5.2. Sum and Difference Formulas
Among of the most useful trigonometric formulas are the so-called sum and difference formulas. These are formulas for trigonometric functions applied to the sum or difference of two angles, such as sin(θ + φ) and cos(θ - φ). Although the derivation of these formulas is rather complicated, the effort is well worth it!
The sum formulas for sine and cosine are as follows:
- (5.2.1) sin(θ + φ) = sin θ cos φ + cos θ sin φ
- (5.2.2) cos(θ + φ) = cos θ cos φ - sin θ sin φ
To prove these identities, we consider the following diagram:
Figure 5.2.3: Sum Formula Diagram
First we prove (5.2.1). The following relations are clear from the diagram:
- |OS| = cos(θ + φ) = |OW| - |SW|
- |PT| = |OR| = cos θ
- |OV| = |OQ| cos φ = cos φ
- |QV| = |OQ| sin φ = sin φ
- |OW| = |OV| cos θ = cos θ cos φ
- |SW| = |VZ|
From the above relations, it follows that cos(θ + φ) = cos θ cos φ - |VZ|. Thus, we need to show that |VZ| = sin θ sin φ. But we see that |VZ| = |QV| sin ∠VQZ = sin ∠VQZ sin φ. Thus it suffices to show that ∠VQZ = θ. But we have ∠VQZ = π/2 - ∠QVZ and ∠QVZ = π/2 - ∠OVA, whence it follows that ∠VQZ = ∠OVA. But ∠OVA and ∠VOW = θ are alternate interior angles formed by parallel lines, whence ∠VQZ = ∠OVA = ∠VOW = θ and the result follows.
Next we prove (5.2.2), for which we use the following relations, which are also clear from Figure 5.2.3.
- |OU| = sin(θ + φ) = |OA| + |AU|
- |OT| = |PR| = sin θ
- |OA| = |OV| sin θ = sin θ cos φ
- |AU| = |QZ| = |QV| cos ∠VQZ = cos θ sin φ
From the first and fourth equation, it is clear that we have sin(θ + φ) = sin θ cos φ + cos θ sin φ, proving the formula.
There is one more useful addition formula, for the tangent function. Specifically we have
- (5.2.4) tan(θ + φ) = (tan θ + tan φ) / (1 - tan θ tan φ)
This formula follows easily from the addition formulas for sine and cosine. The trick is to write tan(θ + φ) as sin(θ + φ) / cos(θ + φ), apply the addition formulas (5.2.1) and (5.2.2) to the numerator and denominator respectively, and finally multiply the numerator and denominator by sec θ sec φ. We leave the details as an exercise.
Another formula which follows easily from (5.2.4) is the following. We leave the proof of this formula as an exercise as well.
- (5.2.5) arctan x + arctan y = arctan((x + y) / (1 - xy))
Difference formulas follow easily from the sum formulas. We have the following:
- (5.2.6) sin(θ - φ) = sin θ cos φ - cos θ sin φ
- (5.2.7) cos(θ - φ) = cos θ cos φ + sin θ sin φ
- (5.2.8) tan(θ - φ) = (tan θ - tan φ) / (1 + tan θ tan φ)
- (5.2.9) arctan x - arctan y = arctan((x - y) / (1 + xy))
To prove (5.2.6) for instance, we merely substitute -φ for φ in (5.2.1). Thus we have sin(θ - φ) = sin(θ + (-φ)) = sin θ cos φ + cos θ sin (-φ) = sin θ cos φ - cos θ sin φ, where we used the equation sin(-φ) = -sin φ in the last step. The proofs of the other formulas are similar.
One reason the sum and difference formulas are of interest is that they allow us to compute the values of trigonometric functions at some angles for which we were not able to compute them before, as the following examples show:
Example 1: Use formula (5.2.1) to compute sin(5π/12).
Solution: Since 5π/12 = π/4 + π/6, we use (5.2.1) with θ = π/4 and φ = π/6. We obtain
sin(5π/12) = sin(π/4 + π/6) = sin π/4 cos π/6 + cos π/4 sin π/6
= (√2/2)(√3/2) + (√2/2)(1/2)
= (√6 + √2) / 4.
Example 2: Use formula (5.2.6) to compute sin(π/12).
Solution: Since π/12 = π/4 - π/6, we use (5.2.6) with θ = π/4 and φ = π/6. We obtain
sin(π/12) = sin(π/4 - π/6) = sin π/4 cos π/6 + cos π/4 sin π/6
= (√2/2)(√3/2) + (√2/2)(1/2)
= (√6 - √2) / 4.