Add two known formulas

Write the formulas for $\sin(a+b)$ and $\sin(a-b)$. Adding them together, the terms with $\cos a\sin b$ cancel each other out:

$$\sin(a+b)+\sin(a-b)=2\sin a\cos b.$$

Now do $u=a+b$ and $v=a-b$. Then $a=(u+v)/2$ and $b=(u-v)/2$. This produces the first identity below; the other cases come from adding or subtracting the sine and cosine formulas:

$$\sin u+\sin v=2\sin\frac{u+v}{2}\cos\frac{u-v}{2},$$

$$\sin u-\sin v=2\cos\frac{u+v}{2}\sin\frac{u-v}{2},$$

$$\cos u+\cos v=2\cos\frac{u+v}{2}\cos\frac{u-v}{2},$$

$$\cos u-\cos v=-2\sin\frac{u+v}{2}\sin\frac{u-v}{2}.$$

These identities are valid for all reals $u,v$. In the latter case, the negative sign is essential.

Go the other way

Rearranging the same formulas:

$$2\sin a\cos b=\sin(a+b)+\sin(a-b),$$

$$2\cos a\sin b=\sin(a+b)-\sin(a-b),$$

$$2\cos a\cos b=\cos(a+b)+\cos(a-b),$$

$$2\sin a\sin b=\cos(a-b)-\cos(a+b).$$

One product may be more useful for finding zeros; a sum may be more useful for bringing terms together or calculating exact values.

Example: solving by factors

At $\sin3x+\sin x=0$, the transformation gives:

$$2\sin2x\cos x=0.$$

Therefore $\sin2x=0$ or $\cos x=0$. The first family is $x=k\pi/2$ and the second is $x=\pi/2+k\pi$. The latter is already included in the first. The set of all solutions is $\{k\pi/2:k\in\mathbb Z\}$.

Exact calculation example

$$\sin75°+\sin15°=2\sin45°\cos30°=\frac{\sqrt6}{2}.$$

No decimal approximation was necessary. The average of the angles determines one factor, and half the difference determines the other.

In everyday life

  • Beat. Two notes of 440 Hz and 442 Hz sound like a note of 441 Hz that pulses 2 times per second — the sum became the product.
  • Phone keys. Each key adds two frequencies; the control unit separates the sum to recognize the key.
  • AM Radio. Multiplying the sound by the carrier creates the sum and difference frequencies that the radio picks up.
Sum of sines in product: the beat that musicians use to tune.

Exercises with resolution

1. Transform $\cos5x-\cos x$ into product.

Check resolution 1

$\cos5x-\cos x=-2\sin3x\sin2x$.

2. Calculate $2\sin75°\sin15°$.

Check resolution 2

By transformation into sum, it is $\cos60°-\cos90°=1/2$.

In the interactive book

The same subjects, with all the demonstrations and the complete list of exercises, are in the interactive Trigonometry book:

This class is a shorter introduction to the topic; In the book, the theory is accompanied by solved and proposed exercises from the chapter.

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