Question

Given that $\sin (A + B) = \sin A\cos B + \cos A\sin B$, find the value of $\sin {75^o}$.

Answer 1

Hint: Here, we will split ${75^o}$ into ${45^o}$ and ${30^o}$ then substitute in given equation i.e.., $\sin (A + B) = \sin A\cos B + \cos A\sin B$ to find the value of $\sin {75^o}$.

Complete step-by-step answer:
We had been given that $\sin (A + B) = \sin A\cos B + \cos A\sin B \to (1)$
And we need to find the value of $\sin {75^o}$.
Now $\sin {75^o}$ can be written as $\sin (45 + 30)$ so using equation 1
$\sin (45 + 30) = \sin {45^0}.\cos {30^0} + \cos {45^0}.\sin {30^0}$
Now, $\sin {45^0} = \dfrac{1}{{\sqrt 2 }},\cos {45^0} = \dfrac{1}{{\sqrt 2 }},\sin {30^0} = \dfrac{1}{2},\cos {30^0} = \dfrac{{\sqrt 3 }}{2}$ so putting values
We have
$\sin (45 + 30) = \dfrac{1}{{\sqrt 2 }} \times \dfrac{{\sqrt 3 }}{2} + \dfrac{1}{{\sqrt 2 }} \times \dfrac{1}{2} \\\ \sin (45 + 30) = \dfrac{{\sqrt 3 }}{{2\sqrt 2 }} + \dfrac{1}{{2\sqrt 2 }} \\\ \sin (75) = \dfrac{{\sqrt 3 + 1}}{{2\sqrt 2 }} \\\$
Hence, the value of $\sin {75^o} = \dfrac{{\sqrt 3 + 1}}{{2\sqrt 2 }}$.

Note: Whenever we come across such questions simply try to change the required angle in the terms of the formula given, then simple substitution and simplification will give you the answer.