Use of Sum and Difference Formulas in Trigonometry

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 snd 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 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