Sum and Difference Formulas

By the end of this lesson, you will be able to apply the sum and difference formulas for sine, cosine, and tangent to find exact values, simplify expressions, and verify identities.

The Big Idea

Sum and difference formulas extend the set of angles at which we can compute exact trig values beyond the standard unit circle angles. They also form the basis for double-angle and half-angle formulas (next lesson) and for proving many advanced identities.

The Formulas

Sine:

$$\sin(\alpha \pm \beta) = \sin\alpha\cos\beta \pm \cos\alpha\sin\beta$$

Cosine:

$$\cos(\alpha \pm \beta) = \cos\alpha\cos\beta \mp \sin\alpha\sin\beta$$

Note: signs are OPPOSITE for cosine — $+$ in $\alpha + \beta$ gives $-$ in the formula.

Tangent:

$$\tan(\alpha \pm \beta) = \frac{\tan\alpha \pm \tan\beta}{1 \mp \tan\alpha\tan\beta}$$

Example 1 — Exact Value: $\sin(75°)$

$75° = 45° + 30°$

$$\sin(75°) = \sin 45°\cos 30° + \cos 45°\sin 30°$$

$$= \frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2} + \frac{\sqrt{2}}{2} \cdot \frac{1}{2} = \frac{\sqrt{6}}{4} + \frac{\sqrt{2}}{4} = \frac{\sqrt{6}+\sqrt{2}}{4}$$

Example 2 — Exact Value: $\cos(15°)$

$15° = 45° - 30°$

$$\cos(15°) = \cos 45°\cos 30° + \sin 45°\sin 30°$$

$$= \frac{\sqrt{2}}{2} \cdot \frac{\sqrt{3}}{2} + \frac{\sqrt{2}}{2} \cdot \frac{1}{2} = \frac{\sqrt{6}+\sqrt{2}}{4}$$

Note: $\sin(75°) = \cos(15°)$ — they're complementary. ✓

Example 3 — Exact Value: $\tan(7\pi/12)$

$7\pi/12 = 3\pi/12 + 4\pi/12 = \pi/4 + \pi/3$

$$\tan\left(\frac{\pi}{4} + \frac{\pi}{3}\right) = \frac{1 + \sqrt{3}}{1 - \sqrt{3}} = \frac{(1+\sqrt{3})^2}{1-3} = \frac{4+2\sqrt{3}}{-2} = -2 - \sqrt{3}$$

Example 4 — Simplifying an Expression

Find $\sin\alpha\cos\beta + \cos\alpha\sin\beta$: this equals $\sin(\alpha + \beta)$ by recognition.

Simplify: $\cos(\theta + \pi)$:

$\cos\theta\cos\pi - \sin\theta\sin\pi = -\cos\theta - 0 = -\cos\theta$ — confirms the known result.

Example 5 — Finding Exact Value with Quadrant Info

$\sin\alpha = 3/5$ ($\alpha$ in QI), $\cos\beta = -5/13$ ($\beta$ in QII). Find $\cos(\alpha + \beta)$.