Reciprocal Trig Identities Calculator

Master reciprocals fast with clean, accurate trigonometry outputs anywhere. Switch degrees or radians, compute instantly, avoid undefined pitfalls. Type an angle or base value, get reciprocals immediately. Export results as CSV or save a PDF. Perfect for study, teaching, engineering, and quick checks anytime.

Inputs

Domain guards: if a base is zero or undefined, its reciprocal becomes undefined.

Export

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Results

Enter inputs and click Compute to see results here.

Example Data Table (Common Angles)

Angle (°) sin cos tan csc sec cot
0 0 1 0 undefined 1 undefined
30 0.5 0.866025 0.57735 2 1.154701 1.732051
45 0.707107 0.707107 1 1.414214 1.414214 1
60 0.866025 0.5 1.732051 1.154701 2 0.57735
90 1 0 undefined 1 undefined undefined
120 0.866025 -0.5 -1.732051 1.154701 -2 -0.57735
135 0.707107 -0.707107 -1 1.414214 -1.414214 -1
150 0.5 -0.866025 -0.57735 2 -1.154701 -1.732051
180 0 -1 0 undefined -1 undefined

Formulas Used

  • csc θ = 1 / sin θ
  • sec θ = 1 / cos θ
  • cot θ = 1 / tan θ = cos θ / sin θ

Domain notes

  • If sin θ = 0 ⇒ csc θ is undefined.
  • If cos θ = 0 ⇒ sec θ and tan θ are undefined.
  • If tan θ = 0 ⇒ cot θ is undefined.
  • For real inputs, |sin θ| ≤ 1 and |cos θ| ≤ 1.

How to Use This Calculator

  1. Select By angle or By function value.
  2. Enter an angle with units, or a base function value.
  3. Choose desired decimal precision, then press Compute.
  4. Review reciprocals and any domain warnings in the results table.
  5. Use Download CSV or Download PDF to export.

Quadrant Signs and Reciprocal Behavior

Signs of each function over quadrants (ASTC rule):

Quadrant sin θ cos θ tan θ csc θ sec θ cot θ
I (0°–90°)++++++
II (90°–180°)+−−+−−
III (180°–270°)−−+−−+
IV (270°–360°)−+−−+−

Parity: sin(−θ)=−sin θ ⇒ csc(−θ)=−csc θ; cos(−θ)=cos θ ⇒ sec(−θ)=sec θ; tan(−θ)=−tan θ ⇒ cot(−θ)=−cot θ.

Special Values and Undefined Points (0°–360°)

θ (°) sin cos tan csc sec cot
0010undefined1undefined
301/2√3/21/√322/√3√3
45√2/2√2/21√2√21
60√3/21/2√32/√321/√3
9010undefined1undefined0
1800−10undefined−1undefined
270−10undefined−1undefined0
360010undefined1undefined

Undefined occurs when a denominator equals zero: csc when sin=0; sec and tan when cos=0; cot when tan=0 (or sin=0 with finite cos).

FAQs

A reciprocal requires dividing by a base. When a base is zero (e.g., sin θ=0), the reciprocal (csc θ) is undefined. The calculator guards these cases automatically.

For real angles, sin and cos values must lie within [−1, 1]. If you enter values outside this range, the calculator flags a domain note. Tangent may be any real value.

Precision controls decimal places for display. Engineering work often uses 3–6 places. You can increase precision for study or demonstration, noting floating‑point rounding limits.

This tool focuses on reciprocal evaluation. To find angles from values, use inverse functions (arcsin, arccos, arctan) with quadrant analysis, which vary by context.

Click Download PDF. Your browser opens a print‑ready view; choose “Save as PDF.” Print styles optimize spacing and hide on‑screen controls for a clean export.