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:
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)`
`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.
Find the exact value of cos 75° by using 75° = 30° + 45°.
If `sin\ α = 4/5` (in Quadrant I) and `cos\ β = -12/13` (in Quadrant II) evaluate `sin(α − β).`
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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