Find dxdy, if x=a(θ+sinθ),y=a(1−cosθ).
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Step-by-Step Solution
Step 1: Differentiate x with respect to θ
We are given x as a function of θ. To find dθdx, we differentiate the expression for x with respect to θ. The constant a can be factored out. The derivative of θ with respect to θ is 1, and the derivative of sinθ with respect to θ is cosθ.
Step 2: Calculate dx/dθ
Applying the differentiation rules, we get the derivative of x with respect to θ as a(1+cosθ).
Step 3: Differentiate y with respect to θ
Similarly, we differentiate the expression for y with respect to θ. The constant a is factored out. The derivative of 1 is 0, and the derivative of −cosθ is −(−sinθ)=sinθ.
Step 4: Calculate dy/dθ
Applying the differentiation rules, we get the derivative of y with respect to θ as asinθ.
Step 5: Apply the Chain Rule
To find dxdy, we use the chain rule for parametric equations, which states that dxdy=dθdxdθdy. We substitute the expressions we found for dθdy and dθdx.
Step 6: Substitute and Simplify
Substitute the calculated derivatives into the chain rule formula. The constant a cancels out. We can further simplify this expression using trigonometric identities: sinθ=2sin(2θ)cos(2θ) and 1+cosθ=2cos2(2θ).
Step 7: Apply Trigonometric Identities
Using the half-angle identities, sinθ=2sin(2θ)cos(2θ) and 1+cosθ=2cos2(2θ), we substitute these into the expression. The 2 and one cos(2θ) term cancel out, leaving cos(2θ)sin(2θ), which simplifies to tan(2θ).