If tan θ = p/q, find the value of (p sin θ – q cos θ)/ (p sin θ + q cos θ).
Answer :
tan θ = p/q
Consider ∆ABC be right angled at B and ∠BCA = θ
tan θ = BC/AB = p/q
BC = px then AB = qx
AC2 = BC2 + AB2
AC2 = (px)2 + (qx)2
⇒ AC2 = p2x2 + q2x2
⇒ AC2 = x2 (p2 + q2)
AC = √x2 (p2 + q2)
⇒ AC = x(√p2 + q2)
In right angled ∆ABC
sin θ = perpendicular/hypotenuse
⇒ sin θ = BC/AC
sin θ = px/x(√p2 + q2)
sin θ = p/(√p2 + q2)
In right angled ∆ABC
cos θ = base/hypotenuse
⇒ cos θ = AB/AC
cos θ = qx/x(√p2 + q2)
cos θ = q/(√p2 + q2)
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