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    Trigonometry For Dummies
    Trigonometry For Dummies
    Trigonometry For Dummies
    Ebook696 pages6 hours

    Trigonometry For Dummies

    By Mary Jane Sterling

    ()

    Read preview

    • Trigonometry

    • Mathematics

    • Angles

    • Functions

    • Identities

    • Power of Knowledge

    • Journey of Learning

    • Trigonometric Functions

    • Right Triangles

    • Graphing

    • Pythagorean Identities

    • Half-Angle Identities

    About this ebook

    A plain-English guide to the basics of trig

    Trigonometry deals with the relationship between the sides and angles of triangles... mostly right triangles. In practical use, trigonometry is a friend to astronomers who use triangulation to measure the distance between stars. Trig also has applications in fields as broad as financial analysis, music theory, biology, medical imaging, cryptology, game development, and seismology.

    From sines and cosines to logarithms, conic sections, and polynomials, this friendly guide takes the torture out of trigonometry, explaining basic concepts in plain English and offering lots of easy-to-grasp example problems. It also explains the "why" of trigonometry, using real-world examples that illustrate the value of trigonometry in a variety of careers.

    • Tracks to a typical Trigonometry course at the high school or college level
    • Packed with example trig problems
    • From the author of Trigonometry Workbook For Dummies

    Trigonometry For Dummies is for any student who needs an introduction to, or better understanding of, high-school to college-level trigonometry.

    LanguageEnglish
    PublisherWiley
    Release dateFeb 6, 2014
    ISBN9781118827574
    Trigonometry For Dummies

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      Book preview

      Trigonometry For Dummies - Mary Jane Sterling

      Getting Started with Trigonometry

      9781118827413-pp0101.tif

      webextras.eps For Dummies can help you get started with lots of subjects. Visit www.dummies.com to learn more and do more with For Dummies.

      In this part…

      Become acquainted with angle measures and how they relate to trig functions.

      Discover formulas that provide lengths of segments, midpoints, and slopes of lines.

      Become familiar with circles and the relationships between radii, diameters, centers, and arcs.

      Relate infinitely many angle measures to just one reference angle.

      Find a simple conversion method for changing from degrees to radians and vice versa.

      Observe the properties of special right triangles, and use the Pythagorean theorem to formulate the relationships between the sides of these right triangles.

      Chapter 1

      Trouncing Trig Technicalities

      In This Chapter

      arrow Understanding what trigonometry is

      arrow Speaking the language by defining the words

      arrow Writing trig functions as equations

      arrow Graphing for understanding

      How did Columbus find his way across the Atlantic Ocean? How did the Egyptians build the pyramids? How did early astronomers measure the distance to the moon? No, Columbus didn't follow a yellow brick road. No, the Egyptians didn't have LEGO instructions. And, no, there isn't a tape measure long enough to get to the moon. The common answer to all these questions is trigonometry.

      Trigonometry is the study of angles and triangles and the wonderful things about them and that you can do with them. For centuries, humans have been able to measure distances that they can't reach because of the power of this mathematical subject.

      Taking Trig for a Ride: What Trig Is

      What's your angle? That question isn't a come-on such as What's your astrological sign? In trigonometry, you measure angles in both degrees and radians. You can shove the angles into triangles and circles and make them do special things. Actually, angles drive trigonometry. Sure, you have to consider algebra and other math to make it all work. But you can't have trigonometry without angles. Put an angle into a trig function, and out pops a special, unique number. What do you do with that number? Read on, because that's what trig is all about.

      Sizing up the basic figures

      Segments, rays, and lines are some of the basic forms found in geometry, and they're almost as important in trigonometry. As I explain in the following sections, you use those segments, rays, and lines to form angles.

      Drawing segments, rays, and lines

      A segment is a straight figure drawn between two endpoints. You usually name it by its endpoints, which you indicate by capital letters. Sometimes, a single letter names a segment. For example, in a triangle, a lowercase letter may refer to a segment opposite the angle labeled with the corresponding uppercase letter.

      A ray is another straight figure that has an endpoint on one end, and then it just keeps going forever in some specified direction. You name rays by their endpoint first and then by any other point that lies on the ray.

      A line is a straight figure that goes forever and ever in either direction. You only need two points to determine a particular line — and only one line can go through both of those points. You can name a line by any two points that lie on it.

      Figure 1-1 shows a segment, ray, and line and the different ways you can name them using points.

      9781118827413-fg0101.tif

      Figure 1-1: Segment AB, ray CD, and line EF.

      Intersecting lines

      When two lines intersect — if they do intersect — they can only do so at one point. They can't double back and cross one another again. And some curious things happen when two lines intersect. The angles that form between those two lines are related to one another. Any two angles that are next to one another and share a side are called adjacent angles. In Figure 1-2, you see several sets of intersecting lines and marked angles. The top two figures indicate two pairs of adjacent angles. Can you spot the other two pairs? The angles that are opposite one another when two lines intersect also have a special name. Mathematicians call these angles vertical angles. They don't have a side in common. You can find two pairs of vertical angles in Figure 1-2, the two middle figures indicate the only pairs of vertical angles. Vertical angles are always equal in measure.

      9781118827413-fg0102.tif

      Figure 1-2: Intersecting lines form adjacent, vertical, and supplementary angles.

      Why are these different angles so special? They're different because of how they interact with one another. The adjacent angles here are called supplementary angles. The sides that they don't share form a straight line, which has a measure of 180 degrees. The bottom two figures show supplementary angles. Note that these are also adjacent.

      Angling for position

      When two lines, segments, or rays touch or cross one another, they form an angle or angles. In the case of two intersecting lines, the result is four different angles. When two segments intersect, they can form one, two, or four angles; the same goes for two rays.

      These examples are just some of the ways that you can form angles. Geometry, for example, describes an angle as being created when two rays have a common endpoint. In practical terms, you can form an angle in many ways, from many figures. The business with the two rays means that you can extend the two sides of an angle out farther to help with measurements, calculations, and practical problems.

      Describing the parts of an angle is pretty standard. The place where the lines, segments, or rays cross is called the vertex of the angle. From the vertex, two sides extend.

      Naming angles by size

      You can name or categorize angles based on their size or measurement in degrees (see Figure 1-3):

      Acute: An angle with a positive measure less than 90 degrees

      Obtuse: An angle measuring more than 90 degrees but less than 180 degrees

      Right: An angle measuring exactly 90 degrees

      Straight: An angle measuring exactly 180 degrees (a straight line)

      Oblique: An angle measuring more than 180 degrees

      9781118827413-fg0103.tif

      Figure 1-3: Types of angles — acute, obtuse, right, straight, and oblique.

      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:

      By its vertex alone: Often, you name an angle by its vertex alone because such a label is efficient, neat, and easy to read. In Figure 1-4, you can call the angle A.

      By a point on one side, followed by the vertex, and then a point on the other side: For example, you can call the angle in Figure 1-4 angle BAC or angle CAB. This naming method is helpful if someone may be confused as to which angle you're referring to in a picture. Remember: Make sure you always name the vertex in the middle.

      By a letter or number written inside the angle: Usually, that letter is Greek; in Figure 1-4, however, the angle has the letter w. Often, you use a number for simplicity if you're not into Greek letters or if you're going to compare different angles later.

      9781118827413-fg0104.tif

      Figure 1-4: Naming an angle.

      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 C, and you can name the sides 9781118827413-eq01001.tif , 9781118827413-eq01002.tif , and 9781118827413-eq01003.tif , 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 can 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 1-5 for the pictures:

      Acute triangle: A triangle where all three angles are acute.

      Right triangle: A triangle with a right angle (the other two angles must be acute).

      Obtuse triangle: A triangle with an obtuse angle (the other two angles must be acute).

      Isosceles triangle: A triangle with two equal sides; the angles opposite those sides are equal, too.

      Equilateral triangle: A triangle where all three side lengths are equal; all the angles measure 60 degrees, too.

      Scalene triangle: A triangle with no angles or sides of the same measure.

      9781118827413-fg0105.tif

      Figure 1-5: Triangles can have more than one name, based on their characteristics.

      Circling the wagons

      A circle is a geometric figure that needs only two parts to identify it and classify it: 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 1-6 shows these features.

      9781118827413-fg0106.tif

      Figure 1-6: The different features of a circle.

      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 π. The decimal value of π isn't exact — it goes on forever and ever, but most of the time, people refer to it as being approximately 3.14 or 9781118827413-eq01004.tif , whichever form works best in specific computations.

      The formula for figuring out the circumference of a circle is tied to π and the diameter:

      tricrules.eps Circumference of a circle: C = πd = 2πr

      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 π and the radius:

      tricrules.eps Area of a circle: A = πr²

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

      Don't give me that jiva

      The ancient Greek mathematician Ptolemy was born some time at the end of the first century. Ptolemy based his version of trigonometry on the relationships between the chords of circles and the corresponding central angles of those chords. Ptolemy came up with a theorem involving four-sided figures that you can construct with the chords. In the meantime, mathematicians in India decided to use the measure of half a chord and half the angle to try to figure out these relationships. Drawing a radius from the center of a circle through the middle of a chord (halving it) forms a right angle, which is important in the definitions of the trig functions. These half-measures were the beginning of the sine function in trigonometry. In fact, the word sine actually comes from the Hindu name jiva.

      9781118827413-sb0101.tif

      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 C = πd = π · 10 ≈ 3.14 · 10 = 31.4. So, the circumference is about 9781118827413-eq01005.tif feet around. You find the area by using the formula A = πr² = π · 5² = π · 25 ≈ 3.14 · 25 ≈ 78.5, so the area is about 9781118827413-eq01006.tif 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 is a tangent:

      Chords: 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 1-7). A chord always stays inside the circle. The largest chord possible is the diameter — you can't get any longer than that segment.

      Tangent: 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 1-7. 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. Any radius drawn to a tangent is perpendicular to that tangent.

      9781118827413-fg0107.tif

      Figure 1-7: Chords and tangent of a circle.

      Angles in a circle

      There are several ways of drawing an angle in a circle, and each has a special way of computing the size of that angle. Four different types of angles are: central, inscribed, interior, and exterior. In Figure 1-8, you see examples of these different types of angles.

      Central angle

      A central angle has its vertex at the center of the circle, and the sides of the angle lie on two radii of the circle. The measure of the central angle is the same as the measure of the arc that the two sides cut out of the circle.

      Inscribed angle

      An inscribed angle has its vertex on the circle, and the sides of the angle lie on two chords of the circle. The measure of the inscribed angle is half that of the arc that the two sides cut out of the circle.

      Interior angle

      An interior angle has its vertex at the intersection of two lines that intersect inside a circle. The sides of the angle lie on the intersecting lines. The measure of an interior angle is the average of the measures of the two arcs that are cut out of the circle by those intersecting lines.

      Exterior angle

      An exterior angle has its vertex where two rays share an endpoint outside a circle. The sides of the angle are those two rays. The measure of an exterior angle is found by dividing the difference between the measures of the intercepted arcs by two.

      9781118827413-fg0108.tif

      Figure 1-8: Measuring angles in a circle

      Example: Find the measure of angle EXT, given that the exterior angle cuts off arcs of 20 degrees and 108 degrees (see Figure 1-9).

      Find the difference between the measures of the two intercepted arcs and divide by 2:

      .png

      The measure of angle EXT is 44 degrees.

      9781118827413-fg0109.tif

      Figure 1-9: Calculating the measure of an exterior angle.

      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).

      9781118827413-fg0110.tif

      Figure 1-10: A sector of a circle.

      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 that central angle determining a sector has an angle measure of 60 degrees, then the sector takes up 9781118827413-eq01008.tif , or 9781118827413-eq01009.tif , of the degrees all the way around. In that case, the sector has 9781118827413-eq01010.tif 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 diameter of the circle is 9 inches.

      Find the area of the circle.

      The area of the whole circle is A = πr² = π · (4.5)² ≈ 3.14(20.25) ≈ 63.585, or about 9781118827413-eq01011.tif square inches.

      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 9781118827413-eq01012.tif .

      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(63.585) ≈ 14.116. The whole circle has an area of almost 64 square inches, and 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 new and 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 9781118827413-eq01013.tif , 9781118827413-eq01014.tif , 9781118827413-eq01015.tif , 9781118827413-eq01016.tif , 9781118827413-eq01017.tif , and 9781118827413-eq01018.tif . 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.

      Interpreting trig abbreviations

      remember.eps 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 can work with them even if you 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 names and their measures with Greek letters.

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

      technicalstuff.eps Algebra has conventional notation involving superscripts, such as the 2 in x². 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.

      Table 1-1 How You Use Superscripts in Trig

      The first entry in Table 1-1 shows how you can save having to write parentheses every time you want to raise a trig function to a power. This notation is neat and efficient, but it can be confusing if you don't know the code. The second entry shows you how to write the reciprocal of a trig function. It means you should take the value of the function and divide it into the number 1. The last entry in Table 1-1 shows how you write the inverse sine function. Using the –1 superscript between sine and the angle means that you're talking about inverse sine (or arcsin), not the reciprocal of the function. In Chapter 15, I cover the inverse trig functions in great detail, making this business about the notation for an inverse trig function even more clear.

      Functioning with angles

      The functions in algebra use many operations and symbols that are different from the common add, subtract, multiply, and divide signs in arithmetic. For example, take a look at the square-root operation, 9781118827413-eq01020.tif . Putting 25 under the radical (square-root symbol) produces an answer of 5. Other operations in algebra, such as absolute value, factorial, and step-function, are used in trigonometry, too. But the world of trig expands the horizon, introducing even more exciting processes. When working with trig functions, you have a whole new set of values to learn or find. For instance, putting 25 into the sine function looks like this: sin 25. The answer that pops out is either 0.423 or –0.132, depending on whether you're using degrees or radians (for more on those two important trig concepts, head on over to Chapters 4 and 5). You can't usually determine or memorize all the values that you get by putting angle measures into trig functions. So, you need trig tables of values or scientific calculators to study trigonometry.

      In general, when you apply a trig function to an angle measure, you get some real number (if that angle is in its domain). Some angles and trig functions have nice values, but most don't. Table 1-2 shows the trig functions for a 30-degree angle.

      Table 1-2 The Trig Functions for a 30-Degree Angle

      Some characteristics that the entries in Table 1-2 confirm are that the sine and cosine functions always have values that are between and including –1 and 1. Also, the secant and cosecant functions always have values that are equal to or greater than 1 or equal to or less than –1. (I discuss these properties in more detail in Chapter 7.)

      Using the table in the Appendix, you can find more values of trig functions for particular angle measures (in degrees):

      tan 45° = 1

      csc 90° = 1

      sec 60° = 2

      remember.eps I chose these sample values so the answers look nice and whole. Most angles and most functions look much messier than these examples.

      Taming the radicals

      A radical is a mathematical symbol that means, Find the number that multiplies itself by itself one or more times to give you the number under the radical. You can see why you use a symbol such as 9781118827413-eq01026.tif rather than all those words. Radicals represent values of functions that are used a lot in trigonometry. In Chapter 7, I define the trig functions by using a right triangle. To solve for the lengths of a right triangle's sides by using the Pythagorean theorem, you have to compute some square roots, which use radicals. Some basic answers to radical expressions are 9781118827413-eq01027.tif , 9781118827413-eq01028.tif , 9781118827413-eq01029.tif , and 9781118827413-eq01030.tif .

      These examples are all perfect squares, perfect cubes, or perfect fourth roots, which means that the answer is a number that ends — the decimal doesn't go on forever. The following section discusses a way to simplify radicals that aren't perfect roots.

      Simplifying radical forms

      Simplifying a radical form means to rewrite it with a smaller number under the radical — if

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