Question

Find y₂ for the following function:

x = a cosθ, y = a sinθ

Answer 1

x = a cos θ, y = a sin θ

`"dx"/("d"θ)`= a(-sinθ) = -a sinθ …….. (i)

`"dy"/("d"θ)` = a(cosθ)

`therefore y_1 = "dy"/"dx" = ("dy"/("d"θ))/("dx"/("d"θ)) = ("a" cos theta)/(- "a" sin theta)`

`y_1 = "dy"/"dx"` = - cot θ

`y_2 = ("d"^2y)/"dx"^2 = - (- "cosec"^2 theta) ("d"theta)/"dx"`

`= "cosec"^2theta ("d"theta)/"dx"`

`= "cosec"^2theta 1/("dx"/("d"theta))`

`=> "cosec"^2 theta xx ("cosec" theta)/(- "a")`

`= (- 1)/"a" "cosec"^3 theta`