Question

Find the value of \[\sin {{315}^{\circ }}\cos {{315}^{\circ }}+\sin {{420}^{\circ }}\cos {{330}^{\circ }}\]

Answer 1

Hint: We solve this problem first by converting all the angles given to less than \[{{90}^{\circ }}\] because we have the trigonometric table of standard values for the angles less than \[{{90}^{\circ }}\]
We convert each angle as the sum or difference of \[{{360}^{\circ }}\] that we convert each angle as \[{{360}^{\circ }}+\theta \] or \[{{360}^{\circ }}-\theta \] so that we have conversions of trigonometric ratios as
\[\sin \left( {{360}^{\circ }}+\theta \right)=\sin \theta \]
\[\sin \left( {{360}^{\circ }}-\theta \right)=-\sin \theta \]
\[\cos \left( {{360}^{\circ }}+\theta \right)=\cos \theta \]
\[\cos \left( {{360}^{\circ }}-\theta \right)=-\cos \theta \]
By using the above formulas we reduce the given angles into angles less than \[{{90}^{\circ }}\] to find the required value easily.

Complete step by step answer:
We are asked to find the value of \[\sin {{315}^{\circ }}\cos {{315}^{\circ }}+\sin {{420}^{\circ }}\cos {{330}^{\circ }}\]
Let us assume that the required value as
\[\Rightarrow A=\sin {{315}^{\circ }}\cos {{315}^{\circ }}+\sin {{420}^{\circ }}\cos {{330}^{\circ }}......equation(i)\]
Now, let us convert each angle in the above equation into \[{{360}^{\circ }}+\theta \] or \[{{360}^{\circ }}-\theta \]
Now, by converting the first angle that is \[{{315}^{\circ }}\] we get
\[\Rightarrow {{315}^{\circ }}={{360}^{\circ }}-{{45}^{\circ }}\]
Now, by converting the next angle that is \[{{420}^{\circ }}\] we get
\[\Rightarrow {{420}^{\circ }}={{360}^{\circ }}+{{60}^{\circ }}\]
Now, by converting the next angle that is \[{{330}^{\circ }}\] we get
\[\Rightarrow {{330}^{\circ }}={{360}^{\circ }}-{{30}^{\circ }}\]
Now, by substituting the required angles in the equation (i) we get
\[\Rightarrow A=\sin \left( {{360}^{\circ }}-{{45}^{\circ }} \right)\cos \left( {{360}^{\circ }}-{{45}^{\circ }} \right)+\sin \left( {{360}^{\circ }}+{{60}^{\circ }} \right)\cos \left( {{360}^{\circ }}-{{30}^{\circ }} \right)\]
We know that the conversions of trigonometric ratios that are
\[\sin \left( {{360}^{\circ }}+\theta \right)=\sin \theta \]
\[\sin \left( {{360}^{\circ }}-\theta \right)=-\sin \theta \]
\[\cos \left( {{360}^{\circ }}+\theta \right)=\cos \theta \]
\[\cos \left( {{360}^{\circ }}-\theta \right)=-\cos \theta \]
Now, by using these conversions in the above equation we get
\[\begin{align}
  & \Rightarrow A=\left( -\sin {{45}^{\circ }} \right)\left( -\cos {{45}^{\circ }} \right)+\left( \sin {{60}^{\circ }} \right)\left( -\cos {{30}^{\circ }} \right) \\
 & \Rightarrow A=\sin {{45}^{\circ }}\cos {{45}^{\circ }}-\sin {{60}^{\circ }}\cos {{30}^{\circ }} \\
\end{align}\]
We know that from the standard trigonometric table the values of trigonometric ratios for some angles as
\[\begin{align}
  & \sin {{45}^{\circ }}=\cos {{45}^{\circ }}=\dfrac{1}{\sqrt{2}} \\
 & \sin {{60}^{\circ }}=\cos {{30}^{\circ }}=\dfrac{\sqrt{3}}{2} \\
\end{align}\]
Now, by substituting these values in above equation we get
\[\begin{align}
  & \Rightarrow A=\left( \dfrac{1}{\sqrt{2}} \right)\left( \dfrac{1}{\sqrt{2}} \right)-\left( \dfrac{\sqrt{3}}{2} \right)\left( \dfrac{\sqrt{3}}{2} \right) \\
 & \Rightarrow A=\dfrac{1}{2}-\dfrac{3}{4} \\
 & \Rightarrow A=\dfrac{-1}{4} \\
\end{align}\]

Therefore we can conclude that the value of given expression as
\[\therefore \sin {{315}^{\circ }}\cos {{315}^{\circ }}+\sin {{420}^{\circ }}\cos {{330}^{\circ }}=\dfrac{-1}{4}\]

Note: Students may do mistake in converting the given angles into angles less than \[{{90}^{\circ }}\]
Here, we can see that the given angles are all near to \[{{360}^{\circ }}\] so that we can easily convert the given angles in the form \[{{360}^{\circ }}+\theta \] or \[{{360}^{\circ }}-\theta \] to get the angles less than \[{{90}^{\circ }}\]
But we have so many ways of converting the given angles into angles less than \[{{90}^{\circ }}\]
We can convert in the form \[{{450}^{\circ }}+\theta \] or \[{{450}^{\circ }}-\theta \]
But in this process the formulas of conversions will change, that is sine gets converted to cosine and cosine gets converted to sine and also the sign changes.
But even though we will not get the all angles less than \[{{90}^{\circ }}\] because
\[\Rightarrow {{315}^{\circ }}={{450}^{\circ }}-{{135}^{\circ }}\]
Here, we again need to convert the angle \[{{135}^{\circ }}\] in the form of \[{{90}^{\circ }}+\theta \] which increases the solution length.
So, we need to check for nearest value where we get the given angles as the angles less than \[{{90}^{\circ }}\]