# Angle Subtraction Formulas

Definition: Angle subtraction formulas, also known as angle subtraction identities, allow us to find the exact values of less common trigonometric angles.$\dpi{120} \bg_white \large \\sin(a - b) = sin(a)cos(b) - cos(a)sin(b)\\\\ cos(a - b) = cos(a)cos(b) + sin(a)sin(b)\\\\ tan(a - b) =\;\;\;\frac{tan(a) -tan(b)}{ 1 + tan(a)\,tan(b) }$

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

Solution:

Step 1: Find two angles with a difference of 75 whose sine and cosine values are known. As 135 and 60 subtract to give us 75, we can substitute these values into the formula:

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

cos(135 – 60) = cos(135)cos(60) + sin(135)sin(60)

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

cos(135)cos(60) + sin(135)sin(60)

= (-√2/2)(1/2) + (√2/2)(√3/2

=-√2/4 + √6/4

The exact value of cos(75) is -√2/4 + √6/4 or (√6 – √2)/4

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