Trigonometry • unit circle intuition

Tangent grows from the unit circle’s tangent line and the ratio sin(θ)/cos(θ).

As the radius meets the vertical tangent line at x = 1, the intersection height is tan(θ). That same value appears on the tangent graph, including the dramatic growth near angles where cosine is 0 and tangent is undefined.

Tangent line geometry

Pi-labeled tangent graph

Asymptotes shown clearly

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Live relationship

tan(0) = 0

Current angle

0° • 0

Current tangent value

0.000

Teaching cue

At θ = 0, the radius hits the tangent line at height 0, so tan(θ) = 0.

Unit circle + tangent line

The radius meets the vertical tangent line at (1, 0). That vertical distance is the tangent value.

Tangent graph

The same value is plotted on y = tan(θ). Dashed magenta lines mark vertical asymptotes where tangent is undefined.

Green marks the tangent segment on the line x = 1.

Blue marks the corresponding point on y = tan(θ).

Magenta dashed lines show where tangent is undefined.