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(15^{o})
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(15^{o}) = sin (45^{o}  30^{o})

We now use the difference formula for sine.
= sin(45^{o})*cos(30^{o})  cos(45^{o})*sin(30^{o})

Substitute the values of sine and cosine of 45^{o} and 30^{o} in the above to obtain.
sin(15^{o}) = [sqrt(2) / 2][sqrt(3) / 2]  [sqrt(2) / 2][1 / 2]

Common denominator and factoring.
sin(15^{o}) = 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 sin^{2}x + cos^{2}x = 1 to find cos x.
cos x = (+ or ) SQRT(1  sin^{2}x)

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 sin^{2}y + cos^{2}y = 1 to find cos y.
cos y = (+ or ) SQRT(1  sin^{2}y)

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
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Updated: February 2015
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