How this calculator works
On a circle of radius one, cosine is the horizontal coordinate and sine is the vertical coordinate. The other four functions are ratios or reciprocals of those coordinates. Reducing the angle by whole turns gives the same terminal point. Undefined ratios are shown as text rather than very large finite substitutes. Ratio mode uses the standard principal inverse branches, including arccotangent in (0°, 180°).
Formula
(x,y)=(cos θ,sin θ); tan θ=y/x; cot θ=x/y; sec θ=1/x; csc θ=1/y; x²+y²=1.
Worked example
At 30°, x = √3/2 and y = 1/2, so tan = 1/√3 and csc = 2. At 90°, x = 0 and y = 1, so tangent and secant are undefined.
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Frequently asked questions
Are negative angles allowed?
Yes. They turn clockwise. For example -90° reaches the same point as 270°.
How are values on the axes handled?
A denominator of zero gives Undefined. Exact multiples of a quarter turn use exact axis coordinates. Nearby finite values keep their sign and magnitude.
Can a ratio identify every possible angle?
No. Ratio mode returns a principal angle. Other solutions follow from periodicity and symmetry.