Addition and subtraction formulas in trigonometry
In trigonometry, the formulas for adding and subtracting trigonometric functions (sine, cosine, tangent) are as follows:
Sine
$$ \sin (a+b) = \sin a \cos b + \cos a \sin b $$ $$ \sin (a-b) = \sin a \cos b - \cos a \sin b $$
Proof sine addition formula and sine subtraction formula
Cosine
$$ \cos (a+b) = \cos a \cos b - \sin a \sin b $$ $$ \cos (a-b) = \cos a \cos b + \sin a \sin b $$
Proof cosine addition formula and cosine subtraction formula
Tangent
$$ \tan (a+b) = \frac{\tan a + \tan b}{1- \tan a \tan b} $$ $$ \tan (a-b) = \frac{\tan a - \tan b}{1+ \tan a \tan b} $$
Proof tangent addition formula and tangent subtraction formula
A practical example
Let’s take two angles, a=30° and b=60°
$$ \sin a = \sin 30° = \frac{1}{2} $$
$$ \sin b = \sin 60° = \frac{\sqrt{3}}{2} $$
The sine of the sum a+b is not simply the sum of the sines of the two angles:
$$ \sin (a+b) \ne \sin(a) + \sin(b) $$
$$ \sin (30°+60°) \ne \sin(30°) + \sin(60°) = \frac{1}{2} + \frac{\sqrt{3}}{2} = \frac{1 + \sqrt{3}}{2} $$
Note: The sine of 30°+60° is actually the sine of 90°. And as we all know, the sine of 90° is 1. So, it’s a common mistake to try and calculate the sine of the sum of two angles as sin(a+b) = sin(a) + sin(b).
To correctly find the sine of the sum a+b, we use the sine addition formula:
$$ \sin (a+b) = \sin a \cos b + \cos a \sin b $$
Now substituting a=30° and b=60° gives us:
$$ \sin (30°+60°) = \sin 30° \cos 60° + \cos 30° \sin 60° $$
We already know that sin(30°) = 1/2 and sin(60°) = √3/2
$$ \sin (30°+60°) = \frac{1}{2} \cdot \cos 60° + ( \cos 30° ) \cdot \frac{\sqrt{3}}{2} $$
The cosine values for 30° and 60° are cos(30°) = √3/2 and cos(60°) = 1/2
$$ \sin (30°+60°) = \frac{1}{2} \cdot \frac{1}{2} + \frac{\sqrt{3}}{2} \cdot \frac{\sqrt{3}}{2} $$
$$ \sin (30°+60°) = \frac{1}{4} + \frac{3}{4} $$
$$ \sin (30°+60°) = \frac{4}{4} $$
$$ \sin (30°+60°) = 1 $$
The sine of 30°+60° equals 1.
The result is correct.
And that’s how it works.