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:
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sin(α + β) = sin α cos β + cos α sin β
sin(α − β) = 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(α − β) = 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:
LiveMath can demonstrate how these formulas are true for any values of α or β.
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:
Proof of the Tangent of the Sum and Difference of Two Angles
Our proof for these uses the trigonometric identitiy for tan that we met before.
Example 1
Find the exact value of cos 75° by using 75° = 30° + 45°.
Example 2
If sin α = 4/5 (in Quadrant I) and cos β = -12/13 (in Quadrant II) evaluate sin(α − β).
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
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