   Chapter 10.2, Problem 25E

Chapter
Section
Textbook Problem

# Show that the curve x = cos t, y = sin t cos t has two tangents at (0, 0) and find their equations. Sketch the curve.

To determine

To find: The curve for the parametric equations x=cost and y=sintcost , checking that the given parametric equation has two tangents.

Explanation

Given:

The parametric equation for the variable x is as follows.

x=cost

The parametric equation for the variable y is as follows.

y=sintcost

Calculation:

Differentiate the parametric equation for the variable x with respect to t .

dxdt=sint (1)

Differentiate the parametric equation for the variable y with respect to t .

dydt=sin2t+cos2t

dydt=cos2t (2)

The tangent line passes through coordinate point (0,0) at x=0 , y=0

x=0cost=0t=π2,3π2

Substitute π2 for t in equation (1).

dxdt=sin(t)dxdt=sin(π2)dxdt=1

Substitute π2 for t in equation (2).

dydt=cos2tdydt=cos2(π2)dydt=1

Substitute 3π2 for t in equation (1).

dxdt=sin(t)dxdt=sin(3π2)dxdt=1

Substitute 3π2 for t in equation (2).

dydt=cos2tdydt=cos2(3π2)dydt=1

The values of x and y for each step value of t is tabulated in the below table

 t 0 π2 3π2 5π2 x 0 -1 1 −1 y 0 −1 −1 −1

Write the chain rule for dydx .

dydx=dydtdxdt

Substitute (cos2t) for dydt and (sint) for dxdt in the above equation

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