In this post you will learn what is the formula for sin (A โ B) with examples.
Sin (A โ B) Formula :
The formula of sin(A โ B) is sin A cos B โ cos A sin B.
Example : If sin A = \(3\over 5\) and cos B = \(9\over 41\), find the value of sin (A โ B).
Solution : We have,
sin A = \(3\over 5\) and cos B = \(9\over 41\)
\(\therefore\)ย cos A = \(\sqrt{1 โ sin^2 A}\)ย andย sin B = \(\sqrt{1 โ cos^2 B}\)
\(\implies\)ย cos A = \(\sqrt{1 โ {9\over 25}}\) = \(4\over 5\)ย andย sin B = \(\sqrt{1 โ {81\over 1681}}\) = \(40\over 41\)
Now, By using above formula,
sin (A + B) = sin A cos B โ cos A sin B
= \(3\over 5\) \(\times\) \(9\over 41\) โ \(4\over 5\) \(\times\) \(40\over 41\) = \(-133\over 205\)
Example : Find the value of sin 15.
Solution : sin 15 = sin (45 โ 30)
By using above formula,
sin (45 โ 30) = sin 45 cos 30 โ cos 45 sin 30
\(\implies\) sin 15 = \(1\over \sqrt{2}\) \(\times\) \(\sqrt{3}\over 2\) โ \(1\over \sqrt{2}\) \(\times\) \(1\over 2\)
\(\implies\) sin 15 = \(\sqrt{3} โ 1\over 2\sqrt{2}\)
