Question

Find the value of:- \[\sin 25^\circ \cos 65 + \cos 25^\circ \sin 65\]

Answer 1

Hint: In the given question, we need to evaluate the answer of a trigonometric expression involving sines and cosines. We cannot directly add the two expressions as the arguments (or here we can say the angles) of the sines and cosines are different. So, we are going to have to think of some other way by transforming the sines and cosines in the given expression so that their angles become equal and we can apply some basic identity and use its result to evaluate the answer. A thing to note is that the sum of the angles of sines and cosines, when the two are equal, is \[90^\circ \].

Formula Used:
We are going to use the following two formulae for evaluating the answer of this question:
\[\sin A = \cos \left( {90 - A} \right)\]
and, \[{\sin ^2}A + {\cos ^2}A = 1\]

Complete step-by-step answer:
The given expression is \[\sin 25^\circ \cos 65 + \cos 25^\circ \sin 65\].
Now, according to the formula, \[\sin A = \cos \left( {90 - A} \right)\].
Thus, \[\sin 25^\circ = \cos \left( {90 - 25} \right) = \cos 65^\circ \]
And, \[\cos 25^\circ = \sin \left( {90 - 25} \right) = \sin 65^\circ \]
Now, substituting the values into the main expression,
\[\sin 25^\circ \cos 65 + \cos 25^\circ \sin 65 = \cos 65^\circ \times \cos 65^\circ + \sin 65^\circ \times \sin 65^\circ \]
\[ = {\cos ^2}65^\circ + {\sin ^2}65^\circ = 1\]
Hence, the required answer is \[1\].

Note: So, for solving questions of such type, we first write what has been given to us. Then we write down what we have to find. Then we think about the formulae which contains the known and the unknown and pick the one which is the most suitable for the answer. Then we put in the knowns into the formula, evaluate the answer and find the unknown. It is really important to follow all the steps of the formula to solve the given expression very carefully and in the correct order, because even a slightest error is going to make the whole expression wrong and is going to give us an incorrect answer.