles

See also:

• Trigonometry Contents
• Compound Angle Proofs
• Half Angles

Page Contents

1. Compound Angles
1. sin(α±β)
2. cos(α±β)
3. tan(α±β)
2. Double Angles
1. sin(2α)
2. cos(2α)
3. 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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