Arctan 1 = 45° (π/4) — Value, Degrees, Radians

Quick Answer:

Result: $\arctan 1 = 45°$
Notation: $\tan^{-1}(1) = \dfrac{\pi}{4}$ rad
Method shown: principal-value range of the inverse tangent, anchored to the unit circle Approximate value: $0.785398$ radians
Exact form: $\dfrac{\pi}{4}$

The inverse tangent answers a single question: which angle has a tangent of 1? On a right triangle where the opposite and adjacent sides are equal, the slope is exactly 1, and that triangle's acute angle is $45°$. Read off the unit circle, $\arctan 1$ is the angle whose terminal point sits on the line $y = x$ in the first quadrant — the half-way diagonal between the positive $x$-axis and positive $y$-axis.

Quick Reference Table — Arctan of Standard Values

This table covers the standard arctangent values that turn up alongside $\arctan 1$, in both degrees and radians. Every output sits inside the principal range $\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)$.

The pattern reads cleanly: arctangent is an odd function, so $\arctan(-x) = -\arctan(x)$. That is why the $-1$ row is the exact mirror of the $1$ row.

Where Arctan 1 Appears

The value $\arctan 1 = 45°$ shows up anywhere a slope of exactly 1 needs to become an angle. A roof, ramp, or staircase that rises one unit for every one unit it runs forward sits at $45°$ — the steepest "balanced" incline before the climb feels more vertical than horizontal.

It also fixes the principal argument of the complex number $1 + i$, which points along the $45°$ diagonal, and it sets the default rotation in screen graphics when an object is dragged equally far across and up. The National Institute of Standards and Technology uses the same inverse-tangent relationship when converting measured slope ratios into reported angles.

What Is Arctan?

Arctan, written $\arctan x$ or $\tan^{-1}(x)$, is the inverse of the tangent function. Tangent takes an angle and returns a ratio (opposite over adjacent, or $\sin\theta / \cos\theta$); arctangent runs the relationship backward, taking a ratio and returning the angle that produced it.

There is a catch. Tangent repeats every $180°$, so infinitely many angles share the tangent value 1 — $45°$, $225°$, $405°$, and so on. To stay a genuine function, arctangent returns only one of them: the angle inside the principal range $\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)$, equivalently $(-90°, 90°)$. Inside that range the tangent climbs steadily through every real number exactly once, so each input gets a single, unambiguous angle.

How to Find The Value of Arctan 1

Method 1 — From the right triangle

Set $\tan\theta = 1$, which means $\dfrac{\text{opposite}}{\text{adjacent}} = 1$, so the two legs are equal.

A right triangle with equal legs is an isosceles right triangle. Its two acute angles are equal, and they sum to $90°$.

Each acute angle is therefore $45°$.

Final answer: $\arctan 1 = 45° = \dfrac{\pi}{4}$.

Method 2 — From the unit circle

On the unit circle, $\tan\theta = \dfrac{y}{x}$. Setting this equal to 1 requires $y = x$.

In the first quadrant, the point with $y = x$ on the circle is $\left(\frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2}\right)$, which sits at $45°$.

That angle is inside the principal range, so it is the value arctangent returns.

Final answer: $\arctan 1 = \dfrac{\pi}{4}$.

Method 3 — Degree-to-radian conversion

Once you have $45°$, convert with the factor $\dfrac{\pi}{180°}$:

$$45° \times \frac{\pi}{180°} = \frac{\pi}{4} \approx 0.7854 \text{ rad}$$

Common mistakes with arctan 1

Mistake 1: Answering 225° instead of 45°

Where it slips in: A learner notices that $\tan(225°) = 1$ as well and offers $225°$ as the value of $\arctan 1$.

Don't do this: Reporting any angle outside $(-90°, 90°)$ — $225°$ has the right tangent but the wrong range.

The correct way: Arctangent only returns the angle inside its principal range, so $\arctan 1 = 45°$, full stop.

Mistake 2: Confusing arctan 1 with arctan of 1 degree

Where it slips in: The input to arctangent is a pure ratio, not an angle, but the "1" gets misread as $1°$.

Don't do this: Computing $\arctan(1°)$ or feeding the calculator a degree value where a ratio belongs.

The correct way: The 1 in $\arctan 1$ is the tangent value; the answer $45°$ is the angle. Input is a number, output is an angle.

Mistake 3: Calculator in the wrong mode

Where it slips in: Evaluating $\arctan 1$ with the calculator set to radians and expecting $45$.

Don't do this: Reading $0.7854$ off the screen and treating it as degrees.

The correct way: In radian mode the answer is $0.7854$ (that is $\frac{\pi}{4}$); switch to degree mode to read $45$. Check the mode indicator first.

Conclusion

• Arctan 1 equals $45°$, which is $\frac{\pi}{4}$ radians or about $0.7854$.

• The answer comes from the isosceles right triangle (equal legs give a slope of 1) and the unit-circle line $y = x$.

• Arctangent returns only the angle in the principal range $(-90°, 90°)$ — that is why $225°$ is rejected.

• Arctangent is odd, so $\arctan(-1) = -45°$ mirrors $\arctan 1$.

Practice these three before moving on

1. Evaluate $\arctan 1 + \arctan(-1)$.

2. Convert $\arctan 1$ to radians without a calculator.

3. Explain in one sentence why $\arctan 1 \neq 225°$.

If #1 didn't give $0$, recall that arctangent is an odd function. Want a live Bhanzu trainer to walk through more inverse-trig problems? Book a free demo class — online globally.