A man standing directly opposite to one side of a road of width $x$ metre views a circular shaped traffic green signal of diameter $a$ metre on the other side of the road. The bottom of the green signal is $b$ metre height from the horizontal level of the viewer's eye. If $\alpha

Question

A man standing directly opposite to one side of a road of width $x$ metre views a circular shaped traffic green signal of diameter $a$ metre on the other side of the road. The bottom of the green signal is $b$ metre height from the horizontal level of the viewer's eye. If $\alpha$ denotes the angle subtended by the diameter of the green signal at the viewer's eye, then prove that

$$\alpha=	an^{-1}\left(\dfrac{a+b}xight)-	an^{-1}\left(\dfrac bxight).$$

Accepted Answer

> [!TLDR]
>
> Drop the two lines of sight from the viewer's eye to the bottom and to the top of the signal; each makes a right triangle with the horizontal distance $x$, and the subtended angle $\alpha$ is the difference of the two angles of elevation.

**Step 1. Set up the horizontal reference.** Let the viewer's eye be at $O$, at the same horizontal level as the base line drawn across the road to the signal's side. The horizontal distance from the viewer to the signal post is the road width $x$.

**Step 2. Heights of the bottom and top of the signal above eye level.** The bottom of the (circular) signal is at height $b$ above the eye's horizontal level. The signal has diameter $a$, so its top is at height $b+a$ above the same level.

**Step 3. Angle of elevation to the bottom.** In the right triangle formed by the horizontal ($x$), the vertical rise ($b$), and the line of sight to the bottom, $	an(	ext{angle to bottom})=\dfrac bx$, so the angle of elevation to the bottom is $	an^{-1}\left(\dfrac bxight)$. …

Function (restricted)	Domain $\to$ Range	Inverse	Domain $\to$ Range
$\sin x$	$\left[-\tfrac\pi2,\tfrac\pi2\right]\to[-1,1]$	$\sin^{-1}x$	$[-1,1]\to\left[-\tfrac\pi2,\tfrac\pi2\right]$
$\cos x$	$[0,\pi]\to[-1,1]$	$\cos^{-1}x$	$[-1,1]\to[0,\pi]$
$\tan x$	$\left(-\tfrac\pi2,\tfrac\pi2\right)\to(-\infty,\infty)$	$\tan^{-1}x$	$(-\infty,\infty)\to\left(-\tfrac\pi2,\tfrac\pi2\right)$
$\cot x$	$(0,\pi)\to(-\infty,\infty)$	$\cot^{-1}x$	$(-\infty,\infty)\to(0,\pi)$
$\csc x$	$\left[-\tfrac\pi2,\tfrac\pi2\right]-\{0\}\to\mathbb R-(-1,1)$	$\csc^{-1}x$	$\mathbb R-(-1,1)\to\left[-\tfrac\pi2,\tfrac\pi2\right]-\{0\}$
$\sec x$	$[0,\pi]-\left\{\tfrac\pi2\right\}\to\mathbb R-(-1,1)$	$\sec^{-1}x$	$\mathbb R-(-1,1)\to[0,\pi]-\left\{\tfrac\pi2\right\}$

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