Q. 16 Give a smooth parametrization r(... [FREE SOLUTION]

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.

Expert verified

A parametrization r(t) for the unit circle $r (t) = \cos t i - \sin t j, 0 鈮� t 鈮� 2 蟺$ .

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

We wish to parameterize the unit circle.

The unit circle is defined by the equation $x^{2} + y^{2} = 1 .$

From elementary trigonometry we recall the identity $(c o s (t))^{2} + (s i n (t))^{2} = 1$ for all $[0, 2 蟺)$ .

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

$r (t) = \cos t i + \sin t j, 0 鈮� t 鈮� 2 蟺$

Since we have to travelling in the clockwise direction.

So $r (t) = \cos t i - \sin t j, 0 鈮� t 鈮� 2 蟺$

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