Mathematics
Unit Circle
The unit circle is a circle with a radius of 1, centered at the origin of a coordinate system. In trigonometry, it is used to define the values of sine and cosine for any angle. The unit circle is a fundamental concept in understanding trigonometric functions and their relationships to angles and coordinates.
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9 Key excerpts on "Unit Circle"
eBook - PDF- Cynthia Y. Young(Author)
- 2023(Publication Date)
- Wiley(Publisher)
The equation for the Unit Circle (radius of 1 centered at the origin) is given by x 2 + y 2 = 1. The term circular function is often used as a synonym for trigonometric function, but it is important to note that a circle is not a function (it does not pass the vertical line test). If we form a central angle θ in the Unit Circle such that the terminal side lies in quadrant I, we can use the previous two definitions of the sine and cosine functions when r = 1 (i.e., on the Unit Circle). Trigonometric Function Right Triangle Trigonometry Cartesian Plane sin θ opposite __________ hypotenuse = y __ 1 = y y __ r = y __ 1 = y cos θ adjacent __________ hypotenuse = x __ 1 = x x __ r = x __ 1 = x 442 CHAPTER 5 Trigonometric Functions of Real Numbers Notice that any point (x, y) on the Unit Circle can be written as (cos θ, sin θ). We can now summarize the exact values for the sine and cosine functions in the following illustration. The following observations are consistent with properties of trigonometric functions we’ve studied already: • sin θ > 0 in quadrant I and quadrant II, where y > 0. • cos θ > 0 in quadrant I and quadrant IV, where x > 0. • The equation of the Unit Circle x 2 + y 2 = 1 leads also to the Pythagorean identity cos 2 θ + sin 2 θ = 1. (x, y) = (cos θ, sinθ) x y 60º 3 45º 4 30º 6 360º 2 0º 0 0 330º 6 11 315º 4 7 300º 3 5 270º 2 3 240º 3 4 225º 4 5 210º 6 7 180º 150º 6 5 135º 4 3 90º 2 120º 3 2 (0, 1) (0, –1) (1, 0) (–1, 0) ( , – ) 2 √3 2 1 ( , ) 2 √3 2 1 ( , – ) 2 2 √2 ( , ) 2 √2 2 √2 ( , – ) 2 √3 √3 2 1 ( – , – ) 2 √3 2 1 ( – , – ) 2 √2 √2 2 √2 ( – , – ) 2 2 1 ( – , ) 2 √3 2 1 ( – , ) 2 √2 2 √2 ( – , ) 2 √3 2 1 ( , ) 2 √3 2 1 STUDY TIP (cos θ, sin θ) represents a point (x, y) on the Unit Circle. Trigonometric Functions: Unit Circle Approach Let (x, y) be any point on the Unit Circle ( x 2 + y 2 = 1).
No longer available |Learn more- James Stewart, Lothar Redlin, Saleem Watson(Authors)
- 2016(Publication Date)
- Cengage Learning EMEA(Publisher)
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. 402 CHAPTER 5 ■ Trigonometric Functions: Unit Circle Approach 5.1 THE Unit Circle ■ The Unit Circle ■ Terminal Points on the Unit Circle ■ The Reference Number In this section we explore some properties of the circle of radius 1 centered at the origin. These properties are used in the next section to define the trigonometric functions. ■ The Unit Circle The set of points at a distance 1 from the origin is a circle of radius 1 (see Figure 1). In Section 1.9 we learned that the equation of this circle is x 2 y 2 1. THE Unit Circle The Unit Circle is the circle of radius 1 centered at the origin in the xy-plane. Its equation is x 2 y 2 1 EXAMPLE 1 ■ A Point on the Unit Circle Show that the point P a !3 3 , !6 3 b is on the Unit Circle. SOLUTION We need to show that this point satisfies the equation of the Unit Circle, that is, x 2 y 2 1. Since a !3 3 b 2 a !6 3 b 2 3 9 6 9 1 P is on the Unit Circle. Now Try Exercise 3 ■ EXAMPLE 2 ■ Locating a Point on the Unit Circle The point PA !3 / 2, y B is on the Unit Circle in Quadrant IV. Find its y-coordinate. SOLUTION Since the point is on the Unit Circle, we have a !3 2 b 2 y 2 1 y 2 1 3 4 1 4 y 1 2 Since the point is in Quadrant IV, its y-coordinate must be negative, so y 1 2 . Now Try Exercise 9 ■ ■ Terminal Points on the Unit Circle Suppose t is a real number. If t 0, let’s mark off a distance t along the Unit Circle, starting at the point 1 1, 0 2 and moving in a counterclockwise direction.
eBook - PDFPre-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
eBook - PDFPrecalculus
A Prelude to Calculus
- Sheldon Axler(Author)
- 2016(Publication Date)
- Wiley(Publisher)
The cosine and sine functions, which we are about to introduce, were invented to help us extend this table to all angles. Before introducing the cosine and sine functions, we state a common assumption about notation in trigonometry. Angles without units If no units are given for an angle, then assume the units are radians. The figure below shows a radius of the Unit Circle corresponding to θ (here θ might be measured in either radians or degrees). The endpoint of this radius is used to define the cosine and sine, as follows. This figure defines cosine and sine. If you understand this figure well, then you can figure out a big chunk of trigonometry. Cosine The cosine of θ, denoted cos θ, is the first coordinate of the endpoint of the radius of the Unit Circle corresponding to θ. Sine The sine of θ, denoted sin θ, is the second coordinate of the endpoint of the radius of the Unit Circle corresponding to θ. The two definitions above can be combined into a single statement, as follows. Cosine and sine The endpoint of the radius of the Unit Circle corresponding to θ has coordinates (cos θ, sin θ ). 308 Chapter 4 Trigonometric Functions Example 1 Evaluate cos π 2 and sin π 2 . Here units are not specified for the angle π 2 . Thus we assume we are dealing with π 2 radians. solution The radius corresponding to π 2 radians has endpoint (0, 1). Thus cos π 2 = 0 and sin π 2 = 1. Using degrees instead of radians, we could write cos 90 ◦ = 0 and sin 90 ◦ = 1. The table below gives the cosine and sine of some special angles. This table is The radius corresponding to π 2 radians has endpoint (0, 1). obtained by breaking the last column of the previous table into two columns, with the first coordinate labeled as cosine and the second coordinate labeled as sine. Compare this table to the first table in this section and make sure you understand what is going on here.
eBook - PDF- Doug French(Author)
- 2004(Publication Date)
- Continuum(Publisher)
Chapter 10 Trigonometry and Circular Functions In every right triangle, if we describe a circle with a centre a vertex of an acute angle and radius the length of the longest side, then the side subtending this acute angle is the right sine of the arc adjacent to that side and opposite the given angle; the third side is equal to the sine of the complement of the arc. (Regiomontanus (1436-1476) in Fauvel and Gray, 1987, p. 245) INTRODUCING SINE AND COSINE The geometrical task of finding lengths and angles in triangles requires familiarity with the circular functions -sine, cosine and tangent -whose role in mathematics is extensive and important, going way beyond the solution of triangles that originally inspired their develop-ment. Understanding the properties of these functions and their application to a variety of problems requires an interplay between geometric and algebraic ideas. Unfortunately, students often come to regard the subject as difficult, seeing it as a matter of remembering a lot of apparently unrelated formulae and procedures. It is essential to develop an understanding of how the three functions relate to right-angled triangles and then, as they are developed further, to appreciate the link to circles, the behaviour and properties of their graphs and the con-nections between identities as a web of interrelated ideas rather than as a set of disparate facts. Many school textbooks, working with right-angled triangles, define sine, cosine and tangent initially as ratios, which, as has been pointed out in Chapter 8, is an idea that students often find difficult. A ratio provides a comparison between two numbers and is not necessarily itself seen as a number that can be interpreted, in particular, as a length. In fact, a length is a ratio because there is an implicit comparison with whatever is being taken as the unit of measure-ment. However, students think of lengths as something simple and concrete that can be represented by numbers.
eBook - PDF- Ron Larson(Author)
- 2021(Publication Date)
- Cengage Learning EMEA(Publisher)
(1, 0) ( , ) x y t t t > 0 θ y x (1, 0) ( , ) x y t t t < 0 θ y x As the real number line wraps around the Unit Circle, each real number t corresponds to a point (x, y) on the circle. For example, the real number 0 corresponds to the point (1, 0). Moreover, the Unit Circle has a circumference of 2π , so the real number 2π also corresponds to the point (1, 0). Each real number t also corresponds to a central angle θ (in standard position) whose radian measure is t. With this interpretation of t, the arc length formula s = r θ (with r = 1) indicates that the real number t is the (directional) length of the arc intercepted by the angle θ , given in radians. Trigonometric functions can help you analyze the movement of an oscillating weight. For example, in Exercise 50 on page 276, you will analyze the displacement of an oscillating weight suspended by a spring using a model that is the product of a trigonometric function and an exponential function. © Richard Megna/Fundamental Photographs 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. 4.2 Trigonometric Functions: The Unit Circle 271 GO DIGITAL The Trigonometric Functions From the preceding discussion, it follows that the coordinates x and y are two functions of the real variable t. These coordinates are used to define the six trigonometric functions of a real number t. sine cosecant cosine secant tangent cotangent Abbreviations for these six functions are sin, csc, cos, sec, tan, and cot, respectively. In the definitions of the trigonometric functions, note that the tangent and secant are not defined when x = 0.
eBook - PDFSixth Form Pure Mathematics
Volume 1
- C. Plumpton, W. A. Tomkys(Authors)
- 2014(Publication Date)
- Pergamon(Publisher)
CHAPTER IV THE CIRCULAR FUNCTIONS 4.1 Definition of an Angle An angle is defined here as the amount of anti-clockwise rotation of a radius vector about an axis through one end of the radius, from a fixed initial position. Using degrees as units of angle measurement, Figs. 36 show angles of 210°, 440° (one revolution +80°) and -120° (clock-wise rotation). For convenience of reference, one revolution of the radius vector is divided into four quadrants as illustrated in Fig. 36 (i). -2nd Quodrant / / 3rd Quadront i 1st Quadront ■£ ► * 4 th Quadrant (i) 210° , c /r, cosine d=x/r, tangent d=y/x, cotangent 0= 1/tangent 0, secant 0 = 1/cosine 0, cosecant 6 = 1/sin 6. The names of these functions will be abbreviated as sin, cos, tan, cot, sec and cosec respectively. These definitions include the definitions of the functions for acute and obtuse angles which suffice for the numerical solution of triangles. Formulae and methods based on those definitions will remain valid with the new ones. Following directly from the defi-nitions : |sin0|^=l, | c o s 0 | ^ l , |sec 01=3*1, |cosec0|^l, and there is no limitation on the values of tan 0 and cot 0. Signs of the Circular Functions. It follows from the definitions above that the signs of the circular functions are as illustrated in Fig. 38 (i).
eBook - PDF- Sheldon Axler(Author)
- 2011(Publication Date)
- Wiley(Publisher)
Unfortunately, the coordinates of the endpoint The endpoint of the radius correspond- ing to π 18 radians is approximately (0.9848, 0.1736). of that radius do not have a nice form—neither coordinate is a rational number or even the square root of a rational number. The cosine and sine functions, which we are about to introduce, were invented to help us extend the table above to all angles. Before introducing the cosine and sine functions, we explain a common assumption about notation in trigonometry: Angles without units If no units are given for an angle, then assume that the units are radians. 530 chapter 9 Trigonometric Functions The figure below shows a radius of the Unit Circle corresponding to θ (here θ might be measured in either radians or degrees): cos Θ, sin Θ Θ 1 This figure defines the cosine and sine. The endpoint of this radius is used to define the cosine and sine, as follows: Cosine The cosine of θ, denoted cos θ, is the first coordinate of the endpoint of the radius of the Unit Circle corresponding to θ. Sine The sine of θ, denoted sin θ, is the second coordinate of the endpoint of the radius of the Unit Circle corresponding to θ. The two definitions above can be combined into a single statement, as follows: Cosine and sine The endpoint of the radius of the Unit Circle corresponding to θ has coordinates (cos θ, sin θ). example 1 Evaluate cos π 2 and sin π 2 . solution The radius corresponding to π 2 radians has endpoint (0, 1). Thus Here units are not specified for the angle π 2 . Thus we assume that we are deal- ing with π 2 radians. cos π 2 = 0 and sin π 2 = 1. Equivalently, using degrees instead of radians we could write cos 90 ◦ = 0 and sin 90 ◦ = 1. section 9.3 Cosine and Sine 531 The table below gives the cosine and sine of some special angles. This table Most calculators can be set to work in ei- ther radians or de- grees.
eBook - ePub- Mary Jane Sterling(Author)
- 2019(Publication Date)
- For Dummies(Publisher)
Part 2Trig Is the Key: Basic Review, the Unit Circle, and Graphs
IN THIS PART …You should be familiar with the basics of trigonometry from earlier math classes — right triangles, trig ratios, and angles, for example. But your Algebra II course may or may not have expanded on those ideas to prepare you for the direction that pre-calculus is going to take you. For this reason, it's assumed that you’ve never seen this stuff before. You won't be left behind when continuing your mathematics journey.This part begins with trig ratios and word problems and then moves on to the Unit Circle: how to build it and how to use it. You’ll some trig equations and make and measure arcs. Graphing trig functions is a major component of pre-calculus, so you'll see how to graph each of the six functions.
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Chapter 6
Basic Trigonometry and the Unit Circle
IN THIS CHAPTERWorking with the six trigonometric ratiosMaking use of right triangles to solve word problemsUsing the Unit Circle to find points, angles, and right triangle ratiosIsolating trig terms to solve trig equationsCalculating arc lengthsAh … trigonometry, the math of triangles! Invented by the ancient Greeks, trigonometry is used to solve problems in navigation, astronomy, and surveying. Think of a sailor lost at sea. All he has to do is triangulate his position against two other objects, such as two stars, and calculate his position using — you guessed it — trigonometry!In this chapter, the basics of right triangle trigonometry are reviewed. Then you see how to apply that knowledge to the Unit Circle, a very useful tool for graphically representing trigonometric ratios and relationships. From there, you can solve trig equations. Finally, these concepts are combined so that you can apply them to arcs. The ancient Greeks didn’t know what they started with trigonometry, but the modern applications are endless!
Index pages curate the most relevant extracts from our library of academic textbooks. They’ve been created using an in-house natural language model (NLM), each adding context and meaning to key research topics.
