Circles in the Plane Help (page 2)
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:
( x – a ) 2 + ( y – b ) 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
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.
Degrees, Minutes, Seconds
The angular degree (°), also called the degree of arc, is the unit of angular measure most familiar to lay people. One degree (1°) is 1/360 of a full circle. An angle of 90° represents a quarter circle, 180° represents a half circle, 270° represents a three-quarter circle, and 360° represents a full circle. A right angle has a measure of 90°, an acute angle has a measure of more than 0° but less than 90°, and an obtuse angle has a measure of more than 90° but less than 180°.
To denote the measures of tiny angles, or to precisely denote the measures of angles in general, smaller units are used. One minute of arc or arc minute, symbolized by an apostrophe or accent (′) or abbreviated as m or min, is 1/60 of a degree. One second of arc or arc second, symbolized by a closing quotation mark (″) or abbreviated as s or sec, is 1/60 of an arc minute or 1/3600 of a degree. An example of an angle in this notation is 30° 15′ 0″, which denotes 30 degrees, 15 minutes, 0 seconds.
Alternatively, fractions of a degree can be denoted in decimal form. You might see, for example, 30.25°. This is the same as 30° 15′ 0″. Decimal fractions of degrees are easier to work with than the minute/second scheme when angles must be added and subtracted, or when using a conventional calculator to work out trigonometry problems. Nevertheless, the minute/second system, like the English system of measurements, remains in widespread use.
Circles in the Plane Practice Problems
A text discussion tells you that θ 1 = π/4. What is the measure of θ 1 in degrees?
There are 2π rad in a full circle of 360°. The value π/4 is equal to 1/8 of 2π. Therefore, the angle θ 1 is 1/8 of a full circle, or 45°.
Suppose your town is listed in an almanac as being at 40° 20′ north latitude and 93° 48′ west longitude. What are these values in decimal form? Express your answers to two decimal places.
There are 60 minutes of arc in one degree. To calculate the latitude, note that 20′ = (20/60)° = 0.33°; that means the latitude is 40.33° north. To calculate the longitude, note that 48′ = (48/60)° = 0.80°; that means the longitude is 93.80° west.
Practice Problems for these concepts can be found at: The Circle Model Practice Test
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