Example Data Table (Common Angles)
| Angle | Radians | sin(θ) | cos(θ) | tan(θ) | (x, y) |
|---|---|---|---|---|---|
| 0° | 0 | 0 | 1 | 0 | (1, 0) |
| 30° | π/6 | 1/2 | √3/2 | √3/3 | (√3/2, 1/2) |
| 45° | π/4 | √2/2 | √2/2 | 1 | (√2/2, √2/2) |
| 60° | π/3 | √3/2 | 1/2 | √3 | (1/2, √3/2) |
| 90° | π/2 | 1 | 0 | Undefined | (0, 1) |
| 180° | π | 0 | -1 | 0 | (-1, 0) |
| 270° | 3π/2 | -1 | 0 | Undefined | (0, -1) |
| 360° | 2π | 0 | 1 | 0 | (1, 0) |
Formula Used
- x = r·cos(θ) and y = r·sin(θ).
- tan(θ) = sin(θ)/cos(θ) (undefined when cos(θ)=0).
- Reciprocals: sec(θ)=1/cos(θ), csc(θ)=1/sin(θ), cot(θ)=1/tan(θ).
- Conversions: rad = deg·π/180 and deg = rad·180/π.
- Reference angle (acute): reduce θ to one turn, then use symmetry by quadrant.
- Extras (θ in radians): s=r·|θ|, A=½·r²·|θ|, c=2r·|sin(θ/2)|.
How to Use This Calculator
- Select an input mode: angle or coordinates.
- Enter θ (and the unit) or enter (x, y).
- Set radius, rounding, and optional features.
- Press Submit to show results above the form.
- Use Download CSV or Download PDF to save outputs.
FAQs
1) What is the unit circle?
The unit circle is a circle with radius 1 centered at the origin. Every angle θ corresponds to a point (cosθ, sinθ) on that circle.
2) Why are cosine and sine the x and y values?
On a circle of radius r, the horizontal component is r·cosθ and the vertical component is r·sinθ. When r=1, these become cosθ and sinθ directly.
3) When is tan(θ) undefined?
tan(θ)=sin(θ)/cos(θ). If cos(θ)=0, the division is impossible, so tan(θ) is undefined. This happens at 90° and 270° (π/2 and 3π/2).
4) What is a reference angle?
A reference angle is the acute angle between the terminal side of θ and the x-axis. It’s always between 0° and 90° (0 and π/2) and helps use symmetry.
5) How do I convert degrees to radians?
Multiply degrees by π/180. For example, 60° becomes 60·π/180 = π/3. To convert radians to degrees, multiply by 180/π.
6) What does “normalize angle” mean?
Normalization wraps angles into one full turn. That means 0° to 360° (or 0 to 2π). It keeps values equivalent while making quadrant and exact matching clearer.
7) How are arc length and sector area computed?
Using radians: arc length is s=r·|θ|, and sector area is A=½·r²·|θ|. The calculator converts degrees to radians automatically before these computations.
8) Can I use coordinates instead of an angle?
Yes. Choose coordinate input and enter (x, y). The calculator finds θ using atan2(y, x), then computes trig values. If you set a radius, the point is projected onto that circle.