# Use of Sum and Difference Formulas in Trigonometry

This is a tutorial on how to use the sum and difference formulas to solve problems in trigonometry. For a list of these formulas, go here.

 Example 1: Find the exact value of sin(15o) Solution to Example 1 15 o is not a special angle. However 15 = 45 - 30 and both 45 and 30 are special angles. Hence sin(15o) = sin (45o - 30o) We now use the difference formula for sine. = sin(45o)*cos(30o) - cos(45o)*sin(30o) Substitute the values of sine and cosine of 45o and 30o in the above to obtain. sin(15o) = [sqrt(2) / 2][sqrt(3) / 2] - [sqrt(2) / 2][1 / 2] Common denominator and factoring. sin(15o) = sqrt(2)[sqrt(3) - 1] / 4 Example 2: Simplify cos(x - pi/2) Solution to Example 2 Use the difference formula for cosine to expand the given expression cos(x - pi/2) = cos x * cos pi/2 + sin x * sin pi/2 cos pi/2 = 0 and sin pi/2 = 1, hence. cos(x - pi/2) = sin x Example 3: Given sin x = 1 / 5 and sin y = -2 / 3, angle x is in quadrant II and angle y is in quadrant III, find the exact value of sin(x + y). Solution to Example 3 Expand sin(x + y) using the sum formula of the sine. sin(x + y) = sin x * cos y + cos x * sin y We know sin x but not cos x, we use the identity sin2x + cos2x = 1 to find cos x. cos x = (+ or -) SQRT(1 - sin2x) Since x is in quadrant II, cos x is negative. cos x = - SQRT(1 - (1/5)2) We know sin y but not cos y, we use the same identity as above sin2y + cos2y = 1 to find cos y. cos y = (+ or -) SQRT(1 - sin2y) Since y is in quadrant III, cos y is negative. cos y = - SQRT(1 - (-2/3)2) = - SQRT(1 - 4/9) = (-1/3)SQRT(5) We now substitute sin x, cos x, sin y and cos y by their values in the formula above. sin(x + y) = sin x * cos y + cos x * sin y = [1/5]*[-(1 / 3)SQRT(5)] + [-(1 / 5)SQRT(24)][-2 / 3] = [-SQRT(5) + SQRT(24)] / 15 Links related to trigonometric formulas Trigonometric Formulas and Their Applications and Trigonometric Identities and Their Applications