Circular Functions
Circular functions are mathematical functions that are defined using the unit circle. The most common circular functions are the sine and cosine functions, which represent the y and x coordinates of points on the unit circle, respectively. These functions are widely used in technology and engineering for modeling periodic phenomena such as sound waves, alternating currents, and oscillations.
5 Key excerpts on "Circular Functions"
eBook - ePub- Mike Aitken, Bill Broadhurst, Stephen Hladky(Authors)
- 2009(Publication Date)
- Garland Science(Publisher)
...However, trigonometry and the trigonometric functions, like sines and cosines, are important to biologists even outside the area of structures, mechanical movements, and forces because they provide the simplest description of things that change in cycles. Cycles, regular repeated patterns, are very much a part of life. We cannot escape them; for a blatant example (see Figure 4.2) you only need to look up, is it light or dark outside. Many waves are also cyclic, notably sound waves, and light. Because cycles and waves are so important, the ability to describe them is an essential part of the equipment needed in our science. Figure 4.2 Number of hours of daylight in the southern UK. To understand the basic trigonometric functions we first need to know some basic geometry to do with circles, angles, and simple shapes made up of straight lines. After introducing these functions, we can return to oscillations and waves and the reasons why the trigonometric functions can be used to describe them. Figure 4.3 A circle indicating radii of length, r, from the center to the circle, an arc, and a diameter of length d = 2 r. 4.1 Circles and angles In geometry, a circle is very special: it is the only shape for which every point on the perimeter is the same distance from a single point called the center. A straight line from the center to the circle is called a radius. A straight line from the circle to the center extended until it hits the circle again is called a diameter. An arc is any portion of the circle and the arc length is the distance from one of its ends to the other: the distance you would walk along the circle if you started at one end of the arc and proceeded to the other. The distance all the way around is called the circumference. From antiquity it has been known that the ratio of circumference to diameter for all circles is always the same...
eBook - ePubSTEM Education by Design
Opening Horizons of Possibility
- Brent Davis, Krista Francis, Sharon Friesen(Authors)
- 2019(Publication Date)
- Routledge(Publisher)
...To appreciate this point, it’s useful to pause to think about things in life that repeat themselves in regular cycles. In everyday life, trigonometric functions are probably the most important examples of periodic functions, and perhaps the most familiar of those is the smoothly undulating sine curve, variations of which appear in many of the margins in this chapter. As illustrated in these margin figures, the sine curve can be used to model a great many phenomena, including changes in daylight hours throughout the year and across different latitudes, moon phases and position on the horizon, ocean waves, planetary motion, sound waves, and electromagnetic radiation. Like other mathematics concepts studied in school, this one is important not because it can be used to answer a textbook question about how high a 5 meter ladder will reach if it forms an 80° angle with the ground (although that’s entirely useful!), but because it affords insight into how, for example, making a sound is related to rocking in a boat, playing on a swing, or decreased daylight in winter. The phrasing here is important. Notice that we didn’t say that the sorts of phenomena highlighted in these margins are “examples of the sine function.” Because they aren’t. They are phenomena that can be modeled with the sine function – and there’s a big difference. The sine function isn’t lurking in planetary orbits or sound waves. It is a concept that enables humans to recognize, cluster, and study a particular sort of regularity in the universe. It is a modeling tool. Above and below are several familiar situations that are often modeled using the concept of multiplication. The vital point here is that mathematics is about humanity’s engagement with the world. Concepts are not mined from a mysterious, ideal realm, but are distilled from encounters with many different forms and events. Consider the more familiar concept of multiplication, for example...
- Lawrence S. Leff, Christina Pawlowski-Polanish(Authors)
- 2021(Publication Date)
- Barrons Educational Services(Publisher)
...Each complete pattern of values is a cycle. When θ varies from 0 to 2π, sin θ and cos θ take on values from −1 to +1. Since an angle formed by adding any multiple of 2π to θ is coterminal with θ, the same pattern of values for sin θ and cos θ repeats every 2π. For example: Sine and cosine are periodic functions with periods of 2π. The period of tangent is π. PERIOD AND AMPLITUDE A function f is periodic if there exists some number p for which f (x + p) = f (x) for all x. • If p is the smallest such positive number, p is called the period of the function. Figure 10.1 shows a periodic function whose period is 8 − 3 = 5, which is the length of the smallest interval of x -values that the function needs to complete one cycle. • The amplitude of a periodic function is one-half of the difference between the maximum and minimum values of the function. For the periodic function in Figure 10.1, the amplitude is. FIGURE 10.1 A periodic function SINE AND COSINE CURVES To discover the graphs of the sine and cosine curves requires going back to the unit circle and “unwrapping it” to plot the values of sine and cosine on the coordinate plane. As point P (x, y) moves counterclockwise in a unit circle, its position is determined by angle θ as shown in Figure 10.2. For any given θ, sin θ specifies the height (y) of P while cos θ gives its horizontal position (x). The process of creating the graph of y = sin x or y = cos x is based on associating θ with its respective sin θ or cos θ and writing them as points in the coordinate plane, where the x -coordinate represents θ and the y -coordinate represents its trigonometric value. Using the unit circle, you will be able to create a table of values for each trigonometric function to then translate onto their graph. FIGURE 10.2 Plotting the points from the table of values onto the coordinate plane gives the graphs of y = cos x and y = sin x, as shown in Figures 10.3 and 10.4...
eBook - ePub- John Bird(Author)
- 2019(Publication Date)
- Routledge(Publisher)
...Chapter 11 Introduction to trigonometry Why it is important to understand: Introduction to trigonometry Knowledge of angles and triangles is very important in engineering. Trigonometry is needed in surveying and architecture, for building structures/systems, designing bridges and solving scientific problems. Trigonometry is also used in electrical engineering: the functions that relate angles and side lengths in right angled triangles are useful in expressing how a.c. electric current varies with time. Engineers use triangles to determine how much force it will take to move along an incline, GPS satellite receivers use triangles to determine exactly where they are in relation to satellites orbiting hundreds of miles away. Whether you want to build a skateboard ramp, a stairway, or a bridge, you can’t escape trigonometry. At the end of this chapter, you should be able to: state the theorem of Pythagoras and use it to find the unknown side of a right angled triangle define sine, cosine and tangent of an angle in a right angled triangle evaluate trigonometric ratios of angles solve right angled triangles sketch sine, cosine and tangent waveforms state and use the sine rule state and use the cosine rule use various formulae to determine the area of any triangle apply the sine and cosine rules to solving practical trigonometric problems 11.1 Introduction Trigonometry is a subject that involves the measurement of sides and angles of triangles and their relationship with each other. There are many applications in engineering and science where a knowledge of trigonometry is needed. 11.2 The theorem of Pythagoras The theorem of Pythagoras * states: ‘In any right-angled triangle, the square on the hypotenuse is equal to the sum of the squares on the other two sides.’ Science and Mathematics for Engineering. 978-0-367-20475-4, © John Bird. Published by Taylor & Francis...
eBook - ePub- Stan Tempelaars(Author)
- 2014(Publication Date)
- Routledge(Publisher)
...Such a system is called a harmonic oscillator. The following chapter is devoted to this. A tuning fork that is not struck too hard is an example of a ‘natural’ harmonic oscillator. 2. It is possible to lay a link between non-sinusoidal vibrations and sinusoidal vibrations. This important reduction of arbitrary vibrations to sinusoidal vibrations is called Fourier analysis and is dealt with in chapter 4. 3. With an important group of systems that are called linear systems, sinusoidal vibrations are given a sort of preference treatment in the sense that the sinusoidal shape in these systems is not affected. In chapter 5 attention is paid to this ‘sine in/sine out’-principle. D. Trigonometric functions Because sinusoidal signal functions are so important in the theory of vibrations, it is necessary to be familiar with the mathematical characteristics of the sine functions. The following summary gives the most important rules for our applications (see also Szabo et al. 1974). 1. Definitions Figure 2.6.7 Rectangular triangle. In fig.2.6.7 you see again the triangle with the original definition of the sine: If this proportion is known then all other possible proportions between the sides of the triangle are set. In principle we can limit ourselves thus to the sine. Still, it is sometimes easy to give a name to some of the other proportions as well. This has been done among others with the two proportions For broader definitions of cos and tan use is made, just as for the sine, of a circle with radius 1. Have a look at the figure to the right and compare the definitions with the previous one. 2. Special cases For certain values of α the values for sine, cos and tan can be derived from the geometric characteristics. Check for yourself: Figure 2.6.8 Definition sine, cosine and tangent. 3...
