# 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

- 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 1Find 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 ApplicationsFree Trigonometry Tutorials and Problems

Trigonometric Identities and Their Applications