Tangent of Three Right Angles

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Theorem

$\tan 270 \degrees = \tan \dfrac {3 \pi} 2$ is undefined

where $\tan$ denotes tangent.


Proof

We have:

\(\ds \tan 270 \degrees\) \(=\) \(\ds \map \tan {360 \degrees - 90 \degrees}\)
\(\ds \) \(=\) \(\ds -\tan 90 \degrees\) Tangent of Conjugate Angle

But from Tangent of Right Angle, $\tan 90 \degrees$ is undefined.

Hence so is $\tan 270 \degrees$.

$\blacksquare$


Also defined as

Some sources give that:

$\tan 270 \degrees = \infty$

but this naïve approach is overly simplistic and cannot be backed up with mathematical rigour.


Also see


Sources