Question

How do you find the exact value of $\sin 33{}^\circ \cos 27{}^\circ +\cos 33{}^\circ \sin 27{}^\circ $.

Answer 1

Hint: To solve the above question we will use the concept of trigonometric identities. We will use the formula $\sin \left( A+B \right)=\sin A\cos B+\sin B\cos A$ to solve the above question. At first, we will compare the equation given in the question $\sin 33{}^\circ \cos 27{}^\circ +\cos 33{}^\circ \sin 27{}^\circ $ with $\sin A\cos B+\sin B\cos A$ to get the value of A and B. Then, we will put the value of A and B in equation $\sin \left( A+B \right)=\sin A\cos B+\sin B\cos A$ and get the required answer.

Complete step-by-step solution:
We can see that the above given question is of trigonometry, so we will use trigonometric identity to solve the above question.
Since, we have to find the value of $\sin 33{}^\circ \cos 27{}^\circ +\cos 33{}^\circ \sin 27{}^\circ $.
We can see that the above expression is similar to the sum of two different angle formulas $\sin \left( A+B \right)=\sin A\cos B+\sin B\cos A$.
So, after comparing the above expression $\sin 33{}^\circ \cos 27{}^\circ +\cos 33{}^\circ \sin 27{}^\circ $ with $\sin A\cos B+\sin B\cos A$ we will get:
$A=33{}^\circ $ and $B=27{}^\circ $.
So, we will put the value of A and B in the formula $\sin \left( A+B \right)=\sin A\cos B+\sin B\cos A$, then we will get:
$\Rightarrow \sin \left( 33{}^\circ +27{}^\circ \right)=\sin 33{}^\circ \cos 27{}^\circ +\sin 33{}^\circ \cos 27{}^\circ $
$\Rightarrow \sin \left( 60{}^\circ \right)=\sin 33{}^\circ \cos 27{}^\circ +\sin 33{}^\circ \cos 27{}^\circ $
Now, from trigonometric table we know that \[\sin 60{}^\circ =\dfrac{\sqrt{3}}{2}\] , so we will put the value $\dfrac{\sqrt{3}}{2}$ in place of \[\sin 60{}^\circ \], then we will get:
 $\Rightarrow \sin 33{}^\circ \cos 27{}^\circ +\sin 33{}^\circ \cos 27{}^\circ =\dfrac{\sqrt{3}}{2}$
So, we can say that exact value of $\sin 33{}^\circ \cos 27{}^\circ +\cos 33{}^\circ \sin 27{}^\circ $ is $\dfrac{\sqrt{3}}{2}$.
This is our required solution.

Note: Students are required to note that when we are given $\sec \theta $, $\operatorname{cosec}\theta $, $\tan \theta $, and $\cot \theta $ in the trigonometric expression then we always change them into $\sin \theta $ and $\cos \theta $ because we know half angle formula, double angle formula for sine and cosine function only so we by changing trigonometric function other than sine and cosine into sine and cosine make our take easier.