Question

What is $\sin {20^ \circ }\cos {4^ \circ } + \cos {20^ \circ }\sin {4^ \circ }$ equal to?

Answer 1

Hint: Trigonometric identities are very helpful in solving such questions, applying those simple formulas helps to simplify the problem to a great extent. Only one thing should be kept in mind is that we need to arrange the problem in a perfect form before applying those identities so that the formula can be very easily applied to solve the problem.

Complete step-by-step answer:
Now we can recall the formula for $\sin \left( {A + B} \right)$:
$\bullet$ $\sin \left( {A + B} \right) = \sin A\cos B + \cos A\sin B$
Since we now know the formula we can very easily apply the same in the problem. If we substitute $A = {20^ \circ }$ and $B = {4^ \circ }$ we get:
$\bullet$ $\sin {20^ \circ }\cos {4^ \circ } + \cos {20^ \circ }\sin {4^ \circ } = \sin \left( {{{20}^ \circ } + {4^ \circ }} \right)$
$\bullet$ $\sin \left( {{{20}^ \circ } + {4^ \circ }} \right) = \sin {24^ \circ }$
Now if we are very specific to the value then:
$\bullet$ $\sin {24^ \circ } = 0.407$
So $\sin {20^ \circ }\cos {4^ \circ } + \cos {20^ \circ }\sin {4^ \circ }$equal to $\sin {24^ \circ }$or simply $0.407$

Note: Remembering the formula for all trigonometric identities is very crucial as many of the questions related to trigonometry can be solved if we are able to apply these formulas directly or by arranging some part of them in a proper order. But it is also important for one to be able to differentiate between all the identities, as many of the identities are almost similar to each other with a slight difference of either $ + $ or $ - $ and $\sin $or $\cos $etc. These are the very crucial areas where one can neglect these details and can go wrong. Sometimes if the identity fails to solve the problem other identities like $\sin \left( {90 - \theta } \right) = \cos \theta $ are very useful in order to set them in a proper way.