Why cos 90° = 0

This page gives the short answer and then four different explanations (unit circle, right-triangle limit, Pythagorean identity, Euler's formula). Click each section to expand the detailed explanation and see a simple visual for the unit-circle view.

(0, 1)Unit circle (radius = 1). Point at 90° has x-coordinate = cos(90°) = 0.

Visual summary: the point for 90° sits directly above the origin, so its x-coordinate is 0.

1) Unit-circle definition (direct)

On the unit circle, each angle θ from the positive x-axis corresponds to the point (cos(θ), sin(θ)). For 90° the point is (0,1), so the x-coordinate — which is the cosine — is 0.

2) Right-triangle / limiting view

 

3) Pythagorean identity

 

4) Euler's formula (complex numbers)