Question

How do you find the value of $\sin (\dfrac{{5\pi }}{{12}})$?

Answer 1

Hint:
If in the question it is not mentioned which formula we need to use we can use any formula but here it is mentioned that we need to find the value of a given angle so we can solve it by using angle conversion formulas.

Complete step by step solution:
Given that –
We have to find the value of $\sin (\dfrac{{5\pi }}{{12}})$then
Let – $y = \sin (\dfrac{{5\pi }}{{12}})$
We know that in the trigonometry for changing any angle in degrees we will put the value of $\pi = {180^ \circ }$now we will put the value of $\pi $ then we will get
$ \Rightarrow y = \sin (\dfrac{{5 \times {{180}^ \circ }}}{{12}})$
Now after calculating all the above calculation we will get our value of $\sin (\dfrac{{5\pi }}{{12}})$ which is
$ \Rightarrow y = \sin (\dfrac{{5 \times {{180}^ \circ }}}{{12}}) = \sin {(5 \times 15)^ \circ }$
Now after the all multiplication we will get our value
$ \Rightarrow y = \sin {(5 \times 15)^ \circ } = \sin {(75)^ \circ }$
By using the conversion formula of trigonometry, we will change it and we will solve it then after completely solving the above equation we will get $\sin {(75)^ \circ } = \sin (30 + 45)$
We know that the formula of $\sin (A + B) = \sin A\cos B + \cos A\sin B$
Now our equation is $\sin {(75)^ \circ } = \sin (30 + 45) = \sin 30\cos 45 + \cos 30\sin 45$
Now we know that the value of $\sin 30 = \dfrac{1}{2}$ and $\sin 45 = \dfrac{1}{{\sqrt 2 }}$,$\cos 30 = \dfrac{{\sqrt 3 }}{2}$ and $\cos 45 = \dfrac{1}{{\sqrt 2 }}$
Now after putting value we will get
$ \Rightarrow \sin {(75)^ \circ } = \sin (30 + 45) = \sin 30\cos 45 + \cos 30\sin 45$
Now we will get
$ \Rightarrow \sin {(75)^ \circ } = \sin (30 + 45) = \dfrac{1}{2} \times \dfrac{1}{{\sqrt 2 }} + \dfrac{{\sqrt 3 }}{2} \times \dfrac{1}{{\sqrt 2 }}$
After completely solving the value we will get
$ \Rightarrow \sin {(75)^ \circ } = \sin (30 + 45) = \dfrac{1}{{2\sqrt 2 }} + \dfrac{{\sqrt 3 }}{{2\sqrt 2 }}$
$ \Rightarrow \sin {(75)^ \circ } = \sin (30 + 45) = \dfrac{{1 + \sqrt 3 }}{{2\sqrt 2 }}$
Therefore, the value of $\sin (\dfrac{{5\pi }}{{12}})$ is the $\dfrac{{1 + \sqrt 3 }}{{2\sqrt 2 }}$ which is the required value of our question.

Note:
Always remember that in the trigonometry chapter in all classes we have to remember some points and special values of angles so you can solve any question very easily and quickly compared to others.