Question

How do you find the exact values of \[{{\sin }^{3}}\left( \dfrac{\pi }{6} \right)\]using the half angle formula?

Answer 1

Hint: In the given question, we have been asked to find the value of a given sine function using the half angle identity. First we need to apply the trigonometric identity of half angle i.e. \[\cos 2a=1-2{{\sin }^{2}}a\]. Putting the values in this formula and simplifying the expression further using mathematical operations such as addition, subtraction, multiplication and division. In this way we will get the exact value of the given function.

Complete step by step solution:
We have given that,
\[{{\sin }^{3}}\left( \dfrac{\pi }{6} \right)\]
As we know that,
Using the trigonometric identity of the half angle identity, i.e.
\[\cos 2a=1-2{{\sin }^{2}}a\]
Here,
\[a=\dfrac{\pi }{6}\ then\ 2a=\dfrac{2\pi }{6}\]
Applying the above identity, we obtained,
\[\cos \left( \dfrac{2\pi }{6} \right)=1-2{{\sin }^{2}}\left( \dfrac{\pi }{6} \right)\]
Simplifying the above expression, we obtained
\[\cos \left( \dfrac{\pi }{3} \right)=1-2{{\sin }^{2}}\left( \dfrac{\pi }{6} \right)\]
By using the trigonometric ratio table;
Trigonometric ratio table used to find the sine and cosine of the angle:

Angles(in degrees)	\[\cos \theta \]
\[{{0}^{0}}\]	1
\[{{30}^{0}}\]	\[\dfrac{\sqrt{3}}{2}\]
\[{{45}^{0}}\]	\[\dfrac{1}{\sqrt{2}}\]
\[{{60}^{0}}\]	\[\dfrac{1}{2}\]
\[{{90}^{0}}\]	0

The value of \[\cos \dfrac{\pi }{3}=\dfrac{1}{2}\].
Applying the values from above, we get
\[\dfrac{1}{2}=1-2{{\sin }^{2}}\left( \dfrac{\pi }{6} \right)\]
Rearranging the terms in the above expression, we get
\[2{{\sin }^{2}}\left( \dfrac{\pi }{6} \right)=1-\dfrac{1}{2}\]
Solving the RHS of the above expression by taking the LCM, we get
\[2{{\sin }^{2}}\left( \dfrac{\pi }{6} \right)=\dfrac{1}{2}\]
Dividing both the sides by 2, we get
\[{{\sin }^{2}}\left( \dfrac{\pi }{6} \right)=\dfrac{1}{4}\]
Transposing the power 2 to the RHS of the expression, we get
\[\sin \left( \dfrac{\pi }{6} \right)=\sqrt{\dfrac{1}{4}}=\pm \dfrac{1}{2}\]
As we know that \[\dfrac{\pi }{6}\] is in the first quadrant, thus the value of the sine function is positive.
Therefore,
\[\sin \left( \dfrac{\pi }{6} \right)=\dfrac{1}{2}\]
Then,
\[{{\sin }^{3}}\left( \dfrac{\pi }{6} \right)={{\left( \dfrac{1}{2} \right)}^{3}}=\dfrac{1}{8}\]
Therefore, the exact value of \[{{\sin }^{3}}\left( \dfrac{\pi }{6} \right)\] using the half angle formula is \[\dfrac{1}{8}\].
Hence, it is the required possible answer.

Note: In order to solve these types of questions, you should always need to remember the properties of trigonometric and the trigonometric ratios as well. It will make questions easier to solve. It is preferred that while solving these types of questions we should carefully examine the pattern of the given function and then you would apply the formulas according to the pattern observed. As if you directly apply the formula it will create confusion ahead and we will get the wrong answer.