Sum and Difference Formulas

Introduction: In this lesson, formulas involving the sum and difference of two angles will be defined and applied to the fundamental trig functions.

The Lesson: For two angles a and b, we have the following relationships:

Sum formulas:

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



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



3. 

Difference formulas:

1. sin(a – b) = sin(a)cos(b) – cos(a)sin(b)



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



3. 

Proofs of these formulas are available in all trig and pre-calculus texts.

Let's Practice:

1. What is the exact value of sin(105º)?

We can use a sum angle formula noticing that 105º = 45º + 60º.

We have sin(105º) = sin(45º + 60º) = sin(45º )cos(60º) + cos(45º )sin(60º).

We know the exact values of trig functions for 60º and 45º. (LINK my filename trigspecialangles)

Therefore, sin(45º )cos(60º) + cos(45º )sin(60º) = .

This is the exact value because we are using the radicals to express exact square roots. A decimal approximation is 0.9659.

2. What is the exact value of tan(15º)?

We can use a difference angle formula noticing that 15º = 45º - 30º.

tan(15º)	= tan(45º - 30º)
	=

This expression, which represents the exact value of tan(15º), can be rewritten as follows so that there is no radical in the denominator.

3. A quadrant four angle A has a tangent of and a quadrant one angle B has a tangent of 2. What is the exact value of the tan(A + B)? What is the exact value of cos(A – B)?

We show diagrams of each angle below

Angle A

Angle B

To work with the cosine, we need the measurements of the hypotenuse of each triangle in the diagrams because for any angle x.

Using the Pythagorean Theorem, the hypotenuse in the diagram for angle A is and for angle B the hypotenuse is .

Therefore we have:

sin(A) = and cos(A) = .
sin(B) = and cos(B) = .

We use these values in the formula for the cos(A – B).

cos(A – B) = cos(A)cos(B) + sin(A)sin(B) = .