Evaluating tangent at zero degrees and ninety degrees

Key takeaway: Use the tangent definition and right-triangle geometry to interpret both special-angle values.

Part a: Evaluate the tangent values

Using a calculator:

• $\tan 0^\circ = 0$
• $\tan 90^\circ$ is undefined

The first value is straightforward. The second is not a real number because calculators usually display an error or an undefined result.

Part b: Explain using the tangent ratio

The tangent of an angle in a right triangle is defined as

$\tan \theta = \frac{\text{opposite}}{\text{adjacent}}$

For $0^\circ$, the opposite side has length 0 while the adjacent side is positive. That gives

$\tan 0^\circ = \frac{0}{\text{positive number}} = 0$

This matches the idea that a $0^\circ$ angle is flat, so there is no rise compared with the run.

For $90^\circ$, the usual right-triangle definition breaks down. If you try to think geometrically, the adjacent side would shrink toward 0, so the ratio becomes

$\frac{\text{opposite}}{0}$

and division by zero is undefined. That is why

$\tan 90^\circ$

is undefined, not zero and not a very large number.

Geometric reasoning

A tangent value measures steepness. At $0^\circ$, the line is horizontal, so steepness is 0. As the angle gets closer and closer to $90^\circ$, the line becomes more vertical, and tangent grows without bound. But at exactly $90^\circ$, there is no finite tangent value because the horizontal side disappears.

Final statement

So the results are:

$\boxed{\tan 0^\circ = 0}$

and

$\boxed{\tan 90^\circ\text{ is undefined}}$

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Pitfalls the pros know 👇 A very common mistake is to say that $\tan 90^\circ$ equals infinity. In standard school mathematics, it is better to say undefined, because the ratio $\frac{\text{opposite}}{\text{adjacent}}$ requires division by zero at $90^\circ$, and division by zero is not defined. Another mistake is to confuse tangent with slope on a graph and assume every vertical line has a numeric tangent. The definition only works for angles where a right-triangle interpretation makes sense, and $90^\circ$ is the boundary case where that interpretation fails.

What if the problem changes? If the question changed to evaluate $\tan 30^\circ$ and $\tan 60^\circ$, both values would be defined and could be found from special right triangles. If the prompt asked for $\tan(-0^\circ)$, the answer would still be 0 because the angle is the same direction as $0^\circ$. If it asked for an angle very close to $90^\circ$, such as $89.9^\circ$, the tangent would be a very large positive number, showing how the ratio grows as the angle approaches vertical.

Tags: undefined tangent, division by zero, right-triangle definition