Evaluate arctan(1/2) in radians and in degrees.

Step-by-step solution

1. 1

   State what the inverse function asks for. $\arctan\left(\tfrac12\right)$ is the unique angle $\theta$ with

   $\tan\theta=\frac{1}{2},\qquad -\frac{\pi}{2}<\theta<\frac{\pi}{2}$

   That principal range is what makes the answer unique — infinitely many angles have tangent $\tfrac12$, differing by multiples of $\pi$.

2. 2

   Bracket the answer before computing. Since $\tan 0=0$ and $\tan\tfrac{\pi}{4}=1$, and tangent increases, the angle must satisfy $0<\theta<\tfrac{\pi}{4}=45^\circ$. It is positive because $\tfrac12>0$.

3. 3

   Recognise there is no special-angle form. A tangent of $\tfrac12$ does not correspond to any of the standard $30^\circ$, $45^\circ$, $60^\circ$ triangles, so the answer is irrational and must be given numerically:

   $\arctan\left(\frac{1}{2}\right)\approx 0.4636476\ \text{rad}$

4. 4

   Convert to degrees. Multiply by $\tfrac{180}{\pi}$:

   $0.4636476\times\frac{180}{\pi}\approx 26.5651^\circ$

5. 5

   Check by taking the tangent back. $\tan(0.4636476)=0.5000000$ $\checkmark$, and the value sits inside the predicted range $0<26.565^\circ<45^\circ$ $\checkmark$. Geometrically this is the acute angle in a right triangle with legs $1$ and $2$, whose hypotenuse is $\sqrt5$.