Trigonometry and More about Circles Help
Introduction to Trigonometry and More abut Circles
Trigonometry (or “trig”) involves angles and distances. Trigonometry scares some people because of the Greek symbology, but the rules are clear-cut. Once you can get used to the idea of Greek letters representing angles, and if you’re willing to draw diagrams and use a calculator, basic trigonometry loses most of its fear-inspiring qualities. You might even find yourself having fun with it.
More about Circles
Circles are defined by equations in which either x or y can be considered the dependent variable. We’ve already looked at the graphs of some simple circles in the xy -plane. Let’s look more closely at this special species of geometric figure.
General Equation of a Circle
The equation in the xy -plane that represents a circle depends on two things: the radius of the circle, and 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 is:
( 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 1 unit, and it is centered at the origin (0,0). This circle gives us a good way to define the common trigonometric functions, which are sometimes called circular functions .
It's Greek To Us
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 italic, lowercase Greek theta (pronounced “THAY-tuh”). It looks like an italic numeral 0 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 italic, lowercase letter phi (pronounced “FIE” or “FEE”). This character looks like an italic lowercase English letter o with a forward slash through it ( Φ ).
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 text in which angles are denoted θ 1 , θ 2 , θ 3 , and so on.
There are two main units by which the measures of angles in a flat plane can be specified: the radian and the degree. The radian was defined in the previous chapter. A quarter circle is π/2 rad, a half circle is π rad, and three quarters of a circle is 3π/2 rad. Mathematicians prefer to use the radian when working with trigonometric functions, and the abbreviation “rad” is left out. So if you see something like θ 1 = π/4, you know the angle θ 1 is expressed in radians.
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 three quarters of a 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 is read as “30degrees, 15minutes, 0seconds.”
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.
Trigonometry—More about Circles 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.
Find practice problems and solutions for these concepts at: Trigonometry Practice Test.
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