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Question
Find `"dy"/"dx"` if : x = a cos3θ, y = a sin3θ at θ = `pi/(3)`
Solution
a cos3θ, y = a sin3θ
Differentiating x and y w.r.t. θ, we get
`"dx"/"dθ" = a"d"/"dθ"(cosθ)^3`
= `a xx 3cos^2θ."d"/"dθ"(cosθ)`
= 3a cos2θ(– sinθ)
= – 3a cos2θ sinθ
and
`"dy"/"dθ" = a"d"/"dθ"(sinθ)^3`
= `a xx 3 sin^2θ."d"/"dθ"(sinθ)`
= 3a sin2θ cosθ
∴`"dy"/"dx" = (("dy"/"dθ"))/(("dx"/"dθ")`
= `(3a sin^2θ cosθ)/(-3a cos^2θ sinθ)`
= – tanθ
∴ `(dx/dy)_("at" θ - pi/3)`
= `-tan pi/(3)`
= `-sqrt(3)`
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