The Hadamard gate \(H\) is given by
\begin{equation*}
H = \frac{1}{\sqrt{2}} \begin{bmatrix}
1 & 1 \\
1 & -1
\end{bmatrix}
\end{equation*}
Since \(H^{2} = I\), the eigenvalues of the Hadamard matrix are \(\pm 1\). Let \(\lvert H_{\pm} \rangle\) be the eigenvector with eigenvalue \(\pm1\). Then
\begin{align*}
\lvert H_{+} \rangle &= \frac{1}{\sqrt{2}} \sqrt{1 + \frac{1}{\sqrt{2}}} \lvert {0} \rangle + \frac{1}{\sqrt{2}} \sqrt{1 - \frac{1}{\sqrt{2}}} \lvert {1} \rangle \\
| H_{-} \rangle &= -\frac{1}{\sqrt{2}} \sqrt{1 - \frac{1}{\sqrt{2}}} \lvert {0} \rangle + \frac{1}{\sqrt{2}} \sqrt{1 + \frac{1}{\sqrt{2}}} \lvert {1} \rangle
\end{align*}
Note that
\begin{align*}
\frac{1}{\sqrt{2}} \sqrt{1 + \frac{1}{\sqrt{2}}} &= \frac{\sqrt{2 + \sqrt{2}}}{2} = \cos\left( \frac{\pi}{8} \right) = 0.923879532511... \\
\frac{1}{\sqrt{2}} \sqrt{1 - \frac{1}{\sqrt{2}}} &= \frac{\sqrt{2 - \sqrt{2}}}{2} = \sin\left( \frac{\pi}{8} \right) = 0.382683432365...
\end{align*}
Thus, the eigenvectors can also be written as
\begin{align*}
\lvert H_{+} \rangle &= \cos\left( \frac{\pi}{8} \right) \lvert {0} \rangle + \sin\left( \frac{\pi}{8} \right) \lvert {1} \rangle & \lvert H_{-} \rangle &= -\sin\left( \frac{\pi}{8} \right) \lvert {0} \rangle + \cos\left( \frac{\pi}{8} \right) \lvert {1} \rangle
\end{align*}
or, equivalently,
\begin{align*}
\lvert H_{+} \rangle &= \cos\left( -\frac{\pi}{8} \right) \lvert {0} \rangle - \sin\left( -\frac{\pi}{8} \right) \lvert {1} \rangle & \lvert H_{-} \rangle &= \sin\left( -\frac{\pi}{8} \right) \lvert {0} \rangle + \cos\left( -\frac{\pi}{8} \right) \lvert {1} \rangle
\end{align*}
The eigen-basis for \(X\) and \(Z\) can also be written in this form:
\begin{align*}
\lvert Z_{+} \rangle &= \lvert {0} \rangle = \cos(0) \lvert {0} \rangle - \sin(0) \lvert {1} \rangle & \lvert Z_{-} \rangle &= \lvert {1} \rangle = \sin(0) \lvert {0} \rangle + \cos(0) \lvert {1} \rangle \\
\lvert X_{-} \rangle &= \lvert {-} \rangle = \cos\left( \frac{\pi}{4} \right) \lvert {0} \rangle - \sin\left( \frac{\pi}{4} \right) \lvert {1} \rangle & \lvert X_{+} \rangle &= \lvert {+} \rangle = \sin\left( \frac{\pi}{4} \right) \lvert {0} \rangle + \cos\left( \frac{\pi}{4} \right) \lvert {1} \rangle
\end{align*}
Now you know.