Naming angles by letters

How do you name an angle? Why does it even need a name? In most cases, you want to be able to distinguish a particular angle from all the others in a picture. When you look at a photo in a newspaper, you want to know the names of the different people and be able to point them out. With angles, you should feel the same way.

You can name an angle in one of three different ways:

Triangulating your position

All on their own, angles are certainly very exciting. But put them into a triangle, and you've got icing on the cake. Triangles are one of the most frequently studied geometric figures. The angles that make up the triangle give them many of their characteristics.

Angles in triangles

A triangle always has three angles. The angles in a triangle have measures that always add up to 180 degrees-no more, no less. A triangle named ABC has angles A, B, and ITLITL, and you can name the sides AB, BC, and AC, depending on which two angles the side is between. The angles themselves can be acute, obtuse, or right. If the triangle has either an obtuse or right angle, then the other two angles have to be acute.

Naming triangles by their shape

Triangles have special names based on their angles and sides. They can also have more than one name-a triangle can be both acute and isosceles, for example. Here are their descriptions, and check out Figure 1-6 for the pictures:

Circling the wagons

A circle is a geometric figure that needs only two parts to identify it and classify it by size: its center (or middle) and its radius (the distance from the center to any point on the circle). Technically, the center isn't a part of the circle, it's just a sort of anchor or reference point. The circle consists only of all those points that are the same distance from the center.

Radius, diameter, circumference, and area

After you've chosen a point to be the center of a circle and know how far that point is from all the points that lie on the circle, you can draw a fairly decent picture. With the measure of the radius, you can tell a lot about the circle: its diameter (the distance from one side to the other, passing through the center), its circumference (how far around it is), and its area (how many square inches, feet, yards, meters-what have you-fit into it). Figure 1-7 shows these features.

Ancient mathematicians figured out that the circumference of a circle is always a little more than three times the diameter of a circle. Since then, they narrowed that "little more than three times" to a value called pi (pronounced "pie"), designated by the Greek letter [pi]. The decimal value of pi isn't exact- it goes on forever and ever, but most of the time, people refer to it as being approximately 3.14 or 22/7, whichever form works best in specific computations.

The formula for figuring out the circumference of a circle is tied to [pi] and the diameter.

TRIG RULES

Circumference of a circle: ITLITL = [pi]d = 2[pi]r

The d represents the measure of the diameter, and r represents the measure of the radius. The diameter is always twice the radius, so either form of the equation works.

Similarly, the formula for the area of a circle is tied to [pi] and the radius.

TRIG RULES

Area of a circle: A = [pi][r.sup.2]

This formula reads, "Area equals pi are squared." And all this time, I thought that pies are round.

Example: Find the radius, circumference, and area of a circle if its diameter is equal to 10 feet in length.

If the diameter (d) is equal to 10, you write this value as d = 10. The radius is half the diameter, so the radius is 5 feet, or r = 5. You can find the circumference by using the formula ITLITL = [pi]d = [pi] x 10 [approximately equal to] 3.14 x 10 = 31.4. So the circumference is about 31 1/2 feet around. You find the area by using the formula A = [pi][r.sup.2] = [pi] x [5.sup.2] = [pi] x 25 [approximately equal to] 3.14 x 25 = 78.50, so the area is about 78 1/2 square feet.

Chord versus tangent

You show the diameter and radius of a circle by drawing segments from a point on the circle either to or through the center of the circle. But two other straight figures have a place on a circle. One of these figures is called a chord, and the other figure is a tangent.

Chords of a circle

A chord of a circle is a segment that you draw from one point on the circle to another point on the circle (see Figure 1-8). This segment always stays inside the circle. The largest chord possible is the diameter-you can't get any longer than that segment.

Tangents to a circle

A tangent to a circle is a line, ray, or segment that touches the outside of the circle in exactly one point, as in Figure 1-9. It never crosses into the circle. A tangent can't be a chord, because a chord touches a circle in two points, crossing through the inside of the circle.

Sectioning sectors

A sector of a circle is a section of the circle between two radii (plural for radius). You can consider this part like a piece of pie cut from a circular pie plate (see Figure 1-10).

You can find the area of a sector of a circle if you know the angle between the two radii. A circle has a total of 360 degrees all the way around the center, so if a sector has an angle measure of 60 degrees between the two radii, the sector takes up 60/360, or 1/6, of the degrees all the way around. In that case, the sector has 1/6 the area of the whole circle.

Example: Find the area of a sector of a circle if the angle between the two radii forming the sector is 80 degrees and the radius of the circle is 9 inches.

1. Find the area of the circle.

The area of the whole circle is A = [pi][r.sup.2] = [pi] x [(4.5).sup.2] [approximately equal to] 3.14 x 20.25 = 63.585, or about 63 1/2 square inches.

2. Find the portion of the circle that the sector represents.

The sector takes up only 80 degrees of the circle. Divide 80 by 360 to get 80/360 = 2/9 [approximately equal to] 0.222

3. Calculate the area of the sector.

Multiply the fraction or decimal from Step 2 by the total area to get the area of the sector: 0.222 x 63.585 [approximately equal to] 14.116. The whole circle has an area of almost 64 square inches, but the sector has an area of just over 14 square inches.

Understanding Trig Speak

Any math or science topic has its own unique vocabulary. Some very nice everyday words have special meanings when used in the context of that subject. Trigonometry is no exception.

Defining trig functions

Every triangle has six parts: three sides and three angles. If you measure the sides and then pair up those measurements (taking two at a time), you have three different pairings. Do division problems with the pairings-changing the order in each pair-and you have six different answers. These six different answers represent the six trig functions. For example, if your triangle has sides measuring 3, 4, and 5, then the six divisions are 3/4, 4/3, 3/5, 5/3, 4/5, and 5/4. In Chapter 7, you find out how all these fractions work in the world of trig functions by using the different sides of a right triangle. And then, in Chapter 8, you take a whole different approach as you discover how to define the trig functions with a circle.

The six trig functions are named sine, cosine, tangent, cotangent, secant, and cosecant. Many people confuse the spoken word sine with sign-you can't really tell the difference when you hear it unless you're careful with the context. You can "go off on a tangent" in some personal dealings, but that phrase has a whole different meaning in trig. Cosigning a loan isn't what trig has in mind, either. The other three ratios are special to trig speak. You can't confuse them with anything else.

REMEMBER

Interpreting trig abbreviations

Even though the word sine isn't all that long, you have a three-letter abbreviation for this trig function and all the others. Mathematicians find using abbreviations easier, and those versions fit better on calculator keys. The functions and their abbreviations are

As you can see, the first three letters in the full name make up the abbreviations, except for cosecant's.

Noting notation

Angles are the main focus in trigonometry, and you often don't know their measure. Many angles and their angle measures have general rules that apply to them. You can name angles by one letter, three letters, or a number, but to do trig problems and computations, mathematicians commonly refer to the angle measures with Greek letters.

The most commonly used letters for angle measures are [alpha] (alpha), [beta] (beta), [gamma] (gamma), and [theta] (theta). Also, many equations use the variable x to represent an angle measure.

TECHNICAL STUFF

Algebra has conventional notation involving superscripts, such as the 2 in [chi square]. In trigonometry, superscripts have the same rules and characteristics as in other mathematics. But trig superscripts often look very different. Table 1-1 presents a listing of many of the ways that trig uses superscripts.

(Continues...)

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Excerpted from Trigonometry For Dummies by Mary Jane Sterling Excerpted by permission.
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