Graphing Trigonometric Functions
Graphing trigonometric functions involves plotting the values of sine, cosine, tangent, and other trigonometric functions on a coordinate plane. These functions create periodic wave-like patterns due to their repetitive nature. Understanding the properties of these graphs, such as amplitude, period, and phase shift, is essential for analyzing and solving trigonometric equations and real-world problems.
5 Key excerpts on "Graphing Trigonometric Functions"
- Lawrence S. Leff, Christina Pawlowski-Polanish(Authors)
- 2021(Publication Date)
- Barrons Educational Services(Publisher)
...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. Notice the graphs have the same basic shape except that the cosine curve is out of phase with the sine curve. For both y = cos x and y = sin x : • period = 2π • amplitude = 1 • domain = (–∞,∞) • range = [–1, 1] FIGURE 10.3 FIGURE 10.4 FREQUENCY The frequency of a trigonometric function is the number of cycles that its graph completes in an interval of 2π radians. Because the sine curve and the cosine curve each complete one cycle every 2π radians, the frequency of each curve is 1. EXERCISE 1 If 0 ≤ x ≤ 2π, determine the interval on which the graph of y = sin x is decreasing and, at the same time, the graph of y = cos x is increasing. SOLUTION Sketch the graphs of y = sin x and y = cos x on the same set of axes, as shown in the accompanying figure. The sine curve is decreasing and the cosine curve is increasing on. AMPLITUDE AND PERIOD OF y = a sin bx AND y = a cos bx In the equations y = a sin bx and y = a cos bx, the number a affects the amplitude and the number b determines the period. For each of these functions: • The amplitude is | a |. For example, the maximum value of y = 2 sin x is +2 and its minimum value is −2, so the amplitude of y = 2 sin x is. Figure 10.5 compares the graphs of y = sin x, y = 2 sin x, and sin x over the interval 0 ≤ x ≤ 2π. • The period is. If y = cos 2 x, then b = 2, so the period is. Therefore, the graph of y = cos 2 x completes one full cycle in π radians. If, then, so the period is. Figure 10.6 compares the graphs of y = cos x, y = cos 2 x, and over the interval 0 ≤ x ≤ 2π...
eBook - ePubSTEM Education by Design
Opening Horizons of Possibility
- Brent Davis, Krista Francis, Sharon Friesen(Authors)
- 2019(Publication Date)
- Routledge(Publisher)
...Instead of the frequently heard complaint, “When are we going to use this?”, it was clear that class participants were aware that the mathematics they were studying was about their worlds. One student explained how he initially struggled with understanding trigonometry. He appreciated being able to take “the info that we found and turn it into a graph, to actually see it and be able to make your own graph rather than having one given to you, so you’re able see how the values affect the graph.” For him, “It helped a lot.” Another student was excited about how trigonometric functions are found in the world. Her experience with the tasks helped her recognize how these concepts might actually be relevant in her everyday life, “which is cool … it just deepens the understanding. It makes you want to be part of the learning process.” After groups of students had adequate time to grapple with their labs, they were invited to compare their observations and equations with the products of other groups working with other phenomena. It didn’t take long for them to realize that the phenomena at the different stations all had periodic cycles as well as maximum and minimum values. (These concepts are explored in the sidebars in this chapter.) Occurrences as diverse as spinning Ferris wheels, bouncing springs, rotating bicycle wheels, and swinging metronomes could all be represented with sinusoidal functions. That is, in this class, the mathematics was mainly about modeling, not calculating. Consolidating Key Points Use Google Image to do searches of “basic math,” “basics,” “school math,” “math,” and similar terms...
eBook - ePub- Mike Aitken, Bill Broadhurst, Stephen Hladky(Authors)
- 2009(Publication Date)
- Garland Science(Publisher)
...In science, these same functions describe at least approximately the oscillations of objects such as atoms in a molecule, the propagation of sound and light waves, and the hours of daylight throughout the year. Indeed, even those cycles that are not sinusoidal are often analyzed as sums of sine and cosine functions, an advanced topic called frequency analysis, which will not be considered in this book. The properties of the trigonometric functions are easiest to understand when they are considered as functions of an angle α. Note it is conventional to use Greek letters like α, β, γ, θ, and ϕ to stand for angles. We could use any symbols we like but it is easier to communicate if we use the symbols that others expect. The definitions of sin(α) and cos(α) when α is an angle are shown in Figure 4.10. This shows a circle with radius r with a line drawn from the center to the circle at angle α relative to the x axis. We can draw a right angle triangle with base x, height y, and hypotenuse r, as shown. The trigonometric functions, angle, sine, cosine, and tangent, are defined as a n g l e : α = arc length/ r, s i n e : sin (α) = y / r, c o s i n e : cos (α) = x / r, t a n g e n t : tan (α) = sin (α) / cos (α) = y / x. (EQ4.13) Figure 4.10 Definitions of the sine, cosine, and tangent functions when the argument, α, is an angle. sin(α) = y / r, cos(α) = x / r, and tan(α) = y / x. The line, whose length is y, is perpendicular to the x axis. The definitions are very simple – the hard part is actually finding accurate values for x / r and y / r when we know the angle, α. The formulae that enable us to calculate the trigonometric functions are complicated but fortunately these have been programmed into scientific calculators or used to print tables. All we need to do is look up the values. A short table is provided in Appendix 3...
- J. Rosebush, Stu Schwartz(Authors)
- 2016(Publication Date)
- Research & Education Association(Publisher)
...PART II FUNCTIONS, GRAPHS, AND LIMITS Chapter 2 Analysis of Graphs I. ANALYSIS OF GRAPHS A. Basic Functions—you need to know how to graph the following functions and any of their transformations by hand. 1. Polynomials, absolute value, square root functions 2. Trigonometric functions 3. Inverse trigonometric functions and their domain and range 4. Exponential and Natural Logarithmic functions 5. Rational functions 6. Piecewise functions 7. Circle Equations i. Upper semicircle with radius a and center at the origin:. This is a function. For example, ii. Lower semicircle with radius a and center at the origin:. This is a function. For example, iii. Circle with radius a and center at the origin: x 2 + y 2 = a 2. This is not a function since some x -values correspond to more than one y -value. For example, x 2 + y 2 = 9 iv. Circle with radius a and center at (b, c): (x – b) 2 + (y – c) 2 = a 2. This is not a function either. For example, (x – 2) 2 + (y + 3) 2 = 9 8. Summary of Basic Transformations of Functions A. Making changes to the equation of y = f (x) will result in changes in its graph. The following transformations occur most often. B. For trigonometric functions, f (x) = a sin(bx + c) + d or f (x) = a cos(bx + c) + d, a is the amplitude (half the height of the function), b is the frequency (the number of times that a full cycle occurs in a domain interval of 2 π units), is the horizontal shift and d is the vertical shift. Keep in Mind.... The reciprocal of sin(x),, is equivalent to csc(x), whereas sin –1 (x) is the inverse of sin(x), which is the reflection of sin(x) in the line y = x. When changing a function by adding a positive constant to x, the graph will shift to the left, not the right. The graph shifts to the right a units when a is subtracted from x. When graphing a function on the calculator (TI-83 or TI-84), make sure that all the plots are turned off; otherwise you risk getting an error and not being able to graph...
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...
