cos (A - B)

cos (A - B)

How to solve cos (A - B) problems: formula, proof, example, and its solution.

Formula

cos (A - B) = cos A cos B + sin A sin B

cos (A - B) = cos A cos B + sin A sin B

Proof

cos (A - B): Proof of the Formula

On a unit circle,
draw two terminal sides
whose central angles are A and B.

Then the endpoints of the terminal sides are
(cos A, sin A) and (cos B, sin B).

Point on a circle (using sine and cosine)

The length of the red side
is the distance between the endpoints.

Distance formula

Square of a difference

cos2 A + sin2 A = 1
cos2 B + sin2 B = 1

Pythagorean identities

So (red side)2 = 2 - 2 cos A cos B - 2 sin A sin B.

Next, on another unit circle,
draw two terminal sides
whose centeral angles are A - B and 0.

Then the endpoints of the terminal sides are
(cos (A - B), sin (A - B)) and (1, 0).

Point on a circle (using sine and cosine)

The length of the red side
is the distance between the endpoints.

Distance formula

[cos (A - B) - 1]2 = cos2 (A - B) + 2 cos (A - B) + 12

Square of a difference

cos2 (A - B) + sin2 (A - B) = 1

Pythagorean identities

So (red side)2 = 2 - 2 cos (A - B).

The red sides of the above triangles are congruent:
their opposite angles are both A - B.

So 2 - 2 cos (A - B) = 2 - 2 cos A cos B - 2 sin A sin B.

So cos (A - B) = cos A cos b + sin A sin B.

Example

Find the value of the given expression. cos 15 degrees

cos 15º = cos (60º - 45º)
= cos 60º cos 45º + sin 60º sin 45º

Draw a 45-45-90 triangle and a 30-60-90 triangle.

Cosine: CAH.
So cos 60º = 1/2.
And cos 45º = 1/√2.

Sine: SOH.
So sin 60º = √3/2.
And sin 45º = 1/√2.

So cos 60º cos 45º + sin 60º sin 45º
= (1/2)⋅(1/√2) + (√3/2)⋅(1/√2).

Rationalizing a denominator