Sum and difference formulas for sine and cosine

10 - 1 Formulas for Cos(a + b) and sin(a + b)

Formulas for Cos(a+ b)

We will prove the sum formula for the cosine in class. I will state it here and use it to prove other formulas later. The two main purposes for these formulas are:

1) finding exact values of other trig expressions

2) simplifying expressions to find other identities

Sum and Difference Formulas for Sine And Cosine

Cos(a ± b) = cos a cos b ∓ sin a sin b

sin (a ± b) = sin a cos b ± sin b cos a

All of these formulas should be memorized. Plus you need to recall the trig values for 0, 30, 45, 60 and 90!

Demo: sin(A + B) (Manipula Math)

Sample problems

1) Find the exact value of cos 15^o

Solution:

Think of two angles that you know the value of, that either add or subtract and give you 15. There are many. One that comes to mind is 45 and 30.

let a = 45 and b = 30

Use the above formula for the difference of the cosine

cos 15 = cos(45 - 30) = cos 45 cos 30 + sin 45 sin 30

2) Find the exact value of sin 50^o cos 10^o + sin 10^o cos 50^o

Solution:

Match the problem to one of the formulas. Do you see that it is the sum formula for the sine function?

= sin (50 + 10) = sin 60^o
= √(3)/2

3) Suppose sin a = 4/5 and sin b = 5/13, where both a and b are in the first quadrant. Find cos (a - b)

Solution:

We need the difference formula for cosine

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

We know the sin a and sin b, we need to find the cos a and cos b. We can use basic trig to find it. First, draw a picture for each.

Look at the diagrams. Do you see how we got the missing values? That's right, using x, y and r and their relationship to the sine and cosine.

cos a = x/r = 3/5

cos b = x/r = 12/13

Put the values in the formula:

Cos(a - b) = (3/5)(12/13) + (4/5)(5/13)

= (36/65) + (20/65)

= 56/65

4) Verify that cos(x - p ) = - cos x

Solution:

Use the difference formula for cosine:

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

with a = x and b = p

cos(x - p) = cos x cos p + sin x sin p

remembering that cos p = -1 and sin p = 0, we have:

= -1(cos x) + 0(sin x)

= - cos x

That should give you some background in the add/subtract formulas. Let's take a look at the tangent formulas!