From the solution to the problem Strange Rectangle (November 2001) you know angles $SPQ$ and $SRQ$ are right angles, angle $PQR$ is 45 degrees and angle $PSQ$ is 135 degrees.

For the first part you need to know

$ \tan 60^{\circ} = \sqrt3$

When you take $x=\sqrt 3$ and $y=1$ and use the triangles in the diagram you can find the exact values of the sine, cosine and tangent of these angles in surd form.

Look carefully at $AS$ and $SD$ for the second part. Here you take $x=\sqrt 2 + 1$ and $y=1$ to find the exact values of the trigonometric ratios for $22.5$ degrees and $67.5$ degrees.