sin(x + y) = sin x cos y + cos x sin y
cos(x + y) = cos x cos y - sin x sin y
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
sin(x - y) = sin x cos y - cos x sin y
cos(x - y) = cos x cos y + sin x sin y
tan(x - y) = [tan x - tan y] / [1 + tan x tan y]
Example 2
Simplify cos(x - π/2)
Solution to Example 2
Use the difference formula (formula 2 above) for cosine to expand the given expression
cos(x - π/2) = cos x cos π/2 + sin x sin π/2
cos π/2 = 0 and sin π/2 = 1, hence.
cos(x - π/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
sin x cos y = (1/2) [ sin(x + y) + sin(x - y) ]
cos x sin y = (1/2) [ sin(x + y) - sin(x - y) ]
cos x cos y = (1/2) [ cos(x + y) + cos(x - y) ]
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
sin x + sin y = 2 sin[ (x + y) / 2 ] cos[ (x - y) / 2 ]
sin x - sin y = 2 cos[ (x + y) / 2 ] sin[ (x - y) / 2 ]
cos x + cos y = 2 cos[ (x + y) / 2 ] cos[ (x - y) / 2 ]
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)
Questions
question 1
Find the exact value of sin(105°)
question 2
Factor the expression cos(5 x) - cos(3 x)
question 3
Given that sin(x) = -1 / 6 and cos(y) = -1 / 3 and the coterminal sides of x and y are in quadrant III. Fin the exact value of cos(x + y).
question 4
Factor cos(x) + sin(x) [hint: change cosine into sine or sine into cosine first then use one the formulas]
Solutions to the Above Questions
question 1
sin(105°) = (√ 2 + √ 6) / 4
question 2
cos(5 x) - cos(3 x) = 2 sin(4 x) sin(x)
question 3
cos(x + y) = (√ 35 - √ 8) / 18
question 4
cos(x) + sin(x) = 2 cos(π / 4) cos (x - π / 4)
