Question

Find the value of $\sin {75}^{\circ }$using sum or difference identity.

Answer 1

Find the value of the given trigonometric ratio

Given trigonometric ratio: $\sin {75}^{\circ }$

$\sin {75}^{\circ }$can be expressed as, $\sin {75}^{\circ }=\sin ({45}^{\circ }+{30}^{\circ })$

We know that $\mathit{s}\mathit{i}\mathit{n}\mathbf{}\mathbf{(}\mathit{A}\mathbf{+}\mathit{B}\mathbf{)}\mathbf{=}\mathit{s}\mathit{i}\mathit{n}\mathbf{}\mathit{A}\mathit{c}\mathit{o}\mathit{s}\mathbf{}\mathit{B}\mathbf{+}\mathit{c}\mathit{o}\mathit{s}\mathbf{}\mathit{A}\mathit{s}\mathit{i}\mathit{n}\mathbf{}\mathit{B}$

So by applying the above sum identity we get,

$\sin {75}^{\circ }=\sin {45}^{\circ }\cos {30}^{\circ }+\cos {45}^{\circ }\sin {30}^{\circ }$

As, $\sin {45}^{\circ }=\frac{1}{\sqrt{2}},\cos {30}^{\circ }=\frac{\sqrt{3}}{2}=,\cos {45}^{\circ }=\frac{1}{\sqrt{2}}$ and $\sin {30}^{\circ }=\frac{1}{2}$.

By substituting the values we get,

$$\begin{gathered} \sin {75}^{\circ }=\left(\frac{1}{\sqrt{2}}\times \frac{\sqrt{3}}{2}\right)+\left(\frac{1}{\sqrt{2}}\times \frac{1}{2}\right) \\ =\frac{\sqrt{3}}{2\sqrt{2}}+\frac{1}{2\sqrt{2}} \\ =\frac{\sqrt{3}+1}{2\sqrt{2}} \end{gathered}$$

Hence, the value of $\sin {75}^{\circ }$is $\frac{1+\sqrt{3}}{2\sqrt{2}}$.

Alternate solution:

$\sin {75}^{\circ }$can be expressed as, $\sin {75}^{\circ }=\sin ({135}^{\circ }-{60}^{\circ })$

We know that $\sin (A-B)=\sin A\cos B-\cos A\sin B$

So by applying the above sum identity we get,

$\sin {75}^{\circ }=\sin {135}^{\circ }\cos {60}^{\circ }-\cos {135}^{\circ }\sin {60}^{\circ }$

As, $\sin {135}^{\circ }=\frac{1}{\sqrt{2}},\cos {60}^{\circ }=\frac{1}{2},\cos {135}^{\circ }=\frac{-1}{\sqrt{2}}$ and $\sin {60}^{\circ }=\frac{\sqrt{3}}{2}$.

By substituting the values we get,

$$\begin{gathered} \sin {75}^{\circ }=\left(\frac{1}{\sqrt{2}}\times \frac{1}{2}\right)-\left(\frac{-1}{\sqrt{2}}\times \frac{\sqrt{3}}{2}\right) \\ =\frac{1}{2\sqrt{2}}+\frac{\sqrt{3}}{2\sqrt{2}} \\ =\frac{1+\sqrt{3}}{2\sqrt{2}} \\ \therefore \sin {75}^{\circ }=\frac{1+\sqrt{3}}{2\sqrt{2}} \end{gathered}$$