Sum and Difference Identities and Formulas

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Drawing of the trigonometric ratios of the sum angle

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.

See an example or the proof.

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.

See an example or the proof.

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):
    Calculation of the sine of the sum angle (30º + 60º)
  • Cosine of the Sum of Two Angles, α and β (30º + 60º):
    Calculation of the cosine of the sum angle (30º + 60º)
  • Tangent of the Sum of Two Angles, α and β (30º + 60º):
    Calculation of the tangent of the sum angle (30º + 60º)

These results are actually, the trigonometric ratios of 90º.

Proof of the Angle Sum Formulas

Proof of Sum Formula for Sine

Drawing within the angle sum for demonstration

The Sine of the sum of angles α and β, is the DF segment:

Calculation of the segment DF equal to the sine of the sum angle

First, let’s calculate DE segment:

Calculation of the segment DE equal to the sine of the sum angle

And now, EF segment:

Calculation of the segment EF equal to the sine of the sum angle

Substituting in previous formula:

Sine formula of sum angle

So, we obtain the angle sum formula for the sine.

Proof of Sum Formula for Cosine

Drawing within the angle sum for demonstration

As with the sine, the cosine of the angle sum is the AF segment. Then, we have:

Calculation of the segment AF equal to the cosine of the sum angle

Now, let’s calculate AG segment:

Calculation of the segment AG equal to the cosine of the sum angle

And EH segment:

Calculation of the segment EH equal to the cosine of the sum angle

Then, substituting in previous formula (2), that is:

Cosine formula of sum angle

So, we get the angle sum formula for the cosine.

Proof of Sum Formula for Tangent

Drawing within the angle sum for demonstration

The tangent of the angle sum is equal to the sine of the angle sum divided by the cosine of the angle sum.

Calculation of the tangent of the angle sum as division between sine and cosine

If we divide numerator and denominator by (cos α ⋅ cos β):

Calculation of the tangent of the sum angle by simplifying the fractions

We simplify and we will obtain the following formula:

Tangent formula of sum angle

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):
    Calculation of the sine of the difference angle (60º - 30º)
  • Cosine of the Difference of Two Angles, α and β (60º – 30º):
    Calculation of the cosine of the difference angle (60º - 30º)
  • Tangent of the Difference of Two Angles, α and β (60º – 30º):
    Calculation of the tangent of the difference angle (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

Drawing within the angle difference for demostration

Let the angle sum formula for sine:

Sine formula of sum angle

If we apply the transformation β = -β, we will have the difference formula for sine:

Calculation of the sine of the difference angle

Proof of Difference Angle Formula for Cosine

Drawing within the angle difference for demonstration

From the sum formula for cosine, we can obtain that of the difference angle.

Cosine formula of sum angle

As we did with the sine, we can substitute β for -β and then we obtain the following formula:

Calculation of the cosine of the difference angle

Proof of Difference Angle Formula for Tangent

Let the angle sum formula for tangent:

Tangent formula of sum angle

If we substitute β for -β, we will get the angle difference formula for tangent:

Calculation of the tangent of the difference angle

AUTHOR: Bernat Requena Serra

YEAR: 2021


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