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Elementary Trigonometry ELEMENTARY TRIGONOMETRY Michael Corral Schoolcraft College Schoolcraft College Elementary Trigonometry Michael Corral (unable to fetch text document from uri [status: 0 (UnableToConnect), message: "Error: TrustFailure (Ssl error:1000007d:SSL routines:OPENSSL_internal:CERTIFICATE_VERIFY_FAILED)"]) TABLE OF CONTENTS This is a text on elementary trigonometry, designed for students who have completed courses in high-school algebra and geometry. Though designed for college students, it could also be used in high schools. The traditional topics are covered, but a more geometrical approach is taken than usual. Also, some numerical methods (e.g. the secant method for solving trigonometric equations) are discussed. A brief tutorial on using Gnuplot to graph trigonometric functions is included. 1: RIGHT TRIANGLE TRIGONOMETRY ANGLES Trigonometry is the study of the relations between the sides and angles of triangles. The word “trigonometry” is derived from the Greek words trigono (τρ´ιγωνo), meaning “triangle”, and metro (µǫτρω´ ), meaning “measure”. Though the ancient Greeks, such as Hipparchus and Ptolemy, used trigonometry in their study of astronomy between roughly 150 B.C. - A.D. 200, its history is much older. 1.1: ANGLES 1.2: TRIGONOMETRIC FUNCTIONS OF AN ACUTE ANGLE 1.3: APPLICATIONS AND SOLVING RIGHT TRIANGLES 1.4: TRIGONOMETRIC FUNCTIONS OF ANY ANGLE 1.5: ROTATIONS AND REFLECTIONS OF ANGLES 1.E: RIGHT TRIANGLE TRIGONOMETRY ANGLES (EXERCISES) 2: GENERAL TRIANGLES Though the methods described will work for right triangles, they are mostly used to solve oblique triangles, that is, triangles which do not have a right angle. There are two types of oblique triangles: an acute triangle has all acute angles, and an obtuse triangle has one obtuse angle. As we will see, Cases 1 and 2 can be solved using the law of sines, Case 3 can be solved using either the law of cosines or the law of tangents, and Case 4 can be solved using the law of cosines. 2.1: THE LAW OF SINES 2.2: THE LAW OF COSINES 2.3: THE LAW OF TANGENTS 2.4: THE AREA OF A TRIANGLE 2.5: CIRCUMSCRIBED AND INSCRIBED CIRCLES 2.E: GENERAL TRIANGLES (EXERCISES) 3: IDENTITIES 3.1: BASIC TRIGONOMETRIC IDENTITIES 3.2: SUM AND DIFFERENCE FORMULAS 3.3: DOUBLE-ANGLE AND HALF-ANGLE FORMULAS 3.4: OTHER IDENTITIES 3.E: IDENTITIES (EXERCISES) 4: RADIAN MEASURE 4.1: RADIANS AND DEGREES 4.2: ARC LENGTH 4.3: AREA OF A SECTOR 4.4: CIRCULAR MOTION- LINEAR AND ANGULAR SPEED 4.E: RADIAN MEASURE (EXERCISES) 5: GRAPHING AND INVERSE FUNCTIONS 5.1: GRAPHING THE TRIGONOMETRIC FUNCTIONS 5.2: PROPERTIES OF GRAPHS OF TRIGONOMETRIC FUNCTIONS 5.3: INVERSE TRIGONOMETRIC FUNCTIONS 5.E: GRAPHING AND INVERSE FUNCTIONS (EXERCISES) 6: ADDITIONAL TOPICS 6.1: SOLVING TRIGONOMETRIC EQUATIONS 6.2: NUMERICAL METHODS IN TRIGONOMETRY 1 10/6/2021 6.3: COMPLEX NUMBERS 6.4: POLAR COORDINATES 6.E: ADDITIONAL TOPICS (EXERCISES) BACK MATTER INDEX GLOSSARY 2 10/6/2021 CHAPTER OVERVIEW 1: RIGHT TRIANGLE TRIGONOMETRY ANGLES Trigonometry is the study of the relations between the sides and angles of triangles. The word “trigonometry” is derived from the Greek words trigono (τρ´ιγωνo), meaning “triangle”, and metro (µǫτρω´ ), meaning “measure”. Though the ancient Greeks, such as Hipparchus and Ptolemy, used trigonometry in their study of astronomy between roughly 150 B.C. - A.D. 200, its history is much older. 1.1: ANGLES ∘ In elementary geometry, angles are always considered to be positive and not larger than 360 . You also learned that the sum of the angles in a triangle equals 180◦, and that an isosceles triangle is a triangle with two sides of equal length. Recall that in a right triangle one of the angles is a right angle. Thus, in a right triangle one of the angles is 90◦ and the other two angles are acute angles whose sum is 90◦ (i.e. the other two angles are complementary angles). 1.2: TRIGONOMETRIC FUNCTIONS OF AN ACUTE ANGLE For a right triangle △ABC, with the right angle at C and with lengths a, b, and c. For the acute angle A, call the leg BC its opposite side, and call the leg AC its adjacent side. Recall that the hypotenuse of the triangle is the side AB. The ratios of sides of a right triangle occur often enough in practical applications to warrant their own names, so we can define the six trigonometric functions of A. 1.3: APPLICATIONS AND SOLVING RIGHT TRIANGLES Throughout its early development, trigonometry was often used as a means of indirect measurement, e.g. determining large distances or lengths by using measurements of angles and small, known distances. Today, trigonometry is widely used in physics, astronomy, engineering, navigation, surveying, and various fields of mathematics and other disciplines. In this section we will see some of the ways in which trigonometry can be applied. Your calculator should be in degree mode for these examples. 1.4: TRIGONOMETRIC FUNCTIONS OF ANY ANGLE To define the trigonometric functions of any angle - including angles less than 0° or greater than 360° - we need a more general definition of an angle. We say that an angle is formed by rotating a ray OA about the endpoint O (called the vertex), so that the ray is in a new position, denoted by the ray OB. The ray OA is called the initial side of the angle, and OB is the terminal side of the angle. 1.5: ROTATIONS AND REFLECTIONS OF ANGLES Now that we know how to deal with angles of any measure, we will take a look at how certain geometric operations can help simplify the use of trigonometric functions of any angle, and how some basic relations between those functions can be made. The two operations on which we will concentrate in this section are rotation and reflection. 1.E: RIGHT TRIANGLE TRIGONOMETRY ANGLES (EXERCISES) These are homework exercises to accompany Corral's "Elementary Trigonometry" Textmap. This is a text on elementary trigonometry, designed for students who have completed courses in high-school algebra and geometry. Though designed for college students, it could also be used in high schools. The traditional topics are covered, but a more geometrical approach is taken than usual. Also, some numerical methods (e.g. the secant method for solving trigonometric equations) are discussed. 1 10/6/2021 1.1: Angles Recall the following definitions from elementary geometry: a. An angle is acute if it is between 0° and 90°. b. An angle is a right angle if it equals 90°. c. An angle is obtuse if it is between 90° and 180°. d. An angle is a straight angle if it equals 180°. Figure 1.1.1 Types of angles ∘ In elementary geometry, angles are always considered to be positive and not larger than 360 . For now we will only consider such angles. The following definitions will be used throughout the text: a. Two acute angles are complementary if their sum equals 90◦ . In other words, if 0◦ ≤ ∠A, ∠B ≤ 90◦ then ∠A and ∠B are complementary if ∠A +∠B = 90◦ . b. Two angles between 0◦ and 180◦ are supplementary if their sum equals 180◦ . In other words, if 0◦ ≤ ∠A, ∠B ≤ 180◦ then ∠A and ∠B are supplementary if ∠A +∠B = 180◦ . c. Two angles between 0◦ and 360◦ are conjugate (or explementary) if their sum equals 360◦ . In other words, if 0◦ ≤ ∠A, ∠B ≤ 360◦ then ∠A and ∠B are conjugate if ∠A +∠B = 360◦ . Figure 1.1.2 Types of pairs of angles Instead of using the angle notation ∠A to denote an angle, we will sometimes use just a capital letter by itself (e.g. A, B, C) or a lowercase variable name (e.g. x, y, t). It is also common to use letters (either uppercase or lowercase) from the Greek alphabet, shown in the table below, to represent angles: Table 1.1 The Greek alphabet In elementary geometry you learned that the sum of the angles in a triangle equals 180◦ , and that an isosceles triangle is a triangle with two sides of equal length. Recall that in a right triangle one of the angles is a right angle. Thus, in a right triangle one of the angles is 90◦ and the other two angles are acute angles whose sum is 90◦ (i.e. the other two angles are complementary angles). GPL Michael Corral 1.1.1 9/8/2021 https://math.libretexts.org/@go/page/3243 α + 3α + α = 180◦ ⇒ 5α = 180◦ ⇒ α = 36◦ ⇒ X = 36◦, Y = 3 × 36◦ = 108◦, Z = 36◦ QED Figure 1.1.4 By knowing the lengths of two sides of a right triangle, the length of the third side can be determined by using the Pythagorean Theorem: The square of the length of the hypotenuse of a right triangle is equal to the sum of the squares of the lengths of its legs. Figure 1.1.5 Similar triangles △ ABC, △ CBD, △ ACD Recall that triangles are similar if their corresponding angles are equal, and that similarity implies that corresponding sides are proportional. Thus, since △ ABC is similar to △ CBD, by proportionality of corresponding sides we see that c a ¯A¯¯¯¯B¯¯¯ is to ¯C¯¯¯¯B¯¯¯ (hypotenuses) as B¯¯¯¯¯C¯¯¯ is to ¯B¯¯¯¯D¯¯¯ (vertical legs) ⇒ = ⇒ cd = a2 . a d Since △ ABC is similar to △ ACD, comparing horizontal legs and hypotenuses gives b c = ⇒ b2 = c2 − cd = c2 − a2 ⇒ a2 + b2 = c2 . QED c −d b Note: The symbols ⊥ and ∼ denote perpendicularity and similarity, respectively. For example, in the above proof we had ¯¯¯¯¯¯¯¯ ¯¯¯¯¯¯¯¯ CD ⊥ AB and △ ABC ∼ △ CBD ∼ △ ACD. For triangle △ ABC, the Pythagorean Theorem says that a2 + 42 = 52 ⇒ a2 = 25 − 16 = 9 ⇒ a = 3 .
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