Simplify trig equation

Dec 2019
3
0
Ukraine, Sumy
\(\displaystyle
\tan(2\Delta\theta) = \frac{D_1\sin(2\theta_2)-D_2\sin(2\theta_1)}{D_2\cos(2\theta_1)-D_1\cos(2\theta_2)}
\)

I want to have an expression for delta theta. I know i can just take arctan for both sides, but i want right side part to be more simple because it would be computed often. Thanks.
 
Dec 2015
975
128
Earth
\(\displaystyle \sin(2\theta ) = 2\sin(\theta) \cos(\theta) \) and \(\displaystyle \cos(2\theta ) =1-2\sin^2 ( \theta )\).
 
Dec 2019
3
0
Ukraine, Sumy
\(\displaystyle \sin(2\theta ) = 2\sin(\theta) \cos(\theta) \) and \(\displaystyle \cos(2\theta ) =1-2\sin^2 ( \theta )\).
I guess expanding sin and cos will only make expression more complicated.
 

romsek

Math Team
Sep 2015
2,891
1,615
USA
how big do you expect these angles to be? If there are very small $\sin(x) \approx x,~\cos(x)\approx 1$