Question

How do you find the exact value of \[\tan \left( \dfrac{10\pi }{3} \right)\]?

Answer 1

Hint: The given question is the trigonometric expression and in order to solve this solve we have to use the properties of trigonometric functions. First we need to remove the full rotation of \[2\pi \] until the angle is between 0 to \[2\pi \]. Then using the trigonometric ratios table, we will find the exact value of the given expression.

Formula used:
\[\pi \] radian = \[{{180}^{0}}\]
To find the tangent of any angle we need to just divide the sine and cosine of the same angle:\[\tan \theta =\dfrac{\sin \theta }{\cos \theta }=\dfrac{perpendicular}{base}\]

Trigonometric ratio table used to find the sine and cosine of the angle:

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

\[\]

Complete step by step solution:
Here, we have given the function \[\tan \left( \dfrac{10\pi }{3} \right)\] and we need to find out the exact value,
On the trigonometric unit circle,
\[\tan \left( \dfrac{10\pi }{3} \right)=\tan \left( \dfrac{4\pi }{3}+2\pi \right)\]
Removing the full rotation of \[2\pi \]as the angle is between 0 to\[2\pi \].
Therefore,
Now we have,
\[\tan \dfrac{\pi }{3}\]
As we know that,
\[\pi \] radian = \[{{180}^{0}}\]
Therefore, \[\dfrac{\pi }{3}=\dfrac{{{180}^{0}}}{3}={{60}^{0}}\]
Now putting \[\dfrac{\pi }{3}={{60}^{0}}\]
\[\tan {{60}^{0}}=\dfrac{\sin {{60}^{0}}}{\cos {{60}^{0}}}\]
From the trigonometric table we know the value of
\[\sin {{60}^{0}}=\dfrac{\sqrt{3}}{2}\] and \[\cos {{60}^{0}}=\dfrac{1}{2}\]
\[\tan {{60}^{0}}=\dfrac{\dfrac{\sqrt{3}}{2}}{\dfrac{1}{2}}=\dfrac{\sqrt{3}}{1}=\sqrt{3}\]
Therefore, the value of \[\tan \dfrac{\pi }{3}=\sqrt{3}\].
Hence, the exact value will be \[\sqrt{3}\].

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.