Evaluate csc, sec, and cot at any angle. Understand their relationship to sin, cos, and tan through step-by-step reciprocal derivations, exact values at standard angles, and interactive graphs showing all three reciprocal curves with asymptotes.
Input
The graph shows all three reciprocal functions over [−2π, 2π]. The current angle θ is marked with a vertical line. Dashed curves show the parent functions (sin, cos, tan) for reference.
Results
| Function | Decimal value | Exact / note |
|---|---|---|
| sin θ | — | — |
| csc θ | — | — |
| cos θ | — | — |
| sec θ | — | — |
| tan θ | — | — |
| cot θ | — | — |
csc θ = 1/sin θ sec θ = 1/cos θ cot θ = 1/tan θ
Each reciprocal function is the multiplicative inverse of its primary function. A quick way to remember the pairs is the "co-" pairing rule:
Mnemonic: "The pair that DOESN'T share a 'co-' at the same spot are reciprocals."
|sin θ| ≤ 1 ⟹ |csc θ| = 1/|sin θ| ≥ 1
Because sin θ and cos θ are bounded between −1 and 1, their reciprocals can never lie strictly between −1 and 1:
cot θ has no such restriction because tan θ is unbounded, so cot θ spans all real numbers (−∞, ∞).
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