# 2. Sin, Cos and Tan of Sum and Difference of Two Angles

by M. Bourne

The sine of the sum and difference of two angles is as follows:

Tan of Sum and Difference of Two Angles

sin(α + β) = sin α cos β + cos α sin β

sin(αβ) = sin α cos β − cos α sin β

The cosine of the sum and difference of two angles is as follows:

cos(α + β) = cos α cos β − sin α sin β

cos(αβ) = cos α cos β + sin α sin β

### Proofs of the Sine and Cosine of the Sums and Differences of Two Angles

We can prove these identities in several different ways.

Here is a proof using the unit circle:

Here is an easier proof, using complex numbers:

## Tangent of the Sum and Difference of Two Angles

We have the following identities for the tangent of the sum and difference of two angles:

tan(alpha+beta)=(tan\ alpha+tan\ beta)/(1-tan\ alpha\ tan\ beta)

and

tan(alpha-beta)=(tan\ alpha-tan\ beta)/(1+tan\ alpha\ tan\ beta)

### Proof of the Tangent of the Sum and Difference of Two Angles

Our proof for these uses the trigonometric identity for tan that we met before.

### Example 1

Find the exact value of cos 75° by using 75° = 30° + 45°.

Recall the 30-60 and 45-45 triangles from Values of Trigonometric Functions:

We use the exact sine and cosine ratios from the triangles to answer the question as follows:

{:(cos 75^"o",=cos(30^("o")+45^("o"))),(,=cos 30^("o")\ cos 45^("o")-sin 30^("o")\ sin 45^("o")),(,=sqrt3/2(1)/sqrt2-1/2(1)/sqrt2),(,=(sqrt3-1)/(2sqrt2)):}

This is the exact value for cos 75°.

### Example 2

If sin\ α = 4/5 (in Quadrant I) and cos\ β = -12/13 (in Quadrant II) evaluate sin(α − β).

We use

sin(αβ) = sin α cos β − cos α sin β

We firstly need to find cos α and sin β.

If sin α = 4/5, then we can draw a triangle and find the value of the unknown side using Pythagoras' Theorem (in this case, 3):

We do the same thing for cos β = 12/13, and we obtain the following triangle.

Note 1: We are using the positive value 12/13 to calculate the required reference angle relating to beta.

Note 2: The sine ratio is positive in both Quadrant I and Quadrant II.

Note 3: We have used Pythagoras' Theorem to find the unknown side, 5.

Now for the unknown ratios in the question:

cos\ α = 3/5

sin\ β = 5/13

We are now ready to find the required value, sin(αβ):

{:(sin(alpha-beta),=sin\ alpha\ cos\ beta-cos\ alpha\ sin\ beta),(,=4/5(-12/13)-3/5(5/13)),(,(=-48-15)/65),(,=(-63)/65):}

This is the exact value for sin(αβ).

### Exercises

1. Find the exact value of cos 15° by using 15° = 60° − 45°

2. If sin\ α = 4/5 (in Quadrant I) and cos\ β = -12/13 (in Quadrant II) evaluate cos(β − α).

[This is not the same as Example 2 above. This time we need to find the cosine of the difference.]

3. Reduce the following to a single term. Do not expand.

cos(x + y)cos y + sin(x + y)sin y

4. Prove that

cos(30^"o"+x)=(sqrt3 cos x-sin x)/2

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