If x=a(θ+sinθ) and y=a(1-cosθ) then dy/dx equal to ?
Solution:
Given,
x= a(θ + sin θ)
y= a(1 - cos θ)
Step 1: Differentiate x and y with respect to theta
\(\frac{dx}{dθ}\)= a(1 +cos θ) ...(i)
\(\frac{dy}{dθ}\)= a sin θ ...(ii)
Step 2: Divide dy by dx.
\(\frac{dy}{dx}\)=\( \frac{dy/dθ}{dx/dθ}\)
\(=\frac{a sinθ}{a(1 + cos θ)}\)
Step 3: Use formula sin 2θ= 2sinθ cosθ and 1+ cos2θ= 2cos²θ.
$$=\frac{2a sin θ (\frac{θ}{2}).cos(\frac{θ}{2})}{a \times 2cos²(\frac{θ}{2})}$$
= tan(θ/2)
Hence,
If x=a(θ+sinθ) and y=a(1-cosθ) then dy/dx equals to tan(θ/2).
