Tangent records the slope of the rotating unit-circle direction.
WHAT HAPPENS IF…What happens at θ=90°?
Interactive exploration
Loading the mathematical model…
Connect the picture to the ideatanθ=cosθsinθ,cosθ=0
A point on the unit circle is (cosθ,sinθ). Scaling its direction to reach x=1 gives y=sinθ/cosθ. When cosθ=0 no finite intersection exists. Reversing both sine and cosine after π leaves their ratio unchanged, giving period π.
A detail that matters
The vertical asymptote is not part of the function graph.