Trigonometric Addition and Subtraction Formulas

The trigonometric addition and subtraction formulas are essential identities for working with angles. They allow you to calculate the sine, cosine, and tangent of the sum or difference of two angles using the trigonometric values of the individual angles.

These formulas appear throughout mathematics, physics, engineering, and many other fields where trigonometric relationships are used.

Sine

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

Proofs: sine addition formula and sine subtraction formula.

Cosine

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

Proofs: cosine addition formula and cosine subtraction formula.

Tangent

$$ \tan(a+b)=\frac{\tan a+\tan b}{1-\tan a \tan b} $$ $$ \tan(a-b)=\frac{\tan a-\tan b}{1+\tan a \tan b} $$

Proofs: tangent addition formula and tangent subtraction formula.

A Practical Example

Let's see how the sine addition formula works with two well-known angles:

$$ a=30^\circ \qquad b=60^\circ $$

The corresponding sine values are:

$$ \sin 30^\circ=\frac{1}{2} $$

$$ \sin 60^\circ=\frac{\sqrt{3}}{2} $$

At first glance, it may seem reasonable to think that the sine of the sum is simply the sum of the two sines. However, this is not true:

$$ \sin(a+b) \ne \sin a + \sin b $$

In our example:

$$ \sin(30^\circ+60^\circ) \ne \sin 30^\circ + \sin 60^\circ $$

$$ \sin(30^\circ+60^\circ) \ne \frac{1}{2}+\frac{\sqrt{3}}{2}=\frac{1+\sqrt{3}}{2} $$

Note: Since 30° + 60° = 90°, we already know that $$ \sin(30^\circ+60^\circ)=\sin(90^\circ)=1 $$ This immediately shows that adding the two sine values cannot give the correct result.

To calculate the sine of the sum correctly, we must use the sine addition formula:

$$ \sin(a+b)=\sin a \cos b + \cos a \sin b $$

Substituting a = 30° and b = 60° gives:

$$ \sin(30^\circ+60^\circ)=\sin 30^\circ \cos 60^\circ+\cos 30^\circ \sin 60^\circ $$

Using the known trigonometric values

$$ \sin 30^\circ=\frac{1}{2} \qquad \sin 60^\circ=\frac{\sqrt{3}}{2} $$

we obtain:

$$ \sin(30^\circ+60^\circ)=\frac{1}{2}\cos 60^\circ+\cos 30^\circ\frac{\sqrt{3}}{2} $$

We also know that:

$$ \cos 30^\circ=\frac{\sqrt{3}}{2} \qquad \cos 60^\circ=\frac{1}{2} $$

Substituting these values into the formula:

$$ \sin(30^\circ+60^\circ)=\frac{1}{2}\cdot\frac{1}{2}+\frac{\sqrt{3}}{2}\cdot\frac{\sqrt{3}}{2} $$

$$ \sin(30^\circ+60^\circ)=\frac{1}{4}+\frac{3}{4} $$

$$ \sin(30^\circ+60^\circ)=\frac{4}{4} $$

$$ \sin(30^\circ+60^\circ)=1 $$

As expected, the result is 1, which is exactly the value of $$ \sin(90^\circ) $$

This example illustrates why the addition formulas are so important. Without them, it would not be possible to correctly compute the trigonometric functions of angle sums and differences.