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 a variety of ways.

Here is a relatively simple proof using the unit circle:

The next proof is the standard one that you see in most text books. It also uses the unit circle, but is not as straightforward as the first proof. However, we can still learn a lot from this next proof, especially about the way trigonometric identities work.

Finally, here is an easier proof of the identities, 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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