Inputs
Tip: sec(θ)=1/cos(θ), csc(θ)=1/sin(θ), cot(θ)=1/tan(θ). Undefined results appear when denominators are zero.
Now detects exact values for special angles
Results
| Angle (°) | Angle (rad) | Angle (g) | sec(θ) | csc(θ) | cot(θ) | sin(θ) | cos(θ) | tan(θ) |
|---|---|---|---|---|---|---|---|---|
| Enter values and click Calculate to populate results. | ||||||||
Worked Steps
No calculation yet.
Formulas Used
For an angle θ (in radians for core definitions):
- sec(θ) = 1 / cos(θ)
- csc(θ) = 1 / sin(θ)
- cot(θ) = 1 / tan(θ) = cos(θ) / sin(θ)
Unit conversions:
- deg → rad: rad = deg × π/180
- grad → rad: rad = grad × π/200
- Normalize option wraps the angle to one turn ([0, 2π)).
Undefined cases: When cos(θ)=0, sec(θ) is undefined. When sin(θ)=0, csc(θ) and cot(θ) are undefined.
How to Use
- Enter the angle value and select the unit.
- Optionally enable normalization and adjust decimal precision.
- Click Calculate to compute reciprocal functions and trig companions.
- Review the table and the generated step-by-step derivation.
- Use the buttons above to export results as CSV or PDF.
Example Data
| Angle (°) | Angle (rad) | Angle (g) | sec(θ) | csc(θ) | cot(θ) |
|---|---|---|---|---|---|
| 0 | 0 | 0 | 1 | undefined | undefined |
| 30 | 0.523599 | 33.333333 | 1.154701 | 2 | 1.732051 |
| 45 | 0.785398 | 50 | 1.414214 | 1.414214 | 1 |
| 60 | 1.047198 | 66.666667 | 2 | 1.154701 | 0.577350 |
| 90 | 1.570796 | 100 | undefined | 1 | 0 |
Values rounded to six decimals; see Results for custom precision.
Special Angles Reference (Exact Values)
| Angle (°) | Angle (rad) | sin(θ) | cos(θ) | tan(θ) | sec(θ) | csc(θ) | cot(θ) |
|---|---|---|---|---|---|---|---|
| 0 | 0 | 0 | 1 | 0 | 1 | undefined | undefined |
| 30 | π/6 | 1/2 | √3/2 | √3/3 | 2√3/3 | 2 | √3 |
| 45 | π/4 | √2/2 | √2/2 | 1 | √2 | √2 | 1 |
| 60 | π/3 | √3/2 | 1/2 | √3 | 2 | 2√3/3 | √3/3 |
| 90 | π/2 | 1 | 0 | undefined | undefined | 1 | 0 |
| 120 | 2π/3 | √3/2 | -1/2 | -√3 | -2 | 2√3/3 | -√3/3 |
| 135 | 3π/4 | √2/2 | -√2/2 | -1 | -√2 | √2 | -1 |
| 150 | 5π/6 | 1/2 | -√3/2 | -√3/3 | -2√3/3 | 2 | -√3 |
| 180 | π | 0 | -1 | 0 | -1 | undefined | undefined |
| 210 | 7π/6 | -1/2 | -√3/2 | √3/3 | -2√3/3 | -2 | √3 |
| 225 | 5π/4 | -√2/2 | -√2/2 | 1 | -√2 | -√2 | 1 |
| 240 | 4π/3 | -√3/2 | -1/2 | √3 | -2 | -2√3/3 | √3/3 |
| 270 | 3π/2 | -1 | 0 | undefined | undefined | -1 | 0 |
| 300 | 5π/3 | -√3/2 | 1/2 | -√3 | 2 | -2√3/3 | -√3/3 |
| 315 | 7π/4 | -√2/2 | √2/2 | -1 | √2 | -√2 | -1 |
| 330 | 11π/6 | -1/2 | √3/2 | -√3/3 | 2√3/3 | -2 | -√3 |
Use this table to verify exact outputs shown by the calculator.
Quadrant Sign Rules
| Quadrant | sin(θ) | cos(θ) | tan(θ) | sec(θ) | csc(θ) | cot(θ) |
|---|---|---|---|---|---|---|
| I (0°–90°) | + | + | + | + | + | + |
| II (90°–180°) | + | − | − | − | + | − |
| III (180°–270°) | − | − | + | − | − | + |
| IV (270°–360°) | − | + | − | + | − | − |
Signs for reciprocals follow their base functions: sec=1/cos, csc=1/sin, cot=cos/sin.
Domains and Ranges
| Function | Domain (radians / degrees) | Range |
|---|---|---|
| sec(θ) | θ ≠ π/2 + kπ / θ ≠ 90° + 180°k | (−∞, −1] ∪ [1, ∞) |
| csc(θ) | θ ≠ kπ / θ ≠ 0° + 180°k | (−∞, −1] ∪ [1, ∞) |
| cot(θ) | θ ≠ kπ / θ ≠ 0° + 180°k | All real numbers |
Here, k ∈ ℤ. Excluded angles correspond to zeros of denominators.
Frequently Asked Questions
They are the reciprocals of sine, cosine, and tangent, namely cosecant, secant, and cotangent respectively.
When cosine is zero, secant is undefined; when sine is zero, cosecant and cotangent are undefined.
No. It wraps the angle within a single turn but leaves the trigonometric values invariant.
Yes. Choose the unit, and the calculator converts and computes automatically.
Set the decimal places you prefer. Internally, calculations use full precision and display rounds at the end.
These occur when a denominator approaches zero. The calculator marks such cases as undefined for safety.
Yes. Use the four export buttons to download CSV or PDF for either table.