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 snd 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 subtitute 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: 2 April 2013

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