What is the Formula of Cos(A – B) ?

In this post you will learn what is the formula for cos (A – B) with examples.

Cos (A – B) Formula :

The formula of cos(A – B) is cos A cos B + sin A sin B.

Example : If sin A = $3\over 5$ and cos B = $9\over 41$, find the value of cos (A – B).

Solution : We have,

sin A = $3\over 5$ and cos B = $9\over 41$

$\therefore$  cos A = $\sqrt{1 – sin^2 A}$  and  sin B = $\sqrt{1 – cos^2 B}$

$\implies$  cos A = $\sqrt{1 – {9\over 25}}$ = $4\over 5$  and  sin B = $\sqrt{1 – {81\over 1681}}$ = $40\over 41$

Now, By using above formula,

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

= $4\over 5$ $\times$ $9\over 41$ + $3\over 5$ $\times$ $40\over 41$ = $156\over 205$

Example : Find the value of cos 15.

Solution : cos 15 = sin (45 – 30)

By using above formula,

cos (45 – 30) = cos 45 cos 30 – sin 45 sin 30

$\implies$ cos 15 = $1\over \sqrt{2}$ $\times$ $\sqrt{3}\over 2$ + $1\over \sqrt{2}$ $\times$ $1\over 2$

$\implies$ cos 15 = $\sqrt{3} + 1\over 2\sqrt{2}$