Chapter 9: Q. 35 (page 721)

Finish Example 1 (b) by showing that the graph of the parametric equations $x = \cos t, y = \sin t, t \in [π, 2 π]$ is the bottom half of the unit circle centered at the origin with a counterclockwise direction of motion.

Short Answer

Expert verified

The required graph is

Step by step solution

Given Information

The parametric equations , $x = \cos t, y = \sin t, t \in [π, 2 p i]$ .

Calculation

Consider the parametric equations $, x = \cos t, y = \sin t, t \in [π, 2 π]$ .

The objective is to sketch the parametric curve by eliminating the parameter.

Take the equation: $x = \cos t$ .

Squaring the equation on both sides.

$x^{2} = \cos^{2} t$

Now take the equation $y = \sin t$ .

$y^{2} = \sin^{2} t$

Add the equations $x^{2} = \cos^{2} t$ and $y^{2} = \sin^{2} t$ .

Then, $x^{2} + y^{2} = \cos^{2} t + \sin^{2} t$

$x^{2} + y^{2} = 1 [since \cos^{2} t + \sin^{2} t = 1]$

The equation after eliminating the parameter t is $x^{2} + y^{2} = 1$ .

Now when $t = π$ ,

When $t = 2 π$

(x, y) = (\cos t, \sin t) (x, y) = (\cos π, \sin π) (x, y) = (- 1, 0)

Now when $t = \frac{3 π}{2}$ ,

(x, y) = (\cos t, \sin t) (x, y) = (\cos \frac{3 π}{2}, \sin \frac{3 π}{2}) (x, y) = (0, - 1)

Further calculation

When $t = 2 π$

$(x, y) = (\cos t, \sin t) (x, y) = (\cos 2 π, \sin 2 π) (x, y) = (1, 0)$

The graphical representation by using the points $(- 1, 0) (0, - 1) (1, 0)$ is as follows,

The equation after elimination of the parameter is $x^{2} + y^{2} = 1$ .

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Finish Example 1 (b) by showing that the graph of the parametric equations $x = \cos t, y = \sin t, t \in [π, 2 π]$ is the bottom half of the unit circle centered at the origin with a counterclockwise direction of motion.

Short Answer

Step by step solution

Given Information

Calculation

Further calculation

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Finish Example 1 (b) by showing that the graph of the parametric equations x=cost,y=sint,t∈[π,2π] is the bottom half of the unit circle centered at the origin with a counterclockwise direction of motion.

Short Answer

Step by step solution

Given Information

Calculation

Further calculation

One App. One Place for Learning.

Most popular questions from this chapter

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Finish Example 1 (b) by showing that the graph of the parametric equations $x = \cos t, y = \sin t, t \in [π, 2 π]$ is the bottom half of the unit circle centered at the origin with a counterclockwise direction of motion.