Ken Ward's Mathematics
Pages Trigonometry Compound Angles
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Compound Angles sin(α±β) cos(α±β) tan(α±β) Double Angles sin(2α) cos(2α) tan(2α) Compound Angles (Addition Formulae) SineWe have

already shown that:

[1.1]

By substituting

−β for β in Equation 1.1, we get:

[1.2]

■ CosineBy 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]

■ Similarly, by substituting −β for β in Equation 1.3:

[1.4]

■ Tangent [1.4]

To prove this we divide Equation 1.1 by Equation 1.3:

Which yields:

After dividing throughout by cosα·cosβ.

■ By substituting

−β for β in 1.4, we get:

[1.5]

■ Double AnglesThe formulae for double angles follow from those for compound angles.

SineUsing:

[1.1, repeated]

And setting β to α, we have:

sin(2·α)=sinα·cosα+sinα·cosα=

by [2.1]

■ CosineAs with sine, setting β to α in the following:

[1.3, repeated]

We have:

cos2α=cosα·cosα-sinα·sinα=

cos

^{2} α−sin

^{2} α=

2·cos

^{2} α−1

Because

sin^{2} α=1−cos^{2} α So:

[2.2]

Or:

[ [

2.3]

[

2.4]

■ TangentAs before, setting β to α in the following:

[1.4, repeated]

After simplifying, we obtain the formula:

[2.4]

■ Trigonometry Contents
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