Compute exact sine, cosine, tangent, and reciprocal ratios. Track coterminal angles, references, and unit-circle positions. Download clean reports and compare values with plotted graphs.
Multiple of π mode: enter 1 and 6 for π/6, 5 and 4 for 5π/4, or -7 and 6 for -7π/6.
Decimal mode: use radian numbers. Exact forms appear only when the normalized angle matches a standard unit-circle angle.
| Angle | sin(θ) | cos(θ) | tan(θ) |
|---|---|---|---|
| π/6 | 1/2 | √3/2 | √3/3 |
| π/4 | √2/2 | √2/2 | 1 |
| 2π/3 | √3/2 | -1/2 | -√3 |
| 5π/4 | -√2/2 | -√2/2 | 1 |
| 11π/6 | -1/2 | √3/2 | -√3/3 |
Exact trigonometric ratios come from the unit circle. For a normalized angle θ, the point on the unit circle is (cos θ, sin θ).
From that point, the core relationships are:
The calculator first normalizes the angle into the interval 0 ≤ θ < 2π. It then identifies the axis or quadrant, finds the reference angle, and applies the correct signs to standard unit-circle ratios.
For angles like π/6, π/4, π/3, 2π/3, 5π/4, and 11π/6, the exact values come from known special triangles and unit-circle symmetry.
Exact trigonometric values are useful in algebra, geometry, calculus, and physics because they preserve symbolic precision. Instead of rounding early, you keep clean forms such as √3/2 or √2. That makes later simplification easier and reduces small decimal errors.
Radians also matter because they match the natural geometry of the circle. When an angle is written as a multiple of π, the connection between rotation, symmetry, and the unit circle becomes clearer. Exact ratios in radians are therefore standard in advanced mathematics courses.
This calculator supports both symbolic and decimal entry. Symbolic π input is best when you already know the angle as a fraction of π. Decimal mode is helpful when a value comes from another calculation and you want to check whether it lands on a standard exact angle after normalization.
Standard unit-circle angles do. Common examples are 0, π/6, π/4, π/3, π/2, and their quadrant reflections within one full turn.
Normalization moves every angle into one revolution. That makes the quadrant, reference angle, signs, and exact ratios easier to identify correctly.
That happens when the normalized decimal angle does not match a standard unit-circle angle. The calculator still gives reliable decimal approximations.
Coterminal angles share the same terminal side. They differ by whole multiples of 2π, so their trigonometric ratios remain the same.
Tangent equals sin(θ) divided by cos(θ). When cos(θ) is zero, division by zero occurs, so tangent is undefined.
No. Decimal mode only shows exact symbolic results when the entered value matches a standard special angle after normalization.
The point gives coordinates (cos θ, sin θ). Those coordinates directly determine the sine and cosine of the normalized angle.
Use radians in higher mathematics, trigonometric identities, and calculus. They connect angle measure directly to arc length and circle geometry.
Important Note: All the Calculators listed in this site are for educational purpose only and we do not guarentee the accuracy of results. Please do consult with other sources as well.