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Sum and Difference Identities




 
In these lessons, solutions, and examples we will learn
  • the sum identities and difference identities for sine, cosine and tangent.
  • how to use the sum identities and difference identities to simplify trigonometric expressions.
  • how to use the sum identities and difference identities to prove other trigonometric identities.

Related Topics: More Trigonometric Identities

What are the Sum and Difference Identities?

The Sum and Difference Identities for sin, cos and tan are as follows.

Sum Identities

sin (a + b) = sin a cos b + cos a sin b

cos (a + b) = cos a cos b – sin a sin b

sum identities


Difference Identities

sin (a - b) = sin a cos b - cos a sin b

cos (a - b) = cos a cos b + sin a sin b

difference identities


Example:



Solution:

Given that cos(α + β) = cos α cos β – sin α sin β, then



Example:


Solution:





How to use the sum and difference identities for sin, cos, and tan?
Example:
1. Find sin(105°) exactly
2. Find cos(105°) exactly
3. Find tan(105°) exactly How to use Sum and Difference Identities to find exact trig values?
Example:
1. Find \(\cos \left( {\frac{{3\pi }}{4},\frac{\pi }{3}} \right)\) exactly

2. Find cos(42°)cos(18°) - sin(42°)sin(18°) exactly

3. Find \(\frac{{\tan 80^\circ - \tan 35^\circ }}{{1 + \tan 80^\circ \tan 35^\circ }}\) exactly

4. Find cos(u + v) exactly if sin(u) = 3/5 and sin(v) = 12/13 where u and v are acute angles (quadrant I)


 
How to use the Sum and Difference Identities to Prove Other Identities
Examples:
Prove sin(α + β) - sin(α - β) = 2cosαsinβ
Using the Sum and Difference Identities for Sine, Cosine and Tangent
Example 1:
If sin x = 12/13 and x is in the first quadrant, find tan(2x)

Using the Sum and Difference Identities for Sine, Cosine and Tangent
Example 2:
If tan x = 5/3 and x is in the first quadrant, find sin(2x) Using the Sum and Difference Identities for Sine, Cosine and Tangent
Example 3:
Simplify 1 - 16sin2x cos2x


 

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