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 75º = 30º + 45º.

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

To find these trigonometric function values,

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

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

cos 30º

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

So cos 30º = √3/2.

cos 45º

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

So cos 45º = 1/√2.

Write -.

sin 30º

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

So sin 30º = 1/2.

sin 45º

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

So sin 45º = 1/√2.

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

[√3/2]⋅[1/√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.