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

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

& \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{aligned}$$ By using these standard values to above equation we get $$\begin{aligned} & \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{aligned}$$ Therefore the value of $$\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{aligned} & \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{aligned}$$ By using these standard values to above equation we get $$\begin{aligned} & \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{aligned}$$ Therefore the value of $$\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.