What is the exact value of tan 90°?

The exact value of tan 90° is undefined.

In trigonometry, the tangent of an angle in a right-angled triangle is the ratio of the length of the opposite side to the length of the adjacent side. Mathematically, this is written as tan(θ) = opposite/adjacent. For most angles, this ratio gives a specific number. However, when we consider tan 90°, things get a bit tricky.

At 90°, the angle is such that the opposite side is infinitely long compared to the adjacent side, which is essentially zero. This means we are trying to divide by zero, which is not possible in mathematics. Therefore, tan 90° does not have a finite value and is considered undefined.

If you look at the tangent function on a graph, you'll notice that as the angle approaches 90° from either side, the value of the tangent function increases or decreases without bound, heading towards positive or negative infinity. This behaviour is why we say tan 90° is undefined. Understanding this concept is crucial as it helps in analysing the behaviour of trigonometric functions and their applications in various mathematical problems.