Question

A man standing directly opposite to one side of a road of width x meter views a circular shaped traffic green signal of diameter ‘a’ meter on the other side of the road. The bottom of the green signal Is ‘b’ meter height from the horizontal level of viewer’s eye. If ‘a’ denotes the angle subtended by the diameter of the green signal at the viewer’s eye, then prove that α = `tan^-1 (("a" + "b")/x) - tan^-1 ("b"/x)`

Answer 1

Given Width of the Road = x meter

Diameter of the signal AB = a meter

Height of the signal from the eye level = b meter

In ∆ADC,

DC = x 

AC = AB + BC

= a + b

∠ADC = `phi`

tan Φ = `"AC"/"DC"`

tan Φ = `("a" + "b")/x`

Φ = `tan^-1 (("a" + "b")/x)`

In ∆BDC,

DC = x

BC = b

∠BDC = θ

tan θ = `"BC"/"DC"`

tan θ = `"b"/x`

⇒ θ = `tan^-1 ("b"/x)`

α = `phi - theta`

= `tan^-1 (("a" + "b")/x) - tan^-1 ("b"/x)`