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 = \(\sqrt{3} – 1\over 2\sqrt{2}\)

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