Question

Find the value of $\sin 315^\circ \cos 315^\circ + \sin 420^\circ \cos 330^\circ$.

Answer 1

Here, we need to find the value of the given expression. We will convert the given angle measures into acute angles using trigonometric identities. Then, we will multiply and add the terms to find the required value. Trigonometric identities can only be used in an expression where trigonometric functions are present.

Complete step-by-step answer:
First, we will simplify the given trigonometric ratios.
We can rewrite the given angles as the sum or difference of a multiple of $90^\circ$ or $180^\circ$, and an acute angle.
Rewriting the terms of the expression, we get
$\sin 315^\circ = \sin \left( {360^\circ - 45^\circ } \right)$
$\cos 315^\circ = \cos \left( {360^\circ - 45^\circ } \right)$
$\sin 420^\circ = \sin \left( {360^\circ + 60^\circ } \right)$
$\cos 330^\circ = \cos \left( {360^\circ - 30^\circ } \right)$
The cosine of an angle $360^\circ - x$, is equal to the cosine of angle $x$, where $x$ is an acute angle.
Therefore, we get
$\cos 315^\circ = \cos \left( {360^\circ - 45^\circ } \right) = \cos 45^\circ$
$\cos 330^\circ = \cos \left( {360^\circ - 30^\circ } \right) = \cos 30^\circ$
The sine of an angle $360^\circ - x$, is equal to the negative of the sine of angle $x$, where $x$ is an acute angle.
Therefore, we get
$\sin 315^\circ = \sin \left( {360^\circ - 45^\circ } \right) = - \sin 45^\circ$
The sine of an angle $360^\circ + x$, is equal to the sine of angle $x$, where $x$ is an acute angle.
Therefore, we get
$\sin 420^\circ = \sin \left( {360^\circ + 60^\circ } \right) = \sin 60^\circ$
Substituting $\cos 315^\circ = \cos 45^\circ$, $\cos 330^\circ = \cos 30^\circ$, $\sin 315^\circ = - \sin 45^\circ$, and $\sin 420^\circ = \sin 60^\circ$ in the given expression $\sin 315^\circ \cos 315^\circ + \sin 420^\circ \cos 330^\circ$, we get
$\Rightarrow \sin 315^\circ \cos 315^\circ + \sin 420^\circ \cos 330^\circ = - \sin 45^\circ \cos 45^\circ + \sin 60^\circ \cos 30^\circ$
The sine of an angle measuring $45^\circ$ is equal to $\dfrac{1}{{\sqrt 2 }}$.
The sine of an angle measuring $60^\circ$ is equal to $\dfrac{{\sqrt 3 }}{2}$.
The cosine of an angle measuring $45^\circ$ is equal to $\dfrac{1}{{\sqrt 2 }}$.
The cosine of an angle measuring $30^\circ$ is equal to $\dfrac{{\sqrt 3 }}{2}$.
Substituting $\sin 45^\circ = \dfrac{1}{{\sqrt 2 }}$, $\cos 45^\circ = \dfrac{1}{{\sqrt 2 }}$, $\sin 60^\circ = \dfrac{{\sqrt 3 }}{2}$, and $\cos 30^\circ = \dfrac{{\sqrt 3 }}{2}$ in the equation, we get
$\Rightarrow \sin 315^\circ \cos 315^\circ + \sin 420^\circ \cos 330^\circ = - \dfrac{1}{{\sqrt 2 }} \times \dfrac{1}{{\sqrt 2 }} + \dfrac{{\sqrt 3 }}{2} \times \dfrac{{\sqrt 3 }}{2}$
Multiplying the terms of the expression, we get
$\Rightarrow \sin 315^\circ \cos 315^\circ + \sin 420^\circ \cos 330^\circ = - \dfrac{1}{2} + \dfrac{3}{4}$
The L.C.M. of 2 and 4 is 4.
Rewriting the terms with the denominator 4, we get
$\Rightarrow \sin 315^\circ \cos 315^\circ + \sin 420^\circ \cos 330^\circ = - \dfrac{2}{4} + \dfrac{3}{4}$
Adding the terms of the expression, we get
$\begin{array}{l} \Rightarrow \sin 315^\circ \cos 315^\circ + \sin 420^\circ \cos 330^\circ = \dfrac{{ - 2 + 3}}{4}\\\ \Rightarrow \sin 315^\circ \cos 315^\circ + \sin 420^\circ \cos 330^\circ = \dfrac{1}{4}\end{array}$
Therefore, we get the value of the expression $\sin 315^\circ \cos 315^\circ + \sin 420^\circ \cos 330^\circ$ as $\dfrac{1}{4}$.

Note: A common mistake is to convert $\cos 330^\circ = \cos \left( {360^\circ - 30^\circ } \right)$ to $\sin 30^\circ$. This is incorrect because $360^\circ$ is an even multiple of $90^\circ$. If we rewrite $\cos 330^\circ$ as $\cos \left( {270^\circ + 60^\circ } \right)$, then only it will become $\sin 60^\circ$, which is equal to $\dfrac{{\sqrt 3 }}{2}$. Here, cosine gets converted to sine because $270^\circ$ is an odd multiple of $90^\circ$.