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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Tan of Sum and Difference of Two Angles

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:

Proof


Here is an easier proof, using complex numbers:

Proof


LiveMath can demonstrate how these formulas are true for any values of α or β.

LIVEMath


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:

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mathimage


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.

Proof


Example 1

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

Answer


Example 2

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

Answer


Exercises

 

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

Answer


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.]

Answer


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

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

 

Answer


4. Prove that

mathimage

Answer




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