Find perpendicular distance from the origin to the line joining the points (cosθ, sinθ) and (cosϕ, sinϕ)
The equation of the line joining the points (cosθ,sinθ) and (cosϕ,sinϕ) is given by,
y−sinθ=sinϕ−sinθcosϕ−cosθ(x−cosθ)
x(sinϕ−sinθ)+y(cosϕ−cosθ)+cosθsinϕ−cosθsinθ−sinθcosϕ+sinθcosθ=0
x(sinθ−sinϕ )+y(cosϕ−cosθ)+sin(ϕ−θ)=0
Therefore, the perpendicular distance (d) of the given line from point (0,0) is
d=|(0)(sinθ−sinϕ)+(0)(cosϕ−cosθ)+sin(ϕ−θ)|√(sinθ−sinϕ)2+(cosϕ−cosθ)2
=|sin(ϕ−θ)|√sin2θ+sin2ϕ−2sinθsinϕ+cos2ϕ+cos2θ−2cosϕcosθ
=|sin(ϕ−θ)|√(sin2θ+cos2θ)+(sin2ϕ+cos2ϕ)−2(sinθsinϕ+cosθcosϕ)
=|sin(ϕ−θ)|√2(1−cos(ϕ−θ))
=|sin(ϕ−θ)|√2{2sin2{ϕ−θ2}}
=|sin(ϕ−θ)|∣∣2sin{ϕ−θ2}∣∣