# Ken Ward's Mathematics Pages

## Trigonometry Compound Angles

### Sine

[1.1]

By substituting −β for β in Equation 1.1, we get:
[1.2]

### 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]

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 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]

### 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]

### Tangent

As before, setting β to α in the following:
[1.4, repeated]

After simplifying, we obtain the formula:
[2.4]

Trigonometry Contents

Ken Ward's Mathematics Pages

# Faster Arithmetic - by Ken Ward

Ken's book is packed with examples and explanations that enable you to discover more than 150 techniques to speed up your arithmetic and increase your understanding of numbers. Paperback and Kindle: