Because we know the hypotenuse and the side opposite \(\alpha\text{,}\) we observe that \(\sin(\alpha) = \frac{1}{3}\text{.}\) Rewriting this statement using inverse function notation, we have equivalently that \(\alpha = \arcsin(\frac{1}{3})\text{,}\) which is the exact value of \(\alpha\text{.}\) Since this is not one of the known special angles on the unit circle, we can find a numerical estimate of \(\alpha\) using a computational device. Entering arcsin(1/3)
in Desmos, we find that \(\alpha \approx 0.3398\) radians. Note well: whatever device we use, we need to be careful to use degree or radian mode as dictated by the problem we are solving. We will always work in radians unless stated otherwise.
We can now find the remaining leg’s length and the remaining angle’s measure. If we let \(x\) represent the length of the horizontal leg, by the Pythagorean Theorem we know that
\begin{equation*}
x^2 + 1^2 = 3^2\text{,}
\end{equation*}
and thus \(x^2 = 8\) so \(x = \sqrt{8} \approx 2.8284\text{.}\) Calling the remaining angle \(\beta\text{,}\) since \(\alpha + \beta = \frac{\pi}{2}\text{,}\) it follows that
\begin{equation*}
\beta = \frac{\pi}{2} - \arcsin \left(\frac{1}{3}\right) \approx 1.2310\text{.}
\end{equation*}