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Ken Ward's Mathematics Pages

Trigonometry Compound Angles

See also:

• Trigonometry Contents
• CompoundAngleProofs
• HalfAngles

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:
compoundAngleFormula.gif [1.1]

By substituting −β for β in Equation 1.1, we get:
compoundAngleSinMinus.gif [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:
compoundAngleCosPlus.gif [1.3]
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Similarly, by substituting −β for β in Equation 1.3:
compoundAngleCosMinus.gif [1.4]
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Tangent

compoundAngleTanPlus.gif [1.4]
To prove this we divide Equation 1.1 by Equation 1.3:
compoundTan1.gif
compoundTan2.gif

Which yields:
compoundAngleTanPlus.gif
After dividing throughout by cosα·cosβ.
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By substituting −β for β in 1.4, we get:
compoundAngleTanMinus.gif [1.5]
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Double Angles

The formulae for double angles follow from those for compound angles.

Sine

Using:
compoundAngleFormula.gif [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:
compoundAngleCosPlus.gif [1.3, repeated]

We have:
cos2α=cosα·cosα-sinα·sinα=
cos²α−sin²α=
2·cos²α−1
Because sin²α=1−cos²α

So:
doubleAngleCos.gif [2.2]
Or:
[ [2.3]
cos2aAsSine.gif [2.4]
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Tangent

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

After simplifying, we obtain the formula:
doubleAngleTan.gif [2.4]
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Ken Ward's Mathematics Pages

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