/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 127 Graph: \(x^{2}+y^{2}=1 .\) Then ... [FREE SOLUTION] | 91影视

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Chapter 4: Problem 127

Graph: \(x^{2}+y^{2}=1 .\) Then locate the point \(\left(-\frac{1}{2}, \frac{\sqrt{3}}{2}\right)\) on the graph.

Short Answer

Expert verified
The graph of the equation \(x^{2}+y^{2}=1\) is a circle with a radius 1 centered at the origin. The point \(\left(-\frac{1}{2}, \frac{\sqrt{3}}{2}\right)\) is located on the circle at a counterclockwise angle of \(\frac{2\pi}{3}\) radians from the positive x-axis.

Step by step solution

01

Draw the graph of \(x^{2}+y^{2}=1\)

This equation represents a circle centered at the point (0, 0) with a radius of 1. On a two dimensional graph, draw a circle with the origin as the center and a radius of 1 unit.
02

Convert point to polar coordinates

The point \(\left(-\frac{1}{2}, \frac{\sqrt{3}}{2}\right)\) can be converted to polar coordinates. Any two dimensional point \((x,y)\) can be converted to polar coordinates \((r, 胃)\) using the formulas: \(r = \sqrt{x^2 + y^2}\) and \(胃 = atan2(y,x)\). Using these formulas, we determine that the point in polar coordinates is \((1, \frac{2\pi}{3})\). This means this point is located on the unit circle at an angle of \(\frac{2\pi}{3}\) radians.
03

Plot the point on the circle

Now plot the point on the circle at a counterclockwise angle of \(\frac{2\pi}{3}\) radians from the positive x-axis. This will be the location of the point \(\left(-\frac{1}{2}, \frac{\sqrt{3}}{2}\right)\) on the graph of the equation \(x^{2}+y^{2}=1\).

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