Trigonometry For Dummies

From the Inside Flap Learn to: Solve tricky trig equations Solve tricky trig equations Graph functions and figure out formulas Graph functions and figure out formulas Use trigonometry to solve practical problems Use trigonometry to solve practical problems Your guide to getting an angle on sines, cosines, and tangents Confused by cosines? Perplexed by polynomials? Don’t worry! This friendly guide takes the torture out of trigonometry by explaining everything in plain English, offering lots of easy-to-grasp examples, and adding a dash of humor and fun. You’ll see the ways trig applies to everyday life, and how it’s an important tool for a variety of careers. Start with basics — get up and running with an overview of trig functions, radians, and more Start with basics — get up and running with an overview of trig functions, radians, and more Learn tricky trig functions — take things to the next level with SOH-CAH-TOA and circular functions Learn tricky trig functions — take things to the next level with SOH-CAH-TOA and circular functions Identify identities — dig into reciprocal, ratio, opposite-angle, and Pythagorean identities Identify identities — dig into reciprocal, ratio, opposite-angle, and Pythagorean identities Make trig work — get practical with trig, find out how to use your calculator for complex solutions, and solve trig equations Make trig work — get practical with trig, find out how to use your calculator for complex solutions, and solve trig equations Graph functions — figure out the basics of graphing sine, cosine, tangent, cotangent, and more Graph functions — figure out the basics of graphing sine, cosine, tangent, cotangent, and more Open the book and find: Definitions of trigonometry functions Definitions of trigonometry functions How to get down with Pythagoras and his theorem How to get down with Pythagoras and his theorem Ways to use Cartesian coordinates Ways to use Cartesian coordinates The lowdown on the sine, cosine, and tangent functions The lowdown on the sine, cosine, and tangent functions How to go 'round and 'round with circular functions How to go 'round and 'round with circular functions The scoop on inverse trigonometry The scoop on inverse trigonometry Details on graphing trig functions Details on graphing trig functions Ten basic identities and ten not-so-basic identities Ten basic identities and ten not-so-basic identities Learn to: Solve tricky trig equations Solve tricky trig equations Graph functions and figure out formulas Graph functions and figure out formulas Use trigonometry to solve practical problems Use trigonometry to solve practical problems Your guide to getting an angle on sines, cosines, and tangents Confused by cosines? Perplexed by polynomials? Don’t worry! This friendly guide takes the torture out of trigonometry by explaining everything in plain English, offering lots of easy-to-grasp examples, and adding a dash of humor and fun. You’ll see the ways trig applies to everyday life, and how it’s an important tool for a variety of careers. Start with basics ― get up and running with an overview of trig functions, radians, and more Start with basics ― get up and running with an overview of trig functions, radians, and more Learn tricky trig functions ― take things to the next level with SOH-CAH-TOA and circular functions Learn tricky trig functions ― take things to the next level with SOH-CAH-TOA and circular functions Identify identities ― dig into reciprocal, ratio, opposite-angle, and Pythagorean identities Identify identities ― dig into reciprocal, ratio, opposite-angle, and Pythagorean identities Make trig work ― get practical with trig, find out how to use your calculator for complex solutions, and solve trig equations Make trig work ― get practical with trig, find out how to use your calculator for complex solutions, and solve trig equations Graph functions ― figure out the basics of graphing sine, cosine, tangent, cotangent, and more Graph functions ― figure out the basics of graphing sine, cosine, tangent, cotangent, and more Open the book and find: Definitions of trigonometry functions Definitions of trigonometry functions How to get down with Pythagoras and his theorem How to get down with Pythagoras and his theorem Ways to use Cartesian coordinates Ways to use Cartesian coordinates The lowdown on the sine, cosine, and tangent functions The lowdown on the sine, cosine, and tangent functions How to go 'round and 'round with circular functions How to go 'round and 'round with circular functions The scoop on inverse trigonometry The scoop on inverse trigonometry Details on graphing trig functions Details on graphing trig functions Ten basic identities and ten not-so-basic identities Ten basic identities and ten not-so-basic identities Mary Jane Sterling is the author of Algebra I For Dummies and many other For Dummies titles. She has been teaching mathematics at Bradley University in Peoria, Illinois, for more than 30 years and has loved working with future business executives, physical therapists, teachers, and many others. Excerpt. © Reprinted by permission. All rights reserved. Trigonometry For Dummies By Mary Jane Sterling John Wiley & Sons Copyright © 2014 John Wiley & Sons, LtdAll rights reserved.ISBN: 978-1-118-82741-3 CHAPTER 1 Trouncing Trig Technicalities In This Chapter * Understanding what trigonometry is * Speaking the language by defining the words * Writing trig functions as equations * 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. 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. 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): [check] Acute: An angle with a positive measure less than 90 degrees [check] Obtuse: An angle measuring more than 90 degrees but less than 180 degrees [check] Right: An angle measuring exactly 90 degrees [check] Straight: An angle measuring exactly 180 degrees (a straight line) [check] Oblique: An angle measuring more than 180 degrees 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: [check] 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 . [check] 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. [check] 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. 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 [bar. AB ], [bar. BC ], and [bar. 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 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: [check] Acute triangle: A triangle where all three angles are acute. [check] Right triangle: A triangle with a right angle (the other two angles must be acute). [check] Obtuse triangle: A triangle with an obtuse angle (the other two angles must be acute). [check] Isosceles triangle: A triangle with two equal sides; the angles opposite those sides are equal, too. [check] Equilateral triangle: A triangle where all three side lengths are equal; all the angles measure 60 degrees, too. [check] Scalene triangle: A triangle with no angles or sides of the same measure. 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. 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 ITLpITL (pronounced "pie"), designated by the Greek letter p. The decimal value of p 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 p and the diameter: Circumference of a circle: C = p d = 2p 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 p and the radius: Area of a circle: A = p r 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 C = p d = p · 10 ˜ 3.14 10 = 31.4. So, the circumference is about 31 1/2 feet around. You find the area by using the formula A = p r 2 = p · 52 = p · 25 ˜ 3.14 · 25 ˜ 78.5, 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 is a tangent: [check] 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. [check] 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. 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. 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: 108 - 20/2 = 88/2 = 44 The measure of angle EXT is 44 degrees. 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 that central angle determining a sector has an angle measure of 60 degrees, then 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 diameter of the circle is 9 inches. 1. Find the area of the circle. The area of the whole circle is A = p r 2 = p · (4.5)2 ˜ 3.14(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 ˜ 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(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 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. (Continues...) Excerpted from Trigonometry For Dummies by Mary Jane Sterling . Copyright © 2014 John Wiley & Sons, Ltd. Excerpted by permission of John Wiley & Sons. All rights reserved. No part of this excerpt may be reproduced or reprinted without permission in writing from the publisher.Excerpts are provided by Dial-A-Book Inc. solely for the personal use of visitors to this web site. Read more