All six trig functions of an angle, exact values for special angles, and inverse trig
Updated October 2026
Place the angle on the unit circle, measured counterclockwise from the positive x-axis. The point where it meets the circle is (cos θ, sin θ). The other four follow: tan = sin ÷ cos, cot = cos ÷ sin, sec = 1 ÷ cos, csc = 1 ÷ sin. In a right triangle the same values are the side ratios: sin = opposite ÷ hypotenuse, cos = adjacent ÷ hypotenuse, tan = opposite ÷ adjacent.
For 30° the point is (√3/2, 1/2), so sin 30° = 0.5 and cos 30° = 0.866025. Then tan = 0.5 ÷ 0.866025 = 0.57735, csc = 1 ÷ 0.5 = 2, sec = 1 ÷ 0.866025 = 1.154701 and cot = 0.866025 ÷ 0.5 = 1.732051. Enter the angle, pick degrees or radians, and the results also show the quadrant and reference angle; switch to the inverse mode to turn a value back into an angle.
| Angle | Radians | sin | cos | tan |
|---|---|---|---|---|
| 0° | 0 | 0 | 1 | 0 |
| 30° | π/6 | 1/2 | √3/2 | √3/3 |
| 45° | π/4 | √2/2 | √2/2 | 1 |
| 60° | π/3 | √3/2 | 1/2 | √3 |
| 90° | π/2 | 1 | 0 | undefined |
| Angle | Quadrant | sin | cos | tan |
|---|---|---|---|---|
| 30° | I | 0.5 | 0.866025 | 0.57735 |
| 150° | II | 0.5 | -0.866025 | -0.57735 |
| 210° | III | -0.5 | -0.866025 | 0.57735 |
| 330° | IV | -0.5 | 0.866025 | -0.57735 |
| −30° | IV | -0.5 | 0.866025 | -0.57735 |
| 390° | I | 0.5 | 0.866025 | 0.57735 |
| 90° | on an axis | 1 | 0 | undefined |
Every angle in the table except 90° has a 30° reference angle, so the sizes repeat and only the signs change: all positive in Quadrant I, only sin (and csc) in II, only tan (and cot) in III, only cos (and sec) in IV. Angles past 360° or below 0° wrap around: 390° matches 30° and −30° matches 330°.
Wrong angle unit. sin(30) in radians is −0.988032, not 0.5. Radians = degrees × π ÷ 180, so 30° is π/6, about 0.523599.
Expecting the inverse to find every angle. sin 150° is also 0.5, but arcsin(0.5) returns 30° because arcsin only returns angles from −90° to 90°. arccos returns 0° to 180° and arctan stays strictly between −90° and 90°. Use the quadrant to find the other solutions.
Reading sin⁻¹ as 1/sin. sin⁻¹ x means arcsin x; the reciprocal 1/sin x is csc x.
Values outside −1 to 1. arcsin(1.2) and arccos(1.2) have no real answer, because sine and cosine never leave that range.
Tiny numbers instead of zero. Computers give cos 90° as about 6.1 × 10⁻¹⁷ rather than 0, which would make tan 90° a huge number. The calculator treats anything below 10⁻¹² as zero and reports tan and sec at 90° and 270° as undefined.