Question

Find the value of \[sin{{75}^{\circ }}\] .

Answer 1

Hint:The most important formula that is used in this question is as follows
\[\cos 2\theta =1-2{{\sin }^{2}}\theta \]
In such questions, we first let the angle given to be some variable and then we try to find out such a multiple of the given value of angle whose value is known to us.
The values of angles that we try to look for are as follows
\[{{90}^{\circ }},{{60}^{\circ }},{{180}^{\circ }},{{45}^{\circ }},{{135}^{\circ }},{{150}^{\circ }}\]
Then we solve the problem using the different relations that are there between different trigonometric functions.

Complete step-by-step answer:
As mentioned in the question, we have to find the value of \[sin{{75}^{\circ }}\] without the use of a calculator.
Now, as mentioned in the hint, we will first try to find a multiple of the given value of angle whose value is known to us and in this case the angle is \[{{75}^{\circ }}\] .
Now, we can see that the twice of this angle has a known value, so, we can write as follows
\[\begin{align}
  & \theta ={{75}^{\circ }} \\
 & 2\theta ={{150}^{\circ }} \\
\end{align}\]
Now, on taking the cos function of both the angles that is the left hand side and the right hand side angles.
So, on doing it we get the following result
\[\begin{align}
  & \cos 2\theta =\cos \left( {{150}^{\circ }} \right) \\
 & \cos 2\theta =\cos ({{180}^{\circ }}-{{30}^{\circ }}) \\
 & \cos 2\theta =-\cos ({{30}^{\circ }}) \\
\end{align}\]
(By using the properties of cos function)
\[\begin{align}
  & \cos 2\theta =-\cos ({{30}^{\circ }}) \\
 & \cos 2\theta =-\dfrac{\sqrt{3}}{2} \\
\end{align}\]
Now, using the identity mentioned in the hint, we can write as follows
\[\begin{align}
  & 1-2{{\sin }^{2}}\theta =-\dfrac{\sqrt{3}}{2} \\
 & 2{{\sin }^{2}}\theta =1+\dfrac{\sqrt{3}}{2} \\
 & {{\sin }^{2}}\theta =\dfrac{1}{2}+\dfrac{\sqrt{3}}{4} \\
\end{align}\]
Now, taking square root on both the sides, we get
\[\sin {{75}^{\circ }}=\dfrac{\sqrt{2+\sqrt{3}}}{2}\]
We have taken the positive value as the angle that is asked is an acute angle and sin value of an acute angle is positive.
Hence, this is the required value.

Note: -Another method of doing this question is by splitting the given angle into a difference or sum of angles whose values are known to us i.e Using $Sin(A+B)=SinACosB+SinBCosA$ where $A={45}^{\circ }$ and $B={30}^{\circ }$.Here we should remember \[sin{{45}^{\circ }}\] and \[cos{{30}^{\circ }}\] values to simplify it,Substituting the values in formula we get required answer.Students should remember important trigonometric identity,formulas and standard trigonometric angles for solving these types of problems.