How this calculator works
Cosecant inverts the unit-circle height. It is positive where sine is positive and negative where sine is negative. Its branches reach 1 at 90° and -1 at 270°, while sine’s zeros create vertical asymptotes. The function is odd, has period one turn and has no finite amplitude. Ratio mode uses arccsc x=arcsin(1/x), returning an angle between -90° and 90° with zero excluded.
Formula
csc θ=1/sin θ=hypotenuse/opposite; csc(-θ)=-csc θ; csc(θ+2π)=csc θ. Real values satisfy |csc θ|≥1.
Worked example
csc 30°=2 because sin 30°=1/2. csc 270°=-1, and csc 0° is undefined. A known cosecant of -2 returns -30° on the principal branch.
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Frequently asked questions
Can cosecant lie between zero and one?
No. A real cosecant has magnitude at least one.
Is csc⁻¹ always a reciprocal?
Notation can be ambiguous. This tool calls the inverse angle arccsc and writes the reciprocal explicitly as 1/sin θ.
How does its sign relate to sine?
They have the same sign wherever they are defined, because taking a reciprocal preserves a non-zero number’s sign.