Trigonometry

12 Key excerpts on "Trigonometry"

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  eBook - ePub

  The Britannica Guide to Algebra and Trigonometry

  • Britannica Educational Publishing, William Hosch(Authors)
  • 2010(Publication Date)
  • Britannica Educational Publishing

    (Publisher)

  RIGONOMETRY

  T rigonometry is the branch of mathematics concerned with specific functions of angles and their application to calculations. There are six functions of an angle commonly used in Trigonometry. Their names and abbreviations are sine (sin), cosine (cos), tangent (tan), cotangent (cot), secant (sec), and cosecant (csc).

  Trigonometry developed from a need to compute angles and distances in such fields as astronomy, map making, surveying, and artillery range finding. Problems involving angles and distances in one plane are covered in plane Trigonometry. Applications to similar problems in more than one plane of three-dimensional space are considered in spherical Trigonometry.

  HISTORY OF Trigonometry

  CLASSICAL TRIGONOMETRY

  The word “Trigonometry” comes from the Greek words trigonon (“triangle”) and metron (“to measure”). Until about the 16th century, Trigonometry was chiefly concerned with computing the numerical values of the missing parts of a triangle (or any shape that can be dissected into triangles) when the values of other parts were given. For example, if the lengths of two sides of a triangle and the measure of the enclosed angle are known, the third side and the two remaining angles can be calculated. Such calculations distinguish Trigonometry from geometry, which mainly investigates qualitative relations. Of course, this distinction is not always absolute: the Pythagorean theorem, for example, is a statement about the lengths of the three sides in a right triangle and is thus quantitative in nature. Still, in its original form, Trigonometry was by and large an offspring of geometry. It was not until the 16th century that the two became separate branches of mathematics.

  ANCIENT EGYPT AND THE MEDITERRANEAN WORLD

  Several ancient civilizations—in particular, the Egyptian, Babylonian, Hindu, and Chinese—possessed a considerable knowledge of practical geometry, including some concepts that were a prelude to Trigonometry. The Rhind papyrus, an Egyptian collection of 84 problems in arithmetic, algebra, and geometry dating from about 1800 BCE, contains five problems dealing with the seked . A close analysis of the text, with its accompanying figures, reveals that this word means the slope of an incline—essential knowledge for huge construction projects such as the pyramids. For example, problem 56 asks: “If a pyramid is 250 cubits high and the side of its base is 360 cubits long, what is its seked

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  Mathematics for Scientific and Technical Students

  • H. Davies, H.G. Davies, G.A. Hicks(Authors)
  • 2014(Publication Date)
  • Routledge

    (Publisher)

  Chapter 3

  Trigonometry

     

  3.1 Introduction

  Trigonometry is the branch of Mathematics that deals with the relationships between the sides and the angles of a triangle. Provided that a minimum of

  (i) 3 sides,

  (ii) 2 sides and 1 angle or

  (iii) 1 side and 2 angles

  are known about any triangle, the other unknown sides or angles can be calculated using trigonometrical methods.

  Trigonometry is based on the trigonometrical ratios of sine, cosine and tangent, which are the ratios between the sides of a right-angled triangle. The values of these ratios depend upon the size of the angles, and do not depend on the size of the triangle.

  3.2 Trigonometric ratios

  In Fig. 3.1 sides are given names with respect to the angle θ. The side opposite the right-angle (90°) is called the hypotenuse. The other two sides are named according to their position relative to the angle. The ratios are defined as

  For a known angle the values of these ratios can be obtained from an electronic calculator. For example sin 40°=0.6428 can be obtained by entering 40 and pressing the sin key.

  Fig. 3.1

  Example 3.1  Find the lengths of the unknown sides in the steel bracket ABC shown in Fig 3.2 .

  Fig. 3.2

  Using     with θ = 60°   and  

  Multiply both sides by 21.4:

  Using     again with θ = 60°   and   h = 21.4cm

  Example 3.2 Fig 3.3 shows a voltage diagram for an electronic circuit. Calculate the phase angle ϕ between the two voltages.

  Fig. 3.3

  Note: ϕ = cos-1 0.5965 means that ϕ is an angle which has a cos of 0.5965. The value of ϕ can be obtained from the calculator by entering 0.5965 and pressing the

  cos−1

  key. cos-1 is the inverse of the cosine (see Section 3.15 ).

  3.3 Theorem of Pythagoras

  This is an important theorem and is a useful alternative method of finding the third side of a right-angled triangle when the other two sides are known. In the right-angled triangle in Fig 3.4 the sides are labelled a, b, c according to the angles that they are opposite.

  Fig. 3.4

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  Mathematics for Machine Technology

  • John Peterson, Robert Smith(Authors)
  • 2019(Publication Date)
  • Cengage Learning EMEA

    (Publisher)

  494 Trigonometry 7 UNIT 66 Introduction to Trigonometric Functions OBJECTIVES After studying this unit you should be able to ● ● Identify the sides of a right triangle with reference to any angle. ● ● State the ratios of the six trigonometric functions in relation to given right triangles. ● ● Find functions of angles given in decimal degrees and degrees, minutes, and seconds. ● ● Find angles in decimal degrees and degrees, minutes, and seconds of given functions. Trigonometry is the branch of mathematics that is used to compute unknown angles and sides of triangles. The word Trigonometry is derived from the Greek words for triangle and measurement. Trigonometry is used in the design of products. It is also used in the planning, setting up, and processing of manufactured products. The machines that produce the products could not be made without the use of Trigonometry. It is important that machinists, tool and die makers, drafters, designers, and related occupations apply trigonometric principles. Practical machine shop problems are often solved by using a combination of elements of algebra, geometry, and Trigonometry. Therefore, it is essential to develop the ability to analyze a problem in order to relate and determine the mathematical principles that are involved in its solution. Then the problem must be worked in clear, orderly steps, based on mathematical facts. When solving a problem, it is important to understand the trigonometric operations involved rather than to mechanically “plug in” values. Attempting to solve Trigonometry problems without understanding the principles involved will be unsuccessful, particularly in practical shop applications such as those found later in the text. SECTION SEVEN UNIT 66 INTRODUCTION TO TRIGONOMETRIC FUNCTIONS 495 RATIO OF RIGHT TRIANGLE SIDES In a right triangle, the ratio of two sides of the triangle determines the sizes of the angles, and the angles determine the ratio of two sides.

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  Scientific American Science Desk Reference

  • (Author)
  • 2008(Publication Date)
  • Trade Paper Press

    (Publisher)

  Trigonometry solves problems relating to plane and spherical triangles. Its principles are based on the fixed proportions of sides for a particular angle in a right-angled triangle, the simplest of which are known as the sine, cosine, and tangent (so-called trigonometrical ratios). Trigonometry is of practical importance in navigation, surveying, and simple harmonic motion in physics.

  Trigonometry At its simplest level, Trigonometry deals with the relationships between the sides and angles of triangles. Unknown angles or lengths are calculated by using trigonometrical ratios such as sine, cosine, and tangent.

  Using Trigonometry, it is possible to calculate the lengths of the sides and the sizes of the angles of a right-angled triangle as long as one angle and the length of one side are known, or the lengths of two sides. The longest side, which is always opposite to the right angle, is called the hypotenuse . The other sides are named depending on their position relating to the angle that is to be found or used: the side opposite this angle is always termed opposite and that adjacent is the adjacent . So the following trigonometrical ratios are used:

  sine

  The sine is the function of an angle in a right-angled triangle which is defined as the ratio of the length of the side opposite the angle to the length of the hypotenuse (the longest side). It is usually shortened to sin .

  sine The sine is a function of an angle in a right-angled triangle found by dividing the length of the side opposite the angle by the length of the hypotenuse (the longest side). Sine (usually abbreviated sin) is one of the fundamental trigonometric ratios.

  Various properties in physics vary sinusoidally; that is, they can be represented diagrammatically by a sine wave (a graph obtained by plotting values of angles against the values of their sines). Examples include simple harmonic motion, such as the way alternating current (AC) electricity varies with time.

  sine (left) The sine of an angle; (right) constructing a sine wave. The sine of an angle is a function used in the mathematical study of the triangle. If the sine of angle β is known, then the hypotenuse can be found given the length of the opposite side, or the opposite side can be found from the hypotenuse. Within a circle of unit radius (left), the height P1 ,A1 equals the sine of angle P1 0A1

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  Barron's Math 360: A Complete Study Guide to Pre-Calculus with Online Practice

  • Lawrence S. Leff, Christina Pawlowski-Polanish, Barron's Educational Series, Elizabeth Waite(Authors)
  • 2021(Publication Date)
  • Barrons Educational Services

    (Publisher)

  STUDY UNIT III

  TRIGONOMETRIC ANALYSIS

  Passage contains an image

  9 Trigonometry

  WHAT YOU WILL LEARN

  Trigonometry means “measurement of triangles.” The study of Trigonometry arose from the ancient need to understand the relationships between the sides and angles of triangles. With the development of calculus, Trigonometry progressed from the study of ratios within right triangles to trigonometric functions that could be used to better represent the circular and repeating patterns of behavior that characterize a wide range of physical phenomena in the real world.

  This chapter progresses from considering acute angles in right triangles to a more general view of angles as rotations about the origin in the coordinate plane. By fixing the vertex of such an angle at the origin and keeping one side of the angle aligned with the positive x-axis, we can give meaning to trigonometric functions of angles greater than 90° and less than 0°.

  LESSONS IN CHAPTER 9

  •Lesson 9-1: Degree and Radian Measures

  •Lesson 9-2: Right-Triangle Trigonometry

  •Lesson 9-3: The General Angle

  •Lesson 9-4: Working with Trigonometric Functions

  •Lesson 9-5: Trigonometric Functions of Special Angles

  Lesson 9-1: Degree and Radian Measures

  KEY IDEAS

  Angle measures can be expressed in units of degrees or in real-number units called radians. Degrees are based on fractional parts of a circular revolution. Radian measure compares the length of an arc that a central angle of a circle cuts off to the radius of the circle. The Greek letter θ (theta) is commonly used to represent an angle of unknown measure.

  MEASURING ANGLES IN DEGREES AND MINUTES

  One degree, denoted as 1°, is of one complete revolution about a fixed point.

  Each of the 60 equal parts of a degree is called a minute

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  Trigonometry For Dummies

  • Mary Jane Sterling(Author)
  • 2023(Publication Date)
  • For Dummies

    (Publisher)

  2 Trigonometric Functions IN THIS PART . . . Define the basic trig functions using the lengths of the sides of a right triangle. Determine the relationships between the trig cofunctions and their shared sides. Extend your scope to angles greater than 90 degrees using the unit circle. Investigate the ins and outs of the domains and ranges of the six trig functions. Use reference angles to compute trig functions. Apply trig functions to real-world problems. CHAPTER 6 Describing Trig Functions 91 Chapter 6 Describing Trig Functions B y taking the lengths of the sides of right triangles or the chords of circles and creating ratios with those numbers and variables, our ancestors initi- ated the birth of trigonometric functions. These functions are of infinite value, because they allow you to use the stars to navigate and to build bridges that won’t fall. If you’re not into navigating a boat or engineering, then you can use the trig functions at home to plan that new addition. And they’re a staple for students going into calculus. You may be asking, “What is a function? What does it have to do with trigonom- etry?” In mathematics, a function is a mechanism that takes a value you input into it and churns out an answer, called the output. A function is connected to rules involving mathematical operations or processes. The six trig functions require one thing of you — inputting an angle measure — and then they output a number. These outputs are always real numbers, from infinitely small to infinitely large and everything in between. The results you get depend on which function you use. Although in earlier times, some of the function computations were rather tedious, today’s hand-held calculators, and even phones, make everything much easier. IN THIS CHAPTER » Understanding the three basic trig functions » Building on the basics: The reciprocal functions » Recognizing the angles that give the cleanest trig results » Determining the exact values of functions

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  Trigonometry

  • Ron Larson(Author)
  • 2017(Publication Date)
  • Cengage Learning EMEA

    (Publisher)

  With the development of calculus and the physical sciences in the 17th century, a different perspective arose—one that viewed the classic trigonometric relationships as functions with the set of real numbers as their domains. Consequently, the applications of Trigonometry expanded to include a vast number of physical phenomena, such as sound waves, planetary orbits, vibrating strings, pendulums, and orbits of atomic particles. This text incorporates both perspectives, starting with angles and their measure. Vertex Terminal side Initial side x Initial side Terminal side y Angle Angle in standard position Figure 1.1 Figure 1.2 Rotating a ray (half-line) about its endpoint determines an angle. The starting position of the ray is the initial side of the angle, and the position after rotation is the terminal side, as shown in Figure 1.1. The endpoint of the ray is the vertex of the angle. This perception of an angle fits a coordinate system in which the origin is the vertex and the initial side coincides with the positive x-axis. Such an angle is in standard position, as shown in Figure 1.2. Counterclockwise rotation generates positive angles and clockwise rotation generates negative angles, as shown in Figure 1.3. Labels for angles can be Greek letters such as α (alpha), β (beta), and θ (theta) or uppercase letters such as A, B, and C. In Figure 1.4, note that angles α and β have the same initial and terminal sides. Such angles are coterminal. x Positive angle (counterclockwise) Negative angle (clockwise) y β α y x β α y x Coterminal angles Figure 1.3 Figure 1.4 Angles and their measure have a wide variety of real-life applications. For example, in Exercise 68 on page 131, you will use angles and their measure to model the distance a cyclist travels. Connel/Shutterstock.com Copyright 2018 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part.

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  Precalculus with Limits

  • Ron Larson(Author)
  • 2021(Publication Date)
  • Cengage Learning EMEA

    (Publisher)

  9780357457856_0400 10/19/20 Finals left, © De Visu/Shutterstock.com; right, © Vladi333/Shutterstock.com Trigonometry 4 4.5 Respiratory Cycle (Exercise 80, p. 306) 4.7 Space Shuttle (Exercise 94, p. 326) 4.1 Radian and Degree Measure 4.2 Trigonometric Functions: The Unit Circle 4.3 Right Triangle Trigonometry 4.4 Trigonometric Functions of Any Angle 4.5 Graphs of Sine and Cosine Functions 4.6 Graphs of Other Trigonometric Functions 4.7 Inverse Trigonometric Functions 4.8 Applications and Models GO DIGITAL Chapter 4 Section 5 Exercise 43 NEXT PREV. 33 35 37 39 41 43 45 47 49 51 53 259 Copyright 2022 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s). Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it. 260 Chapter 4 Trigonometry 4.1 Radian and Degree Measure GO DIGITAL Angles and their measure have a wide variety of real-life applications. For example, in Exercise 68 on page 269, you will use angles and their measure to model the distance a cyclist travels. Describe angles. Use radian measure. Use degree measure. Use angles and their measure to model and solve real-life problems. Angles As derived from the Greek language, the word Trigonometry means “measurement of triangles.” Originally, Trigonometry dealt with relationships among the sides and angles of triangles and was instrumental in the development of astronomy, navigation, and surveying. With the development of calculus and the physical sciences in the seventeenth century, a different perspective arose—one that viewed the classic trigonometric relationships as functions with the set of real numbers as their domains.

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  Pre-Calculus All-in-One For Dummies

  Book + Chapter Quizzes Online

  • Mary Jane Sterling(Author)
  • 2023(Publication Date)
  • For Dummies

    (Publisher)

  3 The Essentials of Trigonometry IN THIS UNIT . . . Reviewing angles and angle types, along with their properties. Determining trig function values using degrees and radians. Graphing basic trig functions. Transforming trig function graphs. CHAPTER 8 Circling In on Angles 173 Circling In on Angles I n this chapter, you find right triangles drawn on the coordinate plane (x- and y-axes). Moving right triangles onto the coordinate plane introduces many more interesting concepts such as evaluating trig functions and solving trig equations. Also, you become acquainted with a very handy tool known as the unit circle. The unit circle is extremely important in the real world and in mathematics; for instance, you’re at its mercy whenever you fly in an airplane. Pilots use the unit circle, along with vec- tors, to fly airplanes in the correct direction and over the correct distance. Imagine the disaster that would result if a pilot tried to land a plane a bit to the left of the runway! In this chapter, you work on building the unit circle as you review the basics of angles in radians and degrees as they’re found in triangles. With that information, you can place the triangles onto the unit circle (which is also located in the coordinate plane) to solve the prob- lems at the end of this chapter. (You find more on these ideas as you move into graphing trig functions in Chapter 10.) Introducing Radians and Relating to Degrees When you first studied geometry, you probably measured every angle in degrees, based on a portion of a 360 circle around a point. As it turns out, the number 360 was picked to represent the degrees in a circle only for convenience. Chapter 8 IN THIS CHAPTER » Discovering alternate trig function definitions » Inserting triangles on the unit circle » Calculating trig functions on the unit circle

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  Military schools and courses of instruction in the science and art of war,

  in France, Prussia, Austria, Russia, Sweden, Switzerland, Sardinia, England, and the United States. Drawn from recent official reports and documents. Revised Edition

  • (Author)
  • 2013(Publication Date)
  • Perlego

    (Publisher)

  Resolution of right-angled triangles.—Of the case in which the hypothenuse and a side nearly equal to it are given. Knowing a side and two angles of any triangle, to find the other parts, and also the surface of the triangle.

  Knowing two sides a and b of a triangle and the included angle C, to find the other parts and also the surface of the triangle.—The tang.½(A - B) may be determined; or tang.A and tang.B directly.

  Knowing the three sides a, b, c, to find the angles and the surface of the triangle.—Employment of the formula which gives tang.½A.

  Application to surveying for plans.

  Measurement of bases with rods. Measurement of angles.—Description and use of the circle.—Use of the telescope to render the line of sight more precise.—Division of the circle.—Verniers. Measurement and calculation of a system of triangles.—Reduction of angles to the centres of stations. How to connect the secondary points to the principal system.—Use of the plane table and of the compass. 2. SPHERICAL Trigonometry.

  Fundamental relations (cos.a = cos.b cos.c + sin.b sin.c cos.A) between the sides and the angles of a spherical triangle.

  To deduce thence the relations sin.A : sin.B = sin.a : sin.b; cot.a sin.b - cot.A sin.C = cos.b cos.C, and by the consideration of the supplementary triangle cos.A = -cos.B cos.C + sin.B sin.C cos.a.

  Right-angled triangles.—Formulas cos.a = cos.b cos.c; sin.b = sin.a sin.B; tang.c = tang.a cos.B, and tang.b = sin.c tang.B.

  In a right-angled triangle the three sides are less than 90°, or else two of the sides are greater than 90°, and the third is less. An angle and the side opposite to it are both less than 90°, or both greater.

  Resolution of any triangles whatever:

  1o Having given their three sides a, b, c, or their three angles A, B, C.—Formulas tang.½a and tang.1/2A, calculable by logarithms:

  2o Having given two sides and the included angle, or two angles and the included side.—Formulas of Delambre:

  3o

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  The Ellipse

  A Historical and Mathematical Journey

  • Arthur Mazer(Author)
  • 2011(Publication Date)
  • Wiley

    (Publisher)

  y ) coordinates.

  Figure 5.2 Trigonometric functions as coordinates of a unit circle.

  The angle θ is sometimes expressed in radians and sometimes expressed in degrees. It is worth one’s effort to be able to use both units of measurement.

  5.1.2 Triangles

  For angles between 0˚ and 90˚ (from 0 to π /2radians), the trigonometric functions correspond to ratios of right triangles as illustrated in Figure 5.3 .

  Figure 5.3 Trigonometric functions as triangular ratios.

  The equivalence between the definitions in the preceding two tables is seen by noting the triangle formed between a point on the unit circle, the origin, and the point along the x axis given by the x coordinate of the original point. The ratios that define the trigonometric functions are the same for all similar triangles.

  5.1.3 Examples

  Using the definitions, it is possible to determine the trigonometric functions for some values of θ . Examples are given below. These examples are the first entries into a trigonometric table that is further developed in subsequent sections.

  Example 5.1

  Determine the trigonometric for the value θ = π /4 rad (45˚).

  Solution

  When the angle θ is π/4 rad, x = y along the unit circle (see Figure 5.4 ). With the assistance of the Pythagorean theorem, the values for x and y

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  Precalculus

  • Cynthia Y. Young(Author)
  • 2023(Publication Date)
  • Wiley

    (Publisher)

  4.3.1 Conceptual Understand that right triangle definitions of trigonometric functions for acute angles are consistent with definitions of trigonometric functions for all angles in the Cartesian plane. To define the trigonometric functions in the Cartesian plane, let us start with an acute angle θ in standard position. Choose any point (x, y) on the terminal side of the angle as long as it is 388 CHAPTER 4 Trigonometric Functions of Angles not the vertex (the origin). A right triangle can be drawn so that the right angle is made when a perpendicular segment connects the point (x, y) to the x-axis. Notice that the side opposite θ has length y and the other leg of the right triangle has length x. Words Math The distance r from the origin (0, 0) to the point (x, y) can be found using r = √ _______________ (x − 0) 2 + ( y − 0) 2 r = √ _ x 2 + y 2 the distance formula. Since r is a distance, it is always positive. r > 0 Using our first definition of trigonometric functions in terms of right triangle ratios (Section 4.2), we say that sin θ = opposite _________ hypotenuse . From this picture we see that the sine function can also be defined by the relation sin θ = y _ r . Similar reasoning holds for all six trigonometric functions and leads us to the second definition of the trigonometric functions, in terms of ratios of coordinates and distances in the Cartesian plane. (x, y) θ x y (x, y) θ x y r x y (x, y) θ x y r x y Let (x, y) be any point, other than the origin, on the terminal side of an angle θ in standard position. Let r be the distance from the point (x, y) to the origin; then the six trigonometric functions are defined as sin θ = y _ r cos θ = x _ r tan θ = y _ x (x ≠ 0) csc θ = r _ y ( y ≠ 0) sec θ = r _ x (x ≠ 0) cot θ = x _ y ( y ≠ 0) where r = √ _ x 2 + y 2 , or x 2 + y 2 = r 2 .

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