Question

The way to find the exact value of $\tan {270^ \circ }$.

Answer 1

Quadrant should be determined first. Since our angles greater than ${180^ \circ }$ and less than or equal to ${270^ \circ }$ therefore, it is located in quadrant III. Within the $3^{rd}$ quadrant the values of tan are positive only. A tangent line is a line which intersects a circle in only one point, called the point of tangency.

Complete step by step answer:
For an angle $\theta = {270^ \circ }$,

$$\Rightarrow \theta \to {270^ \circ } \\\ \Rightarrow x \to 0 \\\$$

It depends positively or negatively which quadrant $\theta$ approaches ${270^ \circ }$ from.
Since, division by 0 is undefined,
\Rightarrow $$$$\tan \left( {{{270}^ \circ }} \right) = \dfrac{{y = - 1}}{{x \to 0}}
Since, $\tan \left( \theta \right) = \dfrac{y}{x}$
Hence $\tan {270^ \circ }$ is undefined.

Note: A tangent line was a line which intersects a circle in exactly 1 point, called the point of tangency. Within the unit circle imagine a line tangent to the unit circle at some point. If the hypotenuse is extended beyond the right angled triangle until it intersects this tangent line, the vertical length is interrupted by this extended hypotenuse using the point $\left( {1,0} \right)$ as its endpoint is the tangent.
At zero degrees this tangent length is zero. Hence, $\tan \left( 0 \right) = 0$. As our first quadrant angle increases, the tangent will increase very rapidly. As we meet up to 90 degrees, this length will get incredibly large. At 90 degrees we must say that the tangent is undefined, because after we divide the leg opposite by the leg adjacent we cannot divide by zero.
As we move past 90 degrees into the second quadrant, we should "back" up the hypotenuse extended so it’ll meet with the initial tangent line. Because the length now formed is below the x-axis, since the tangent within the second quadrant is negative.