Question

Find the value of \[\tan {885^ \circ }\] using trigonometric relation?

Answer 1

Hint: We only know the value of trigonometric ratio below 360. For finding any value above 360 we will subtract the intervals of 360 from the value until it comes to less than 360. We are subtracting 360 because any trigonometric ratio will repeat its value after 360.Then try to find its value.

Complete step-by-step solution:
We have to find the value of \[\tan {885^ \circ }\]
We know that $\tan \left({360^\circ+ \theta}\right) = \tan \theta$, hence we will subtract the interval of 360 from 885 until it is less than 360.
885-360=495
We will again subtract 360 from 495 since it is not less than 360
495-360=135
Here we can write the above equations as $\tan {885^ \circ } = \tan\left({2\times {360^\circ}+135^\circ }\right)= \tan {135^\circ}$
Now, we have to find the value of \[\tan {135^ \circ }\]
We know the trigonometric relation like
\[\tan ({90^ \circ } + \;\theta ) = - \cot \;\theta .\]
We will break 135 in 90 and 45 and try to simplify it.
We will substitute \[\theta = {45^ \circ }\] in equation \[\tan ({90^ \circ } + \;\theta ) = - \cot \;\theta .\]
\[\tan ({90^ \circ } + \;{45^ \circ }) = - \cot \;{45^ \circ }\]
We know that \[\cot {45^ \circ } = 1\], so
\[\tan ({90^ \circ } + \;{45^ \circ }) = - 1\]
The value of \[\tan {885^ \circ }\] is -1.

Note: Tan is the ratio of sine and cos function. We can also find the value of $\sin {885^ \circ }$ and $\cos {885^ \circ }$then divide them to get the value of $\tan {885^ \circ }$ . we can also find the value of $\tan {885^ \circ }$ using graph, as graph of tan repeat’s its value after interval of 180 or we can say it is periodic after 180.