Trigonometry Geometry and Expanded First Edition By Paul M Tokorcheck University of Southern California Trigonometry Geometry and The Conception of Space Expanded First Edition By Paul M Tokorcheck University of Southern California Bassim Hamadeh, CEO and Publisher Michael Simpson, Vice President of Acquisitions Jamie Giganti, Senior Managing Editor Jess Busch, Senior Graphic Designer Mark Combes, Senior Field Acquisitions Editor Natalie Lakosil, Licensing Manager Copyright © 2016 by Cognella, Inc. All rights reserved. No part of this publication may be reprinted, reproduced, transmitted, or utilized in any form or by any electronic, mechanical, or other means, now known or hereafter invented, including photocopying, microfilming, and recording, or in any information retrieval system without the written permission of Cognella, Inc. First published in the United States of America in 2016 by Cognella, Inc. Trademark Notice: Product or corporate names may be trademarks or registered trademarks, and are used only for identification and explanation without intent to infringe. Cover image copyright© [date] Depositphotos/[contributor] Interior image copyright © [date] Depositphotos/[contributor] Printed in the United States of America ISBN: 978-1-63487-187-7 (pbk) / 978-1-63487-188-4 (br) For Tami. We get to choose our friends, but not our family. She’s the best of both. Preface This text was first created to support a new course at Iowa this book should not serve as an introduction to these topics State University, meeting for the first time in fall 2014. At in any capacity. the time this project was initiated, our standard trigonome- try course was a prototypical pre-calculus offering, focusing I would like to thank the faculty of Iowa State University on skills that were needed by STEM majors for their future for their support, not only for this project, but for all of the calculus courses. However, the course rosters indicated that work I’ve done with them over the past few years. I’d also a large population of these students were actually from our like to thank Mark Combes and Jamie Giganti of Cognella College of Design. These students required a course that Academic Publishing for their help and guidance. was more visual and more pragmatic, and that placed a greater emphasis on three-dimensional geometry. Last, but certainly not least, I am indebted to the staff of the Smokey Row Coffee Shop in Des Moines, Iowa. The At ISU this course asks for a standard course in college al- bulk of this work was created at a corner table of that fine gebra as a prerequisite. General geometric knowledge (the establishment, fueled by a steady supply of large lattes. Pythagorean Theorem), basic algebra skills (factoring), and a loose knowledge of functions and their graphs is all as- Paul Tokorcheck sumed. Students do not need to be experts in these topics; Carlsbad, CA most are reviewed in this text to a certain extent. However, June 2014 v Contents Sets and Numbers 1 Polar Coordinates 143 Measurement and Angles 11 Cylindrical Coordinates 153 The Six Trigonometric Ratios 23 Spherical Coordinates 165 Standard Triangles 35 Transformations of Graphs 175 The Unit Circle 47 Quadratic Equations and Conic Sections 187 Functions and Inverse Functions 59 Parabolas 197 The Inverse Trigonometric Functions 71 Ellipses 207 Solving Equations for Angles 83 Hyperbolas 221 Using Trigonometric Identities 93 Surfaces and Level Curves 233 Applications of Trigonometry 105 Ellipsoids and Cones 243 The Law of Sines and the Law of Cosines 117 Paraboloids 253 Coordinate Systems and Graphs 129 Hyperboloids 265 vii LESSON 1 Sets and Numbers Our perception of the space around us is very much guided If there is a chasm between the inherent properties of a by our senses. A person who is blind in one eye cannot space and our conceptualization of the space, then this perceive depth and therefore must rely on other visual cues statement already indicates the bridge. By associating a to determine the distance to an object. However, there is no number to a given distance, we are able to compare one denying that space has certain intrinsic properties that are distance to another and create a framework in which we can not subject to our interpretation. For example, two distinct discuss distances in general. This association is not trivial. objects in a space have a positive, yet finite distance between them, and this distance is not dependent on our method of Nevertheless, we also make these associations with respect measurement, or even on our ability to perceive the distance to more concrete objects. In modern language it’s common in the first place. to say that we “multiply two sides of a rectangle to find its area,” though the side of a rectangle is geometric object, not We should note that the last statement described a distance a number. We cannot “multiply two sides” any more than as being both positive and finite. we can multiply two refrigerators. 1 LESSON 1 What we mean by this statement is that we can assign a transfer between them. Before we do, we need to discuss numerical value to indicate the length of each side, which the basic concepts of numbers and sets. is the distance from one end of the side to the other. We then multiply the two numbers, and interpret the product Definition: A set is a collection of objects, each of which as another, entirely different, geometric quantity. is called an element. If x is an element of A, we write x 2 A. These may seem to be insignificant semantic distinctions. However, there are a variety of ways to measure quantities To say that x is not an element of A, we write like length or area, which are sometimes in direct compe- x 2/ A. tition with each other. We’ll not only need to distinguish There are several ways to describe a set. One common between different types of quantities, but also the units we method is simply to list the elements of the set within a use when assigning a number to a quantity. pair of braces, as shown here: How to artfully use a space is the realm of architecture and = f g interior design. But how to accurately measure the dimen- A chair, lamp, sofa, table sions of that space, well, that is very much at the heart of B = fred, blue, black, green, yellowg mathematics and its history. This is a history that dates back thousands of years, from Bronze Age Sumerian farm- Note that both of these sets contain a finite number of ele- ers measuring their fields, and the astronomers of China ments; A contains four elements, and B contains five. and India, to the geometers of ancient Greece. Definition: The number of elements in a set is called its In later lessons we’ll focus more on the idea of position, cardinality. The set with no elements is called empty set, which will require some more modern tools. Position is not or null set. It is denoted by Æ. indicated by a single number, but by a larger set of data. Again, there are various ways to do this. We will learn Of course, when a set contains many elements, listing them how to identify points using several methods, and how to individually becomes impractical. 2 LESSON 1 One way to describe sets with many elements is to include Definition: The union of two sets A and B is defined by an ellipsis, and leave it to the reader to discern the pattern [ f j 2 2 g that we are trying to convey. For example: A B = x x A or x B . The intersection of two sets A and B is defined by A = fbird, cat, fish, dog, ...g B = f..., −5, −3, −1, 1, 3, 5, 7, ...g A \ B = fx j x 2 A and x 2 Bg. While commonly used, this method can unfortunately lead The difference of two sets A and B is defined by to ambiguity. Is set A above meant to include all animals? A n B = fx j x 2 A and x 2/ Bg. Just those kept as pets? The description of set B is perhaps a bit more more clear, but we are still relying on the reader In these definitions, note the use of the words “and” and to recognize that all of the listed numbers are odd, and that “or”. In mathematics, “or” is always intended to be inclu- this is the defining feature of this set. sive: x 2 A [ B if x is an element of A, if x is an element of B, or if x is an element of both. However, for x to be an el- An even better method uses set builder notation. In this ement of the intersection, it must necessarily be an element notation, an element is listed along with a short description of both A and B. of the properties that elements of the set should have. The name of the element and its decription are separated either Example: Let A and B be the following two sets: by a vertical line or by a colon. Here are a few examples that may or may not be sets of numbers: A = f1, 24, −4, 7, 13, −17g , = f g A = fa j a is an even numberg B 0, 3, 6, 9, 12, . B = fb 2 A : b > 5g Then A \ B = f24g. It would be awkward to try to write A [ B within a single pair of set brackets, which is why we C = fcars j made before 1995g have the notation to help.