How these 2 approximation equations are gained of euclild distances?

Question

$$r\gg 0$$

$$r_{1}\approx r-\frac{d}{2}\cos(\theta)\tag{1}$$

$$r_{2}\approx r+\frac{d}{2}\cos(\theta)\tag{2}$$

How the above approximation equations are made?

Answer 1

Drop a perpendicular from $A$ to the line of length $r$. Call the point of intersection $S$. $OS = \frac{d}{2} \cos \theta$. The argument is that $AP$ and $SP$ are approximately equal in length, because $\phi = \angle AOS$ is very small. (Do you see why that follows?)

Similarly for $O$ and $B$.

Answer 2

Use law of cosines and $d\ll r$

$r_1^2=r^2+(d/2)^2-dr\cos(\theta)$

$r_2^2=r^2+(d/2)^2+dr\cos(\theta)$

Divide by $r^2$ and get: $(r_1/r)^2=1+(d/2r)^2-(d/r)\cos(\theta)$

Use $d\ll r$ and expand square root to first order to get $r_1/r\approx 1-(d/2r)\cos(\theta)$ or $r_1\approx r-(d/2)\cos(\theta)$

Similarly $r_2\approx r+(d/2)\cos(\theta)$