Question

What is the value of $\sin 30.\cos 60 + \cos 30.\sin 60$?

Answer 1

Hint: There are two methods to solve this problems the first one involves the use of the trigonometric identity $\sin \left( {A + B} \right) = \sin A\cos B + \cos A\sin B$, the given equation is of the form of the right hand side of this trigonometric identity. The second method is explained at the later end.

Complete step-by-step answer:

Given trigonometric equation is
$\sin 30.\cos 60 + \cos 30.\sin 60$
As we know that $\sin \left( {A + B} \right) = \sin A\cos B + \cos A\sin B$ so use this property in above equation we have,
Where A = 30 and B = 60.
$ \Rightarrow \sin 30.\cos 60 + \cos 30.\sin 60 = \sin \left( {30 + 60} \right) = \sin 90$
Now as we know that the value of sin 90 is 1.
$ \Rightarrow \sin 30.\cos 60 + \cos 30.\sin 60 = \sin \left( {30 + 60} \right) = \sin 90 = 1$
So the required value of the trigonometric equation is 1.
So this is the required answer.

Note: In the second method we can directly use the basic values of trigonometric ratios like the value of $\sin {30^0} = \dfrac{1}{2},\cos {60^0} = \dfrac{1}{2},\cos {30^0} = \dfrac{{\sqrt 3 }}{2}{\text{ and sin6}}{{\text{0}}^0} = \dfrac{{\sqrt 3 }}{2}$. Substitution of these values directly into the given equation will give the answer. It is always advised to remember the trigonometric ratio values at some frequently used angles like 30, 60, 90, and 120.