Graphs of Trigonometric Functions
Graphs of trigonometric functions represent the relationships between angles and the values of trigonometric ratios such as sine, cosine, and tangent. These functions produce periodic wave-like patterns, with specific characteristics such as amplitude, period, and phase shift. Understanding these graphs is essential for analyzing and solving problems in fields like physics, engineering, and mathematics.
7 Key excerpts on "Graphs of 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 - 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...
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...
- Dale R. Patrick, Stephen W. Fardo(Authors)
- 2020(Publication Date)
- River Publishers(Publisher)
...Appendix A Trigonometric Functions Trigonometry is a very valuable form of mathematics for anyone who studies electricity/electronics. Trigonometry deals with angles and triangles, particularly the right triangle, which has one angle of 90°. An electronic example of a right triangle is shown in Figure A-1. This example illustrates how resistance, reactance, and impedance are related in AC circuits. We know that resistance (R) and reactance (X) are 90° apart, so their angle of intersection forms a right angle. We can use the law of right triangles, known as the Pythagorean theorem, to solve for the value of any side. This theorem states that in any right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides. With reference to Figure A-1, we can express the Pythagorean theorem mathematically as: Z 2 = R 2 + X 2 or, Z = R 2 + X 2 Figure A-1. Right triangle illustrating the trigonometric relationships between resistance, reactance, and impedance in AC circuits. By using trigonometric relationships, we can solve problems dealing with phase angles, power factor, and reactive power in AC circuits. The three most used trigonometric functions are the sine, the cosine, and the tangent. These functions show the ratios of the sides of the triangle, which determine the size of the angles. Figure A-2 illustrates how these ratios are expressed mathematically, and their values can be found. Figure A-2. Illustration of the trigonometric relationships of the sides of a right triangle to the angle θ. Trigonometric ratios hold true for angles of any size; however, we use angles in the first quadrant of a standard graph (0° to 90°) as a reference, and in order to solve for angles greater than 90° (second-, third-, and fourth-quadrant angles), we can convert them to first-quadrant angles (see Figure A-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)
...If the foot of the ladder is 2 m from the wall, calculate the height of the building 4. Determine the length x in Figure 11.24 Figure 11.24 5. A vertical tower stands on level ground. At a point 105 m from the foot of the tower the angle of elevation of the top is 19°. Find the height of the tower 11.6 Graphs of Trigonometric Functions By drawing up tables of values from 0° to 360°, graphs of y = sin A, y = cos A and y = tan A may be plotted. Values obtained with a calculator (correct to 3 decimal places - which is more than sufficient for plotting graphs), using 30° intervals, are shown below, with the respective graphs shown in Figure 11.25. Figure 11.25 (a) y = sin A (b) y = cos A (c) y = tan A From Figure 11.25 it is seen that: (i) Sine and cosine graphs oscillate between peak values of ±1 (ii) The cosine curve is the same shape as the sine curve but displaced by 90° (iii) The sine and cosine curves are continuous and they repeat at intervals of 360°; the tangent curve appears to be discontinuous and repeats at intervals of 180° 11.7 Sine and cosine rules To ‘solve a triangle’ means ‘to find the values of unknown sides and angles’. If a triangle is right-angled, trigonometric ratios and the theorem of Pythagoras may be used for its solution, as shown earlier. However, for a non-right-angled triangle, trigonometric ratios and Pythagoras’s theorem cannot be used. Instead, two rules, called the sine rule and the cosine rule, are used. 11.7.1 Sine rule With reference to triangle ABC of Figure 11.26, the sine rule states: a sin A = b sin B = c sin C Figure 11.26 The rule may be used only when: (i) 1 side and any 2 angles are initially given, or (ii) 2 sides and an angle (not the included angle) are initially given. 11.7.2 Cosine rule With reference to triangle ABC of Figure 11.26, the cosine...
eBook - ePub- Thomas Pfaff(Author)
- 2019(Publication Date)
- Chapman and Hall/CRC(Publisher)
...2 Functions and Their Graphs We define and use functions throughout this book, and in this chapter we focus on functions and their associated graphs. We provide examples of commonly used mathematical functions: piecewise, step, parametric, and polar. We broaden the use of function by using the geometric definition to obtain a parabola, discuss functions that return functions, and create a function that returns Pythagorean triples. In each case we graph the function; we keep our graphs basic leaving chapter 3 for further details on graphing, except for our last example where we create a checkerboard graph. The command for creating a function is function() {}, where the variable(s) are listed inside the parenthesis and the function is defined within the braces. In our first example, we define the function f to be x 2 sin (x) and evaluate it at 3 with f(3). There are a number of predefined functions such as abs, sqrt, the trigonometric functions, hyperbolic functions, log for the natural log, log10, log2, and the exponential function exp. So, for example, sin(x) is available to use in our definition of f. Note that * must be used for multiplication as we cannot simply juxtapose objects. R Code > f=function(x){x ^ 2*sin(x)} > f(3) [1] 1.27008 We can plot our function with curve. The first three arguments must be the function, the lower value for the independent variable, and the upper value for the dependent variable. The default range for the dependent variable is selected based on the minimum and maximum of the function on the given interval. There are numerous options, such as ylim for the y limits, lwd for the width (i.e., thickness) of the curve, col for the color of the curve, xlab and ylab for labeling the axis, and lty for the type of line (e.g., dashed, dotted). R Code > curve(f,-5,5) Our next two examples illustrate functions of two and three variables. The first returns the area of a rectangle given the length and width...
