Question

If \[\sin \left( C+D \right)=\sin C.\cos D+\cos C.\sin D\] then the value of \[\sin {{75}^{\circ }}\] is
(a) \[\dfrac{1}{2\sqrt{2}}\left( \sqrt{3}+1 \right)\]
(b) \[\dfrac{1}{2}\left( \sqrt{3}+1 \right)\]
(c) \[\dfrac{\sqrt{3}}{2}\]
(d) \[\dfrac{1}{2}\]

Answer 1

Hint: We solve this problem by converting the given angle \[{{75}^{\circ }}\] into a sum of two angles such that the sine and cosine values of those angles are known. We know the standard values of sine and cosine of angles \[{{30}^{\circ }},{{45}^{\circ }},{{60}^{\circ }},{{90}^{\circ }}\] so, we need to convert \[{{75}^{\circ }}\] into sum of two angles so that we can use the given formula that is
\[\sin \left( C+D \right)=\sin C.\cos D+\cos C.\sin D\]

Complete step-by-step answer:
We are given that
\[\Rightarrow \sin \left( C+D \right)=\sin C.\cos D+\cos C.\sin D.....equation(i)\]
We are asked to find the value of \[\sin {{75}^{\circ }}\]
We know the standard values of sine and cosine of angles \[{{30}^{\circ }},{{45}^{\circ }},{{60}^{\circ }},{{90}^{\circ }}\]
Now, let us to convert \[{{75}^{\circ }}\] into sum of two angles as follows
\[\Rightarrow {{75}^{\circ }}={{30}^{\circ }}+{{45}^{\circ }}\]

Now, by applying the sine function on both sides we get
\[\Rightarrow \sin {{75}^{\circ }}=\sin \left( {{30}^{\circ }}+{{45}^{\circ }} \right)\]
We are given that the formula of composite angles that is
\[\Rightarrow \sin \left( C+D \right)=\sin C.\cos D+\cos C.\sin D\]
By using this formula to above equation we get
\[\Rightarrow \sin {{75}^{\circ }}=\sin {{30}^{\circ }}.\cos {{45}^{\circ }}+\cos {{30}^{\circ }}.\sin {{45}^{\circ }}\]
We know that from the standard table of trigonometric ratios
\[\begin{align}
  & \Rightarrow \sin {{30}^{\circ }}=\dfrac{1}{2} \\
 & \Rightarrow \cos {{30}^{\circ }}=\dfrac{\sqrt{3}}{2} \\
 & \Rightarrow \sin {{45}^{\circ }}=\cos {{45}^{\circ }}=\dfrac{1}{\sqrt{2}} \\
\end{align}\]
By using these standard values to above equation we get
\[\begin{align}
  & \Rightarrow \sin {{75}^{\circ }}=\left( \dfrac{1}{2} \right)\left( \dfrac{1}{\sqrt{2}} \right)+\left( \dfrac{\sqrt{3}}{2} \right)\left( \dfrac{1}{\sqrt{2}} \right) \\
 & \Rightarrow \sin {{75}^{\circ }}=\dfrac{1}{2\sqrt{2}}\left( \sqrt{3}+1 \right) \\
\end{align}\]
Therefore the value of \[\sin {{75}^{\circ }}\] is
\[\therefore \sin {{75}^{\circ }}=\dfrac{1}{2\sqrt{2}}\left( \sqrt{3}+1 \right)\]
So, option (a) is the correct answer.

So, the correct answer is “Option A”.

Note: We can find the value of \[\sin {{75}^{\circ }}\] from the composite angle formula of cosine function.
We know that
\[\Rightarrow \sin \theta =\cos \left( {{90}^{\circ }}-\theta \right)\]
By using the above result we get
\[\Rightarrow \sin {{75}^{\circ }}=\cos {{15}^{\circ }}\]
Let us divide the angle \[{{15}^{\circ }}\] into difference of two angles that is
\[\Rightarrow \sin {{75}^{\circ }}=\cos \left( {{45}^{\circ }}-{{30}^{\circ }} \right)\]
The composite angle formula for cosine function is given as
\[\Rightarrow \cos \left( A-B \right)=\cos A.\cos B+\sin A.\sin B\]
By using this formula to above equation we get
\[\Rightarrow \sin {{75}^{\circ }}=\cos {{45}^{\circ }}\cos {{30}^{\circ }}+\sin {{45}^{\circ }}.\sin {{30}^{\circ }}\]
We know that from the standard table of trigonometric ratios
\[\begin{align}
  & \Rightarrow \sin {{30}^{\circ }}=\dfrac{1}{2} \\
 & \Rightarrow \cos {{30}^{\circ }}=\dfrac{\sqrt{3}}{2} \\
 & \Rightarrow \sin {{45}^{\circ }}=\cos {{45}^{\circ }}=\dfrac{1}{\sqrt{2}} \\
\end{align}\]
By using these standard values to above equation we get
\[\begin{align}
  & \Rightarrow \sin {{75}^{\circ }}=\left( \dfrac{1}{2} \right)\left( \dfrac{1}{\sqrt{2}} \right)+\left( \dfrac{\sqrt{3}}{2} \right)\left( \dfrac{1}{\sqrt{2}} \right) \\
 & \Rightarrow \sin {{75}^{\circ }}=\dfrac{1}{2\sqrt{2}}\left( \sqrt{3}+1 \right) \\
\end{align}\]
Therefore the value of \[\sin {{75}^{\circ }}\] is
\[\therefore \sin {{75}^{\circ }}=\dfrac{1}{2\sqrt{2}}\left( \sqrt{3}+1 \right)\]
So, option (a) is the correct answer.