# What is the Formula of Sin(A – B) ?

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

## Sin (A – B) Formula :

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

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

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

= $$3\over 5$$ $$\times$$ $$9\over 41$$ – $$4\over 5$$ $$\times$$ $$40\over 41$$ = $$-133\over 205$$

Example : Find the value of sin 15.

Solution : sin 15 = sin (45 – 30)

By using above formula,

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

$$\implies$$ sin 15 = $$1\over \sqrt{2}$$ $$\times$$ $$\sqrt{3}\over 2$$ – $$1\over \sqrt{2}$$ $$\times$$ $$1\over 2$$

$$\implies$$ sin 15 = $$\sqrt{3} – 1\over 2\sqrt{2}$$