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If you try to build a right triangle for a 90∘90^\circ angle, both legs fold directly on top of each other and the triangle completely collapses. So how do we find sine, cosine, or tangent when there is no triangle left to use?

What Are Quadrantal Angles?

A quadrantal angle is an angle in standard position whose terminal side lies directly on the xx-axis or yy-axis. These angles are exact integer multiples of 90∘90^\circ (or Ο€2\frac{\pi}{2} radians).
To evaluate these collapsed angles without right triangles, we turn to the unit circle, a circle centered at the origin with a radius of r=1r = 1.

The Coordinate Definitions

Because the radius is r=1r = 1, any point on the unit circle gives trigonometric values directly through its coordinates: cos⁑θ=x\cos\theta = x and sin⁑θ=y\sin\theta = y.
The remaining four trigonometric functions are ratios of these coordinates: tan⁑θ=yx\tan\theta = \frac{y}{x}, csc⁑θ=1y\csc\theta = \frac{1}{y}, sec⁑θ=1x\sec\theta = \frac{1}{x}, and cot⁑θ=xy\cot\theta = \frac{x}{y}.
What happens when one of those coordinate values is zero? Let's evaluate an angle on the vertical axis.

Evaluating at 90 Degrees

For ΞΈ=Ο€2\theta = \frac{\pi}{2} (90∘90^\circ), the terminal ray points straight up along the positive yy-axis to the coordinate (0,1)(0, 1).
Substituting x=0x = 0 and y=1y = 1 yields sin⁑π2=1\sin\frac{\pi}{2} = 1, cos⁑π2=0\cos\frac{\pi}{2} = 0, and tan⁑π2=10\tan\frac{\pi}{2} = \frac{1}{0}, which is undefined because division by zero is mathematically impossible.