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 = \(\sqrt{3} – 1\over 2\sqrt{2}\)
