Addition and Subtraction Formulas

The Tempting Mistake

Here is the first thing everyone wants to do:

$\sin(A + B) \stackrel{?}{=} \sin A + \sin B$

It feels right. It is completely wrong.

Watch the two bars. The green one is the true value of $\sin(A+B)$. The yellow one is the naive $\sin A + \sin B$, and it sails straight past 1, which no real sine can ever do. Sine does not distribute over addition.

See It With Numbers

Take $A = 30°$ and $B = 60°$.

• True: $\sin(30° + 60°) = \sin 90° = 1$
• Naive: $\sin 30° + \sin 60° = 0.5 + 0.866 = 1.366$

$\sin(A + B) \neq \sin A + \sin B$. The angle inside cannot be split apart.

So we need a real formula.

The Addition Formulas

Here they are. The angle does split, just not the lazy way:

$\sin(A + B) = \sin A \cos B + \cos A \sin B$

$\cos(A + B) = \cos A \cos B - \sin A \sin B$

Notice the pattern:

• Sine mixes the two functions: $\sin\cos + \cos\sin$
• Cosine keeps like with like: $\cos\cos - \sin\sin$

The Sign Rule

Subtraction uses the very same formulas. You only flip one sign:

$\sin(A - B) = \sin A \cos B - \cos A \sin B$

$\cos(A - B) = \cos A \cos B + \sin A \sin B$

The trick to remembering the signs:

• Sine keeps the sign of the angle: $+$ stays $+$, $-$ stays $-$
• Cosine flips the sign: $+$ becomes $-$, $-$ becomes $+$

Sine agrees, cosine disagrees.

Tangent Too

Since $\tan = \frac{\sin}{\cos}$, the addition rule carries over to tangent:

$\tan(A \pm B) = \frac{\tan A \pm \tan B}{1 \mp \tan A \tan B}$

The top uses the same sign as the angle. The bottom uses the opposite sign.

The Payoff: Exact Values

The real reward is finding exact values for angles that are not on the standard list.

$75°$ is not a “nice” angle. But $75° = 45° + 30°$, and both of those are nice.

No calculator, and the answer is exact. That is what these formulas buy you.

All Six in One Place

Learn the two addition formulas and the sign rule. Everything else follows.