Question

How do you evaluate $\sin 25\cos 65+\cos 25\sin 65$ ?

Answer 1

The easiest way to solve these problems is by using the basic trigonometric formula for compound angles. The formulae that needs to be used here is
$\sin \left( A+B \right)=\sin A\cos B+\cos A\sin B$
After putting the values of $A$ as $25$ and $B$ as $65$ , the entire expression gets simplified to a single term in $\text{sine}$ , that can be solved easily.

Complete step by step answer:
We start the solution by using one of the basic trigonometric formula of compound angles, that is,
$\sin \left( A+B \right)=\sin A\cos B+\cos A\sin B$
Where, $A$ and $B$ are two angles.
In a reverse manner, we can write the equation as,
$\sin A\cos B+\cos A\sin B=\sin \left( A+B \right)....equation1$
In $equation1$ , if we put $A$ as $25$ and $B$ as $65$ , the LHS becomes
$\sin 25\cos 65+\cos 25\sin 65$
Which is the exact same expression given in the problem. Therefore, $equation1$ , this can be evaluated to
$\begin{aligned} & \sin \left( 25+65 \right) \\\ & =\sin 90 \\\ \end{aligned}$
As no such unit of the angle used is mentioned in the question, therefore we assume it to be degrees.
We all know that the value of $\sin \left( {{90}^{\circ }} \right)=1$ .
Therefore, the given expression becomes
$\sin 25\cos 65+\cos 25\sin 65=1$
Thus, we can conclude that the given expression is simplified to $1$ .

Note:
First of all, we need to remember all the formula regarding trigonometry of compound angles to solve these types of problems. Often the students get confused between the formula and interchange one with the other, or the signs. We can also solve the problem by converting $\cos {{65}^{\circ }}$ and $\sin {{65}^{\circ }}$ to $\sin {{25}^{\circ }}$ and $\cos {{25}^{\circ }}$ respectively using the property of complementary angles. Then, the expression becomes ${{\sin }^{2}}{{25}^{\circ }}+{{\cos }^{2}}{{25}^{\circ }}$ which is nothing but $1$ .