Sum and Difference Identities and Formulas
Angle sum and difference identities express trigonometric functions of sums of angles α ± β in terms of functions of α and β.
Using the sum and difference formulas allows us to find the exact values of the sine, cosine, or tangent of an angle. This is because we can rewrite the given in terms of two angles that have known trig values, for example, using the called special angles of the unit circle. Therefore, it is possible to find the sine, cosine, or tangent, of a given angle if we can break it up into the sum or difference of two of the aforementioned special angles (30º, 45º, 60º, 90º, 120º and so).
They are also useful to find the exact value of an expression involving an inverse trigonometric function, to verify identities, or to solve application problems.
Angle Sum Identities and Formulas
Let α and β be two angles. Angle sum identities express trigonometric functions of sums of angles α + β in terms of functions of α and β.
- Angle Sum Identity for Sine:
- Angle Sum Identity for Cosine:
- Angle Sum Identity for Tangent:
Angle Difference Identities and Formulas
Let α and β be two angles. Angle difference identities express trigonometric functions of difference of angles α – β in terms of functions of α and β.
- Angle Difference Identity for Sine:
- Angle Difference Identity for Cosine:
- Angle Difference Identity for Tangent:
Example of the Angle Sum Formulas
Let be two angles, α = 30º and β = 60º. The angle sum identities are:
- Sine of the Sum of Two Angles, α and β (30º + 60º):
- Cosine of the Sum of Two Angles, α and β (30º + 60º):
- Tangent of the Sum of Two Angles, α and β (30º + 60º):
These results are actually, the trigonometric ratios of 90º.
Proof of the Angle Sum Formulas
Proof of Sum Formula for Sine
The Sine of the sum of angles α and β, is the DF segment:
First, let’s calculate DE segment:
And now, EF segment:
Substituting in previous formula:
So, we obtain the angle sum formula for the sine.
Proof of Sum Formula for Cosine
As with the sine, the cosine of the angle sum is the AF segment. Then, we have:
Now, let’s calculate AG segment:
And EH segment:
Then, substituting in previous formula (2), that is:
So, we get the angle sum formula for the cosine.
Proof of Sum Formula for Tangent
The tangent of the angle sum is equal to the sine of the angle sum divided by the cosine of the angle sum.
If we divide numerator and denominator by (cos α ⋅ cos β):
We simplify and we will obtain the following formula:
Example of the Angle Difference Formulas
Let be two angles, α = 60º and β = 30º. Their angle difference identities are:
- Sine of the Difference of Two Angles, α and β (60º – 30º):
- Cosine of the Difference of Two Angles, α and β (60º – 30º):
- Tangent of the Difference of Two Angles, α and β (60º – 30º):
These results are actually, the trigonometric ratios of 30º.
Proof of the Angle Difference Formulas
You can easily get the trigonometric identities of the difference angle from the trigonometric identities of the angle sum by substituting -β for β.
It’s known that:
- sin (-β) = -sin β
- cos (-β) = cos β
- tan (-β) = -tan β
Proof of Difference Angle Formula for Sine
Let the angle sum formula for sine:
If we apply the transformation β = -β, we will have the difference formula for sine:
Proof of Difference Angle Formula for Cosine
From the sum formula for cosine, we can obtain that of the difference angle.
As we did with the sine, we can substitute β for -β and then we obtain the following formula:
Proof of Difference Angle Formula for Tangent
Let the angle sum formula for tangent:
If we substitute β for -β, we will get the angle difference formula for tangent: