   Chapter 10.2, Problem 5E

Chapter
Section
Textbook Problem

Find an equation of the tangent to the curve at the point corresponding to the given value of the parameter.5. x = t cos t, y = t sin t; t = π

To determine

To find: The equation of the tangent for the parametric equations x=tcost and y=tsint .

Explanation

Given:

The parametric equation for the variable x is as follows.

x=tcost (1)

The parametric equation for the variable y is as follows.

y=tsint (2)

The value of the given parameter t is π .

Calculation:

Apply differentiation formula (ddx(uv)=udvdx+vdudx) and differentiate the parametric equation (1) for x with respect to t .

x=tcostdxdt=t(sint)+cost=tsint+cost

Apply differentiation formula (ddx(uv)=udvdx+vdudx) and differentiate the parametric equation (2) for y with respect to t .

y=tsintdydt=tcost+sint

Write the chain rule to expand the term dydx .

dydx=dydtdxdt

Substitute (tcost+sint) for dydt and (tsint+cost) for dxdt in the above equation.

dydx=(tcost+sint)(tsint+cost)=tcost+sinttsint+cost (3)

Write the equation for tangent

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