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Chapter 14: Q. 16 (page 1106)

Give a smooth parametrization r(t) for the unit circle, starting and ending at (1, 0) and travelling in the clockwise direction.

Short Answer

Expert verified

A parametrization r(t) for the unit circle r(t)=costi-sintj,0≤t≤2π.

Step by step solution

01

Step 1. Given Information

Give a smooth parametrization r(t) for the unit circle, starting and ending at (1, 0) and travelling in the clockwise direction.

02

Step 2. A parametric curve is defined as a collection of points given by two continuous functions x(t) and y(t), that is, the points on the curve are the collection of points (x(t), y(t)) where x and y are continuous functions of t. 

We wish to parameterize the unit circle.

The unit circle is defined by the equationx2+y2=1.

From elementary trigonometry we recall the identity (cos(t))2+(sin(t))2=1for all [0,2π).

This directly gives us our first parametrization of the unit circle;

r(t)=costi+sintj,0≤t≤2π

Since we have to travelling in the clockwise direction.

Sor(t)=costi-sintj,0≤t≤2π

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