How this calculator works
Tangent compares vertical change with horizontal change. In a right triangle this is opposite divided by adjacent; on the unit circle it is y/x. The graph increases through each zero and has vertical asymptotes where the horizontal coordinate vanishes. Adding 180° changes the signs of both coordinates, preserving the quotient. Its range is unbounded, so tangent has no finite amplitude. Ratio mode returns arctangent strictly between -90° and 90°.
Formula
tan θ=sin θ/cos θ=opposite/adjacent; tan(θ+π)=tan θ; tan(-θ)=-tan θ. Tangent is undefined where cos θ=0.
Worked example
tan 45°=1, representing equal rise and run. A rise/run ratio of 1/√3 gives 30°. At 90° the adjacent component is zero, so tangent is undefined.
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Frequently asked questions
Is a tangent of one a 100% slope?
Yes. As a rise/run ratio, 1 corresponds to 100%, or a 45° angle. This calculator displays the ratio rather than a percent.
Why is the period only 180°?
Sine and cosine both reverse sign after half a turn; their quotient stays the same.
Why does tan 90° not show a very large number?
Exactly at the asymptote the denominator is zero. Nearby angles have large finite values, but 90° itself is undefined.