Graphing Sine and Cosine
Graphing sine and cosine functions is essential in understanding various mathematical relationships and their applications in everyday life. These functions are derived from right-angled triangles and are closely associated with circular motion, where the radius of a circle represents the hypotenuse. The sine function is defined as the ratio of the length of the opposite side to the hypotenuse, while the cosine function is the ratio of the adjacent side to the hypotenuse. Both functions can be expressed in radians or degrees, with a full circle measuring 360 degrees or 2π radians.
By varying the angle, θ, sine and cosine graphs produce oscillating patterns, reflecting their periodic nature. For instance, the cosine function reaches its maximum value of 1 when θ is 0 and its minimum value of -1 at π radians (180 degrees). Additionally, these functions can be transformed using specific parameters, impacting their amplitude, period, horizontal translation, and vertical shift. Understanding these transformations enriches our comprehension of the sine and cosine graphs and their significance across different branches of science and mathematics. Overall, graphing these functions provides valuable insights into the mathematical principles that govern various physical phenomena.
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Graphing Sine and Cosine
Sine and cosine functions are at the forefront of the most important types of mathematical relationships studied. There exists a link between virtually everything and sinusoidal functions. With its description in everything from household electronics to motion, the very nature of reality depends on the behavior of these graphs. Therefore, by analyzing and interpreting these types of graphs, we paint a better picture of the world around us.
In the early nineteenth century, a mathematician named Joseph Fourier made the discovery that sine and cosine waves may be used to approximate a general periodic waveform. This discovery led to revolutions in modern physics, with the knowledge that one can reconstruct a signal based on its very mathematical nature. Leonhard Euler also began to discover the significance of these relationships, which led him to the famous Euler’s identity, which relates exponential functions and complex numbers to trigonometry. It has thus proven that in every branch of mathematics and the natural science, these graphs make some form of appearance.
Overview
Sine and cosine functions are defined using right-angled triangles, which can be positioned around a circle, and where the radius of the circle represents the hypotenuse of the right-angled triangle. By definition, a circle is a set of lines from a single point (called the origin), such that each of the lines (radii) are of the same length.
The sine of angle θ is defined as the opposite side to the angle, divided by the hypotenuse. Quite similarly, the cosine of angle θ is defined as the adjacent side to the angle divided by the hypotenuse. The angle θ is most commonly measured in radians or degrees. Radians describe the exact position of a point around the circumference of a circle, whereas degrees are an approximation. However, since one degree is very small, there is generally no noticeable difference between the two. A circle will have 360 degrees or 2pi radians. These quantities, therefore, are equal to each other, which allows for a conversion factor between radians and degrees.
By varying the value of θ and recording the cosine or sine of an angle, an oscillating graph can be produced because the coordinates on a unit circle vary between. On the graph of a simple cosine wave, when θ is zero, the function is maximized since cos (0) = 1. Hence, distance from the center of the circle cannot exceed 1. Similarly, when theta is at pi radians or 180 degrees, the minimized value of is at –1.
As with any function, a sine and cosine graph may be transformed. Their transformation properties do, however, have special names because they often describe something useful. If f(x) is a sine and cosine function, then the transformed function g(x) = a(f(b(x – c)) + d with:
An amplitude of |a|
A period of
(radians)
A horizontal translation c
A vertical shift of d
A minimum value of d - |a|
A maximum value of d + |a|
Bibliography
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Euler, Leonard. Elements of Algebra. Trans. John Hewlett. New York, NY: Cambridge UP, 2009.
Posamentier, Alfred S, and Robert L. Bannister. Geometry, Its Elements and Structure. Mineola, NY: Dover, 2014.
Tiner, John Hudson. The World of Mathematics: From Ancient Record Keeping to the Latest Advances in Computers. Green Forest, AR: New Leaf, 2013.
Stankowski, James F., ed. Geometry and Trigonometry. New York: Rosen, 2015. Print.
Merzbach, Uta C., and Carl B. Boyer. A History of Mathematics. 3rd ed. Hoboken: Wiley, 2011.