Question

How do you use the half angle identity to find the exact value of $\sin \left( {\dfrac{\pi }{8}} \right)?$

Answer 1

Hint:
In the given question they have asked for the exact value of a function using half angle identity. So for sine function we have an trigonometric half angle identity as $2{\sin ^2}x = 1 - \cos 2x$, in which we can replace $x = \dfrac{\pi }{8}$ and simplify to get the correct answer.

Complete step by step solution:
In the given question they have asked for the exact value of a function that $\sin \left( {\dfrac{\pi }{8}} \right)$ using half angle identity.
So for sine function we have an trigonometric half angle identity given by:
$2{\sin ^2}x = 1 - \cos 2x$ which can be further simplified for the sake of substitution as,
${\sin ^2}x = \dfrac{{1 - \cos 2x}}{2}$.
Here we have squares of sine function but they have asked for only sine function so we can even simplify further to get the form which they have asked in the question.
Taking square root on both side for the above equation, we get
$\sin x = \sqrt {\dfrac{{1 - \cos 2x}}{2}} $
In the above equation we need to replace $x$ in order to get the required answer. so by referring to given function that is $\sin \left( {\dfrac{\pi }{8}} \right)$ we can replace $x$ by $x = \dfrac{\pi }{8}$ .
On substituting the value of $x$ in the above equation, we get
$\sin \left( {\dfrac{\pi }{8}} \right) = \sqrt {\dfrac{{1 - \cos \left( {2.\dfrac{\pi }{8}} \right)}}{2}} $
$ \Rightarrow \sin \left( {\dfrac{\pi }{8}} \right) = \sqrt {\dfrac{{1 - \cos \left( {\dfrac{\pi }{4}} \right)}}{2}} $
We know that $\cos \left( {\dfrac{\pi }{4}} \right) = \dfrac{{\sqrt 2 }}{2}$ from trigonometric functions standard values
On substituting $\cos \left( {\dfrac{\pi }{4}} \right) = \dfrac{{\sqrt 2 }}{2}$ in the above expression, we get
$ \Rightarrow \sin \left( {\dfrac{\pi }{8}} \right) = \sqrt {\dfrac{{1 - \dfrac{{\sqrt 2 }}{2}}}{2}} $
On simplification, we get
$ \Rightarrow \sin \left( {\dfrac{\pi }{8}} \right) = \sqrt {\dfrac{{2 - \sqrt 2 }}{4}} $
We know that the square root of four is two so we get
$ \Rightarrow \sin \left( {\dfrac{\pi }{8}} \right) = \dfrac{{\sqrt {2 - \sqrt 2 } }}{2}$

Therefore the exact value of $\sin \left( {\dfrac{\pi }{8}} \right)$ is $\dfrac{{\sqrt {2 - \sqrt 2 } }}{2}$.

Note:
In case of trigonometric functions we need to know all the formulas because they can ask from anywhere. The entire problem stands on a single half angle identity formula. So remember the formulas. When simplifying the formula be careful you simplify to what they have asked for.