Trigonometry and Angles Study Guide
Trigonometry and Angles
In this lesson, we examine the concept of an angle. There are two ways to measure the size of an angle: with degrees or with radians. We study both, along with the ways to convert from one to the other.
An angle is formed when two line segments (pieces of straight lines) meet at a point. In Figure 1.1, consists of and , which meet at point Y. Point Y is called the vertex of the angle.
The measure of an angle is not given by the size of the lines that come together, but by how much one of the lines would have to pivot around the vertex to line up with the other. In Figure 1.2, for example, would have to rotate much further around vertex B to be in line with than would have to rotate around E to line up with and . It is for this reason that the measure of is greater than that of .
There are two ways to describe a rotation that goes all the way around, like the one in Figure 1.3. The first way is to call this 360 degrees. The number 360 was chosen by the ancient Babylonians, who counted things in groups of 60. Modern people generally count things in groups of 10, although we still count minutes and seconds in groups of 60 because of Babylonian influences on our civilization.
In fact, one-sixtieth of one degree is called a minute, and one-sixtieth of a minute is called a second. Astronomers and navigators sometimes still measure angles in degrees, minutes, and seconds, although mathematicians generally use decimals.
A straight angle, made by rotating halfway around, has a measure of 180 degrees, written 180°. Half of this is called a right angle, measuring 90°. Because right angles form the corners of squares and rectangles, a 90° angle is often indicated by putting a small square in the angle, as shown in Figure 1.4.
The other way to measure a full rotation is to use the circumference of a circle (the distance around). The formula for the circumference (C) of a circle is C = 2π · r, where r is the radius of the circle. If we suppose the radius of the circle is one inch, one centimeter, or one of whatever units we use to measure distance, the circumference of the circle is C = 2π of those units, where π is the Greek symbol pi, pronounced "pie" Such a circle is called a unit circle (Figure 1.5). No one has ever been able to calculate the number π exactly. A decent approximation is π ≈ 3.14, although π is actually a tiny bit more than this.
The squiggle equals sign, ≈, means "is approximately equal to." Many of the numbers that occur in trigonometry are irrational, which means that they cannot be written out in a finite amount of space. Thus, √2 ≈ 1.414 means that when you multiply 1.414 by itself, you get a number reasonably close to 2. To get exactly 2, you would need an infinite number of decimal places, which is really asking too much! Sometimes close enough is good enough.
If we want to be perfectionists and use exact numbers, then we will have to use symbols to represent irrational numbers. Thus, √2 is the number that multiplies by itself to get exactly 2. The exact distance around a circle with a one-unit radius is 2π.
The radian measure of an angle is the distance it goes around the unit circle, or circle with a radius equal to one. For example, the radian measure of a straight angle is π, and the radian measure of a right angle , as shown in Figure 1.6. These measures don't roll off the tongue like degree measures do. However, because radian measure is a distance, measured in the units of length being used, it is sometimes essential for mathematical calculations. A degree is a measure of rotation, not distance.
How many degrees are in one-third of a right angle? How many radians is this? A right angle consists of 90°. One-third of this is thus
A right angle has a radian measure of .One-third of this is (see Figure 1.7).
Converting between Degrees and Radians
A full 360° rotation has a radian measure of 2π. Thus, we can write that as
360 degrees = 2π radians
If we divide both sides by 360°, we will get , which simplifies to . Because multiplying by 1 doesn't change anything, we can use this to convert an angle measured in degrees to one measured in radians.
How many radians is an angle of 150°? We multiply 150° by and get
Notice that when we divide degrees by degrees, these units cancel out altogether. If we want to convert from radian measure to degrees, we multiply by .
How many degrees are in an angle with radian measure ? We multiply
Practice problems for this study guide can be found at:
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