![]() ![]() Retrieved – via Canadian Center of Science and Education. "On the hidden beauty of trigonometric functions". On the Shoulders of Giants: The Great Works of Physics and Astronomy. ![]() Greenbelt, Maryland, US: NASA Goddard Space Flight Center. ^ a b Maor, Eli (1998), Trigonometric Delights, Princeton University Press, pp. 25–27, ISBN 978-0-4.The chord function can be related to the modern sine function, by taking one of the points to be (1,0), and the other point to be ( cos θ, sin θ), and then using the Pythagorean theorem to calculate the chord length: crd θ = ( 1 − cos θ ) 2 + sin 2 θ = 2 − 2 cos θ = 2 sin ( θ 2 ). o o We will now look at the first quadrant and find the coordinates where the terminal side of the 30o, 45o, and 60o angles intersects the unit circle. Since the unit circle has radius 1, these coordinates are easy to identify they are listed in the table below. The angle θ is taken in the positive sense and must lie in the interval 0 < θ ≤ π (radian measure). quadrantal angles intersects the unit circle. The chord of an angle is the length of the chord between two points on a unit circle separated by that central angle. The chord function is defined geometrically as shown in the picture. Ptolemy used a circle of diameter 120, and gave chord lengths accurate to two sexagesimal (base sixty) digits after the integer part. In the 2nd century AD, Ptolemy compiled a more extensive table of chords in his book on astronomy, giving the value of the chord for angles ranging from 1 / 2 to 180 degrees by increments of 1 / 2 degree. The first known trigonometric table, compiled by Hipparchus in the 2nd century BC, is no longer extant but tabulated the value of the chord function for every 7 + 1 / 2 degrees. In trigonometry Ĭhords were used extensively in the early development of trigonometry. How Does the Unit Circle Work The unit circle is a circle with its center at O ( 0, 0 ) and a radius of 1. The midpoints of a set of parallel chords of a conic are collinear ( midpoint theorem for conics). If the line extensions (secant lines) of chords AB and CD intersect at a point P, then their lengths satisfy AP.A chord that passes through the center of a circle is called a diameter and is the longest chord of that specific circle.Equal chords are subtended by equal angles from the center of the circle.Chords are equidistant from the center if and only if their lengths are equal.$\,1\,$ (inclusive), since it's a point on the unit circle.Among properties of chords of a circle are the following: Notice immediately that both coordinates are always numbers between $\,-1\,$ and The terminal point has coordinates: an $\,x\,$-value, and a $\,y\,$-value.The only thing that matters, for determining the values of the trigonometric functions, is the location of the terminal point! Once you have the terminal point, you can forget how you got there! This point is called the terminal point for the angle. Focus attention on the point where the terminal (ending) side of the angle intersects the unit circle. ![]() ![]() Let $\,\theta\,$ indicate the desired angle. Negative angles are swept out in a clockwise direction (start by going down).Positive angles are swept out in a counterclockwise direction (start by going up).Always start at the positive $\,x\,$-axis.The angle must be ‘laid out’ in the circle in a standard way: In the unit circle approach to trigonometry, angles can have any real number measure.The equation of this circle is $\,x^2 + y^2 = 1\.$ (‘Unit’ refers to the radius being equal to $\,1\.$ ) In trigonometry, this is called the ‘unit circle’. Circle, centered at the origin, of radius $\,1\.$ ![]()
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