Sum 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 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 sin^{2}x + cos^{2}x = 1 to find cos x.
cos x = (+ or ) √(1  sin^{2}x)
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 sin^{2}y + cos^{2}y = 1 to find cos y.
cos y = (+ or ) √(1  sin^{2}y)
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)
More References and Links
Trigonometric Formulas and Their Applications
Free Trigonometry Tutorials and Problems
Trigonometric Identities and Their Applications
