A handful of inverse tangent values come up again and again in exams and textbooks. They all come from two special right triangles, so once you understand those triangles you never need to memorise a table.
The 45-45-90 triangle
Both legs are equal, so the ratio opposite ÷ adjacent is 1:
- arctan(1) = 45° = π/4
The 30-60-90 triangle
The sides are in the ratio 1 : √3 : 2.
- Looking from the 30° angle: opposite = 1, adjacent = √3, so arctan(1/√3) = 30° = π/6. Note that 1/√3 = √3/3 ≈ 0.57735.
- Looking from the 60° angle: opposite = √3, adjacent = 1, so arctan(√3) = 60° = π/3.
The core table
| x | Decimal | arctan x | Radians |
|---|---|---|---|
| 0 | 0 | 0° | 0 |
| 2 − √3 | 0.267949 | 15° | π/12 |
| √2 − 1 | 0.414214 | 22.5° | π/8 |
| √3/3 | 0.577350 | 30° | π/6 |
| 1 | 1 | 45° | π/4 |
| √3 | 1.732051 | 60° | π/3 |
| √2 + 1 | 2.414214 | 67.5° | 3π/8 |
| 2 + √3 | 3.732051 | 75° | 5π/12 |
Negative inputs simply give negative angles, because arctan is an odd function: arctan(−√3) = −60°.
Where 15° and 75° come from
Using the tangent subtraction formula, tan(45° − 30°) = (1 − 1/√3) / (1 + 1/√3), which simplifies to 2 − √3. Similarly tan(45° + 30°) = 2 + √3.
Where 22.5° comes from
The half-angle formula tan(θ/2) = (1 − cos θ) / sin θ with θ = 45° gives (1 − √2/2) / (√2/2) = √2 − 1.
A useful pattern
Notice that each pair in the table adds up to 90°:
- arctan(2 − √3) + arctan(2 + √3) = 15° + 75° = 90°
- arctan(1/√3) + arctan(√3) = 30° + 60° = 90°
That is the identity arctan(x) + arctan(1/x) = π/2 for x > 0, and it is a quick way to check your answers.
The inverse tangent calculator recognises all of these values automatically and shows the exact form, such as π/6, next to the decimal answer.
Try it yourself: open the inverse tangent calculator to check any value from this article with full step-by-step working.