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Circles in the Plane Help

By — McGraw-Hill Professional
Updated on Aug 30, 2011

Introduction to Circles in the Plane

Circles are not technically functions as represented in the Cartesian coordinate system, but they are often encountered in mathematics and science. They are defined by equations in which either x or y can be considered the dependent variable.

Equation Of A Circle

The equation that represents a circle depends on the radius of the circle, and also on the location of its center point.

Suppose r is the radius of a circle, expressed in arbitrary units. Imagine that the center point of the circle in Cartesian coordinates is located at the point x = a and y = b, represented by the ordered pair (a,b). Then the equation of that circle looks like this:

( xa ) 2 + ( yb ) 2 = r 2

If the center of the circle happens to be at the origin, that is, at (0,0) on the coordinate plane, then the general equation is simpler:

x 2 + y 2 = r 2

The Unit Circle

Consider a circle in rectangular coordinates with the following equation:

x 2 + y 2 = 1

This is called the unit circle because its radius is one unit, and it is centered at the origin (0,0). This circle is significant, because it gives us a simple basis to define the common trigonometric functions, which are called circular functions. We’ll define these shortly.

It’s Greek To Us

In geometry, and especially in trigonometry, mathematicians and scientists have acquired the habit of using Greek letters to represent angles. The most common symbol for this purpose is an italicized, lowercase Greek theta (pronounced “THAY-tuh”). It looks like a numeral zero leaning to the right, with a horizontal line through it ( θ ).

When writing about two different angles, a second Greek letter is used along with θ. Most often, it is the italicized, lowercase letter phi (pronounced “fie” or “fee”). It looks like a lowercase English letter o leaning to the right, with a forward slash through it ( ø ). You should get used to these symbols, because if you have anything to do with engineering and science, you’re going to find them often.

Sometimes the italic, lowercase Greek alpha (“AL-fuh”), beta (“BAY-tuh”), and gamma (“GAM-uh”) are used to represent angles. These, respectively, look like this: α, β, γ. When things get messy and there are a lot of angles to talk about, numeric subscripts are sometimes used with Greek letters, so don’t be surprised if you see angles denoted θ 1 , θ 2 , θ 3 , and so on.

Measurements—Radians, Degrees, Minutes, and Seconds

Radians

Imagine two rays emanating outward from the center point of a circle. The rays each intersect the circle at a point. Call these points P and Q. Suppose the distance between P and Q, as measured along the arc of the circle, is equal to the radius of the circle. Then the measure of the angle between the rays is one radian (1 rad).

There are 2π rad in a full circle, where π (the lowercase, non-italic Greek letter pi, pronounced “pie”) stands for the ratio of a circle’s circumference to its diameter. The value of π is approximately 3.14159265359, often rounded off to 3.14159 or 3.14. A quarter circle is π /2 rad, a half circle is π rad, and a three-quarter circle is 3π/2 rad. Mathematicians generally prefer the radian when working with trigonometric functions, and the “rad” is left out. So if you see something like θ 1 = π/4, you know the angle θ 1 is expressed in radians.

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