Improving the estimate of the Sun's rotational speed

To estimate the Sun's rotational speed in the AU lab, we took two points on the Sun's edges and used them to estimate the Sun's rotation speed.  See the picture!

In particular, we took the two points with the greatest difference in speed and assumed they were on a line perpendicular to the Sun's rotation axis, and hence gave a good estimate of the rotation speed.  This is because, for a solid body rotating at some angular frequency \(\omega\), the rotation speed of a point on the surface is
$$v=r\omega,$$
with r as shown on the picture below.

Since we are not accounting for any angle dependence in our computation (i.e., we used \(r=R_{Sun}\)), we have taken it that \(\gamma=90^{\circ}\).  But in reality it may not be.  Using two pairs of points we can determine its actual value, and thence correct the rotation velocity estimate.  For the pair of points c and d, we have the velocity difference \(\Delta v\), which we define as \(\Delta v_{ij}=v_i-v_j\), to be

$$\Delta v_{cd}=2r\omega=2R_{Sun}\omega\sin\gamma,$$
while for the pair of points e and f we have
$$\Delta v_{ef}=2x\omega=2R_{Sun}\omega\cos\gamma.$$

Note that \(x=R_{Sun}\sin(90^{\circ}-\gamma)=R_{Sun}\cos\gamma\); you should convince yourself that the angles are really as I've drawn them.  In so doing, you may want to remember that vertical angles are equal.

With our two equations for two different \(\Delta v\)s, we can solve for \(\gamma\) as
$$\gamma=\tan^{-1}\bigg[\frac{\Delta v_{cd}}{\Delta v_{ef}}\bigg].$$

The true rotation velocity difference is then just
$$\Delta v_{true}=\Delta v_{cd}\frac{1}{\sin\gamma}=\Delta v_{cd}\bigg[\frac{\sqrt{\Delta v_{cd}^2+\Delta v_{ef}^2}}{\Delta v_{cd}}\bigg]=\sqrt{\Delta v_{cd}^2+\Delta v_{ef}^2}.$$

Note the true velocity difference is larger than or equal to our uncorrected result, since \(\sin \gamma \leq 1\); the last result also clearly shows this!  To get to the last result, I used the identity that
$$\sin\bigg[ \tan^{-1}\left(\frac{X}{Y}\right)\bigg]=\frac{X}{\sqrt{X^2+Y^2}}.$$