Definition: Angle addition formulas, also known as angle addition identities, allow us to find the exact values of less common trigonometric angles.

$\dpi{120} \bg_white \\sin(a + b) = sin(a)cos(b) + cos(a)sin(b)\\\\ cos(a + b) = cos(a)cos(b) - sin(a)sin(b)\\\\ tan(a + b) =\;\;\;\frac{tan(a) + tan(b)}{ 1 - tan(a)\,tan(b) }$

Example:  Find the exact value of sin(75).

Solution:

Step 1: Find two angles with a sum of 75 whose sine and cosine values are known. As 30 and 45 add up to 75, we can substitute these values into the formula:

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

sin(30 +45) = sin(30)cos(45) + cos(30)sin(45)

Step 2: Simplify and solve to find the exact value:

sin(30 +45) = sin(30)cos(45) + cos(30)sin(45)

= (1/2)(√2/2) + (√3/2)(√2/2)

= √2/4 + √6/4

The exact value of sin(75) is √2/4 + √6/4 or (√2 +√6)/4