Addition and subtraction formulas

Addition Formulas

To add or subtract angles, we use the following formulas (which you should know off by heart):

cos(a+b) = cos(a) * cos(b) - sin(a) * sin(b) cos(a–b) = cos(a) × cos(b) + sin(a) × sin(b) sin(a+b) = sin(a) × cos(b) + cos(a) × sin(b) sin(a–b) = sin(a) × cos(b) – cos(a) × sin(b)

Example

Let’s calculate cos(75 degrees) by writing 75 = 45 + 30:

cos(75) = cos(45) * cos(30) - sin(45) * sin(30) = (√2)/2 * (√3)/2 - (√2)/2 * (1/2) = √6/4 - √2/4 = (√6 - √2)/4

This exact result would be difficult to obtain without breaking the angle down into two well-known angles.

Common pitfall

Many pupils mix up the signs in the four formulas. Remember that for the cosine, the sign in the formula is opposite to the sign between a and b (cos(a+b) contains a minus sign, cos(a-b) contains a plus sign), whilst for the sine, the sign is the same as in the original angle.