Question

How do you find the exact functional value $\sin (60^\circ + 45^\circ )$ using the cosine sum or difference identity?

Answer 1

Hint : All the trigonometric functions are related to each other through several identities. Bigger angles are expressed as a sum of smaller angles and their values are found using the identities. In the given question, we have to find the sine of $60^\circ + 45^\circ$ by using the sum or difference identity, so first; we will apply the appropriate identity and then plug in the known values to get the correct answer.

Complete step-by-step answer :
Given,
$\sin (60^\circ + 45^\circ )$
We know that –
$\sin (a + b) = \sin a\cos b + \cos a\sin b \\\ \Rightarrow \sin (60^\circ + 45^\circ ) = \sin 60^\circ \cos 45^\circ + \cos 60^\circ \sin 45^\circ \\\ \Rightarrow \sin (60^\circ + 45^\circ ) = \dfrac{{\sqrt 3 }}{2} \times \dfrac{1}{{\sqrt 2 }} + \dfrac{1}{2} \times \dfrac{1}{{\sqrt 2 }} \\\ \Rightarrow \sin (60^\circ + 45^\circ ) = \dfrac{{\sqrt 3 + 1}}{{2\sqrt 2 }} \\\$
Hence, the exact functional value of $\sin (60^\circ + 45^\circ )$ is $\dfrac{{\sqrt 3 + 1}}{{2\sqrt 2 }}$ .
So, the correct answer is “ $\dfrac{{\sqrt 3 + 1}}{{2\sqrt 2 }}$ ”.

Note : We know the value of the sine and cosine function when the angle lies between 0 and $\dfrac{\pi }{2}$ .So we simply plugged the values of $60^\circ$ and $45^\circ$ . We have applied the identity that the sine of the sum of two angles a and b is equal to the sum of the product of the sine of angle a and cosine of angle b and the product of the cosine of angle a and the sine of angle b that is $\sin (a + b) = \sin a\cos b - \cos a\sin b$. Similarly, $\sin (a - b) = \sin a\cos b - \cos a\sin b$ . Many such identities can be used to solve similar questions.