Question

The value of $\tan 150^\circ = ?$
A.$\dfrac{1}{{\sqrt 3 }}$
B.$ - \dfrac{1}{{\sqrt 3 }}$
C.$\sqrt 3 $
D.$ - \sqrt 3 $

Answer 1

Hint: We cannot directly find the value of $\tan 150^\circ $. So, we can write 150 as 180 minus 30. That is $\tan \left( {\pi - 30^\circ } \right)$. Now, as 30 degree is being subtracted from $\pi $, $\tan \left( {\pi - 30^\circ } \right)$ will lie between $\dfrac{\pi }{2}$ and $\pi $, that is 2nd quadrant. We know that $\tan \theta $ is negative in the 2nd quadrant. So, the value of $\tan 150^\circ $ will be equal to the negative value of $\tan 30^\circ $.

Complete step-by-step answer:
In this question, we are asked to find the value of $\tan 150^\circ $. Now, we don’t have direct value for $\tan 150^\circ $. So, we need to use some trigonometric relations and formulas to find the value of $\tan 150^\circ $.
Now, we can write 150 as 180 minus 30. Therefore, we get
$ \Rightarrow \tan 150^\circ = \tan \left( {180^\circ - 30^\circ } \right)$ - - - - - - - - - - - - - - - (1)
Now, we know that 180 degrees is equal to $\pi $. Therefore, equation (1) becomes
$ \Rightarrow \tan 150^\circ = \tan \left( {\pi - 30^\circ } \right)$ - - - - - - - - - - - - - - - (2)
Now, as $30^\circ $ is being subtracted from $\pi $, $\tan \left( {\pi - 30^\circ } \right)$ will lie between $\dfrac{\pi }{2}$ and $\pi $, that is 2nd quadrant. We know that in the 2nd quadrant only sin and cosec are positive.
Hence, in the 2nd quadrant the value of $\tan \left( {\pi - 30^\circ } \right)$ will be negative.
$ \Rightarrow \tan 150^\circ = \tan \left( {\pi - 30^\circ } \right)$
$
   = -\tan 30^\circ \\
   = -\dfrac{1}{{\sqrt 3 }} \\
   = - \dfrac{1}{{\sqrt 3 }} \;
 $
Hence, option B is the correct answer.
So, the correct answer is “Option B”.

Note: $ \Rightarrow $1st quadrant = All positive
$ \Rightarrow $2nd quadrant = $\sin \theta $ and $\cos ec\theta $ are positive.
$ \Rightarrow $3rd quadrant = $\tan \theta $ and $\cot \theta $ are positive.
$ \Rightarrow $4th quadrant = $\cos \theta $ and $\sec \theta $ are positive.