Sum, Difference and Product of Trigonometric Formulas Questions

The sum, difference and product formulas involving sin(x), cos(x) and tan(x) functions are used to solve trigonometry questions through examples and questions with detailed solutions.

Sum Formulas in Trigonometry

1. sin(x + y) = sin x cos y + cos x sin y
   
2. cos(x + y) = cos x cos y - sin x sin y
   
3. tan(x + y) = [tan x + tan y] / [1 - tan x tan y]

Example 1
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 1
Expand sin(x + y) using the sum formula of the sine (formula 1 above).
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 -) √(1 - sin²x)
Since x is in quadrant II, cos x is negative.
cos x = - √(1 - (1/5)²) = - (1/5) √24
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 -) √(1 - sin²y)
Since y is in quadrant III, cos y is negative.
cos y = - √(1 - (-2/3)²)
= - √(1 - 4/9)
= (-1/3)√(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)√(5)] + [-(1 / 5)√(24)][-2 / 3]
= [-√(5) + √(24)] / 15

Difference Formulas in Trigonometry

1. sin(x - y) = sin x cos y - cos x sin y
   
2. cos(x - y) = cos x cos y + sin x sin y
   
3. tan(x - y) = [tan x - tan y] / [1 + tan x tan y]

Example 2
Simplify cos(x - ?/2)
Solution to Example 2
Use the difference formula (formula 2 above) for cosine to expand the given expression
cos(x - ?/2) = cos x cos ?/2 + sin x sin ?/2
cos ?/2 = 0 and sin ?/2 = 1, hence.
cos(x - ?/2) = sin x

Example 3
Find the exact value of sin(15°)
Solution to Example 3
15 ° is not a special angle. However 15° = 45° - 30° and both 45° and 30° are special angles. Hence
sin(15°) = sin (45° - 30°)
We now use the difference formula for sine.
= sin(45°) cos(30°) - cos(45°) sin(30°)
Substitute the values of sine and cosine of 45° and 30° in the above to obtain.
sin(15°) = [√(2) / 2][√(3) / 2] - [√(2) / 2][1 / 2]
Common denominator and factoring.
sin(15°) = √(2)[√(3) - 1] / 4

Product to Sum Formulas in Trigonometry

1. sin x cos y = (1/2) [ sin(x + y) + sin(x - y) ]
   
2. cos x sin y = (1/2) [ sin(x + y) - sin(x - y) ]
   
3. cos x cos y = (1/2) [ cos(x + y) + cos(x - y) ]
   
4. sin x sin y = (1/2) [ cos(x - y) - cos(x + y) ]

Example 4
Simplify 2 cos(3 x)cos(2 x) - cos(x)
Solution to Example 4
Use the product formula (formula 3 above) to write cos(3 x) cos(2 x) as a sum in the given expression
2 cos(3 x) cos(2 x) - cos(x) = 2 ( (1/2) cos(3 x + 2 x) + cos (3 x - 2 x)) - cos(x)
Simplify.
cos(5 x) + cos (x) - cos(x) = cos(5 x)

Sum to Product Formulas in Trigonometry

1. sin x + sin y = 2 sin[ (x + y) / 2 ] cos[ (x - y) / 2 ]
   
2. sin x - sin y = 2 cos[ (x + y) / 2 ] sin[ (x - y) / 2 ]
   
3. cos x + cos y = 2 cos[ (x + y) / 2 ] cos[ (x - y) / 2 ]
   
4. cos x - cos y = 2 sin[ (x + y) / 2 ] sin[ (x - y) / 2 ]

Example 5
Factor the expression sin(4x) - sin(2x)
Solution to Example 5
Use the difference to product formula (formula 2 above) to write the given expression as product
sin(4x) - sin(2x) = 2 cos ( (4x + 2x) / 2) sin((4x - 2x) / 2)
Simplify.
sin(4x) - sin(2x) = 2 cos (3 x) sin(x)

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