Chapter 6.3, Problem 41E

Solution

a)

To find: an interval of values for $t$ so that the portion of the graph is in the first quadrant.

The required range is : $0\le t\le \frac{\pi }{2}$ .

Given information:

The parametric equations are:

  $x=2\cos t$

  $y=2\sin t$

Calculation:

The graph is as follows:

  

For the portion of the graph to be in the first quadrant both $x$ and $y$ are positive. Both the values are positive when $t$ lies between $0$ and $\frac{\pi }{2}$ .

Hence, the required range is : $0\le t\le \frac{\pi }{2}$ .

b)

To find: an interval of values for $t$ so that the portion of the graph is above the $x$ -axis.

The required range is : $0\le t\le \pi$

Given information:

The parametric equations are:

  $x=2\cos t$

  $y=2\sin t$

Calculation:

The graph is as follows:

  

For the portion of the graph to be above the $x$ -axis, $t$ lies between $0$ and $\pi$ .

Hence the required range is : $0\le t\le \pi$

c)

To find: an interval of values for $t$ so that the portion of the graph is to the left of the $y$ -axis.

The required range is : $\frac{\pi }{2}\le t\le \frac{3\pi }{2}$ .

Given information:

The parametric equations are:

  $x=2\cos t$

  $y=2\sin t$

Calculation:

The graph is as follows:

  

For the portion of the graph to be above the $x$ -axis, $t$ lies between $\frac{\pi }{2}$ and $\frac{3\pi }{2}$ .

Hence, the required range is : $\frac{\pi }{2}\le t\le \frac{3\pi }{2}$ .