cos (A - B)

How to find cos (A - B) by using its formula: formula, 1 example, and its solution.

Formula

Formula

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

For cosine,
cos and sin are separated: cos cos, sin sin
and the middle sign changes: (-) → (+).

cos (A + B)

Example

Example

Solution

Set 15º = 45º - 30º.

You can also set
15º = 60º - 45º.
You'll get the same answer.

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

To find these trigonometric function values,

draw a 45-45-90 triangle
whose sides are 1, 1, √2,

and a 30-60-90 triangle
whose sides are 1, √3, 2.

cos 45º

Cosine is CAH:
Cosine,
Adjacent side (1),
Hypotenuse (√2).

So cos 45º = 1/√2.

cos 30º

Cosine is CAH:
Cosine,
Adjacent side (√3),
Hypotenuse (2).

So cos 30º = √3/2.

Write +.

sin 45º

Sine is SOH:
Sine,
Opposite side (1),
Hypotenuse (√2).

So sin 45º = 1/√2.

sin 30º

Sine is SOH:
Sine,
Opposite side (1),
Hypotenuse (2).

So sin 30º = 1/2.

So cos 45º cos 30º + sin 45º sin 30º
= [1/√2]⋅[√3/2] + [1/√2]⋅[1/2].

[1/√2]⋅[√3/2] + [1/√2]⋅[1/2]
= (√3 + 1)/2√2

To rationalize the denominator 2√2,
multiply [√2/√2].

(√3 + 1)√2 = √6 + √2

2√2⋅√2 = 2⋅2

2⋅2 = 4

So (√6 + √2)/4 is the answer.