Question

How do you find the value of $\tan {240^ \circ }$ ?

Answer 1

Hint:
First of all find signs of the trigonometric ratios in all the four quadrants. All trigonometric functions are positive in the first quadrant. Express ${240^ \circ }$ as $\left( {{{180}^ \circ } + {{60}^ \circ }} \right)$ and find in which quadrant will this angle lie.

Complete step by step solution:
We know that, signs of trigonometric functions in different quadrants are:
Quadrant 1 – In this quadrant all trigonometric functions are positive. This quadrant lies in the angle less than ${90^ \circ }$ and more than 0
Quadrant 2 – In this quadrant, the trigonometric functions sin and cosec are positive while cos, sec, tan and cot functions are negative. This quadrant lies in the angle less than ${180^ \circ }$ and more than ${90^ \circ }$
Quadrant 3 – This quadrant has tan and cot functions as positive and cos, sin, sec and cosec functions negative. This quadrant lies in the angle less than ${270^ \circ }$ and more than ${180^ \circ }$
Quadrant 4 – cos and sec functions are positive in this quadrant and sin, cosec, tan and cot functions are negative in this quadrant. This quadrant lies in the angle less than ${360^ \circ }$ and more than ${180^ \circ }$
According to the question, the function given in the question is $\tan {240^ \circ }$. So, this function can also be expressed as –
$ \Rightarrow \tan \left( {{{180}^ \circ } + {{60}^ \circ }} \right)$
From the rule of quadrants, we can see that tan is positive in the third quadrant and first quadrant only. From the angle written above we can see that the angle lies in the quadrant which is more than ${180^ \circ }$. So, we can easily conclude that the angle lies in the third quadrant. Therefore, the value of $\tan {60^ \circ }$ will be positive. So, now we know that the value of $\tan {60^ \circ }$ is $\sqrt 3 $.

So, $\sqrt 3 $ is the required value for $\tan {240^ \circ }$.

Note:
The students usually try to calculate ${240^ \circ }$ directly but this is not possible. The above written method is the correct way of solving these types of questions. Many students also make mistakes for positive and negative so they should see the rule of quadrants for signs of trigonometric function by which the students will be able to put positive and negative signs correctly.