Question

Find the value of $\tan {1080^ \circ }$

Answer 1

Hint : First write the given angle in terms of $2n{\pi } + \theta$ , multiple of 360 degrees or $2{\pi }$ radians, where n can be any natural number. And the value of tan of angle $2n{\pi } + \theta$ will give $\tan \theta$ itself. So divide the given angle appropriately.

Complete step-by-step answer :
Here the value of $\tan {1080^ \circ }$ is positive because the angel falls in the first quadrant. And in the first quadrant all of the trigonometric ratios are positive. Tangent is positive in the first and third quadrants. Tangent is also a periodic function. So its value repeats after every $\pi$ radians. Tangent is one of most important functions in trigonometry. It is the inverse of cotangent function. Tangent function is also used to measure slopes of straight lines.
We are given to find the value of $\tan {1080^ \circ }$ .
Tan is the ratio of sine to cosine or in a right angled triangle it is the ratio of opposite side and adjacent side to the given angle.
So here $\tan {1080^ \circ }$ must be written in multiples of 360 degrees.
1080 degrees is 3 times 360 degrees.
This means $\tan {1080^ \circ } = \tan \left( {3 \times {{360}^ \circ }} \right)$
360 degrees is $2{\pi }$ radians.
$\Rightarrow \tan {1080^ \circ } = \tan \left[ {\left( {3 \times 2{\pi }} \right) + 0} \right]$
$\Rightarrow\tan {1080^ \circ } = \tan \left[ {\left( {2 \times 3 \times {\pi }} \right) + 0} \right]$
The RHS of the above equation is in the form of $\tan \left( {2n\pi + {\theta }} \right)$ which is equal to $\tan \theta$ . Here $\theta$ is zero (0).
Therefore, $\tan \left[ {\left( {2 \times 3 \times {\pi }} \right) + {0^ \circ }} \right] = \tan {0^ \circ }$
The value of $\tan {0^ \circ }$ is 0.
Therefore, the value of $\tan {1080^ \circ }$ is 0.
So, the correct answer is “0”.

Note : Tan is the ratio of sine and cosine. So we can find the values of $\sin {1080^ \circ }$ and $\cos {1080^ \circ }$ ; and then divide them to get the value. Or we finally got that $\tan {1080^ \circ }$ as $\tan {0^ \circ }$ , So we can divide $\sin {0^ \circ }$ by $\cos {0^ \circ }$ , to get its value.