# Evaluate the following: $\sin 45^\circ \sin 30^\circ + \cos 45^\circ \cos 30^\circ $

Last updated date: 24th Mar 2023

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Answer

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Hint: Use direct values of angles of sin and cos

As we know the value of$\sin 45^\circ = \cos 45^\circ = \dfrac{1}{{\sqrt 2 }}$, the value of$\sin 30^\circ = \dfrac{1}{2}$, and the value of$\cos 30^\circ = \dfrac{{\sqrt 3 }}{2}$,

so, substitute these values in the given equation we have

$ \Rightarrow \sin 45^\circ \sin 30^\circ + \cos 45^\circ \cos 30^\circ = \left( {\dfrac{1}{{\sqrt 2 }}} \right)\left( {\dfrac{1}{2}} \right) + \left( {\dfrac{1}{{\sqrt 2 }}} \right)\left( {\dfrac{{\sqrt 3 }}{2}} \right)$

$ = \dfrac{1}{{2\sqrt 2 }}\left( {1 + \sqrt 3 } \right)$

So, this is the required answer.

Note: - In such types of questions the key concept we have to remember is that we always remember all the standard angle values of sin and cosine, it will help us a lot in finding the required answer, so directly substitute these values we will get the required answer.

As we know the value of$\sin 45^\circ = \cos 45^\circ = \dfrac{1}{{\sqrt 2 }}$, the value of$\sin 30^\circ = \dfrac{1}{2}$, and the value of$\cos 30^\circ = \dfrac{{\sqrt 3 }}{2}$,

so, substitute these values in the given equation we have

$ \Rightarrow \sin 45^\circ \sin 30^\circ + \cos 45^\circ \cos 30^\circ = \left( {\dfrac{1}{{\sqrt 2 }}} \right)\left( {\dfrac{1}{2}} \right) + \left( {\dfrac{1}{{\sqrt 2 }}} \right)\left( {\dfrac{{\sqrt 3 }}{2}} \right)$

$ = \dfrac{1}{{2\sqrt 2 }}\left( {1 + \sqrt 3 } \right)$

So, this is the required answer.

Note: - In such types of questions the key concept we have to remember is that we always remember all the standard angle values of sin and cosine, it will help us a lot in finding the required answer, so directly substitute these values we will get the required answer.

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