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If you re used to measuring angles in degrees, the radian can seem unnatural at first Why, you might ask, would we want to divide a circle into an irrational number of angular parts Mathematicians do this because it nearly always works out more simply than the degree-measure scheme in algebra, geometry, trigonometry, pre-calculus, and calculus The radian is more natural than the degree, not less! We can define the radian in a circle without having to quote any numbers at all, just as we can define the diagonal of a square as the distance from one corner to the opposite corner The radian is a purely geometric unit The degree is contrived (What s so special about the fraction 1/360, anyhow To me, it would have made more sense if our distant ancestors had defined the degree as 1/100 of a circle)
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The measure of a certain angle q is p /6 What fraction of a complete circular rotation does this represent What is the measure of q in degrees
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A full circular rotation represents an angle of 2p The value p /6 is equal to 1/12 of 2p Therefore, the angle q represents 1/12 of a full circle In degree measure, that s 1/12 of 360 , which is 30
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Let s look again at the equation of a unit circle in the Cartesian xy plane We get it by adding the squares of the variables and setting the sum equal to 1: x2 + y2 = 1 Imagine that q is an angle whose vertex is at the origin, and we measure this angle in a counterclockwise sense from the x axis, as shown in Fig 2-1 Suppose this angle corresponds to a ray that intersects the unit circle at a point P, where P = (x0, y0) We can define the three basic circular functions, also called the primary circular functions, of q in a simple way But before we get into that, let s extend our notion of angles to include negative values, and also to deal with angles larger than 2p
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Offbeat angles In trigonometry, any direction angle, no matter how extreme, can always be reduced to something that s nonnegative but less than 2p Even if the ray OP in Fig 2-1 makes more than one complete revolution counterclockwise from the x axis, or if it turns clockwise instead, its
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A Fresh Look at Trigonometry
P ( x0, y0)
Each axis division is 1/4 unit
Unit circle
Figure 2-1 The unit circle, whose equation is
x2 + y2 = 1, can serve as the basis for defining trigonometric functions In this graph, each axis division represents 1/4 unit
direction can always be defined by some counterclockwise angle of least 0 but less than 2p relative to the x axis Think of this situation another way The point P must always be somewhere on the circle, no matter how many times or in what direction the ray OP rotates to end up in a particular position Every point on the circle corresponds to exactly one nonnegative angle less than 2p counterclockwise from the x axis Conversely, if we consider the continuous range of angles going counterclockwise over the half-open interval [0,2p), we can account for every point on the circle Any offbeat direction angle such as 9p /4 can be reduced to a direction angle that measures at least 0 but less than 2p by adding or subtracting some whole-number multiple of 2p But we must be careful about this A direction angle specifies orientation only The orientation of the ray OP is the same for an angle of 3p as for an angle of p, but the larger value carries with it the idea that the ray (also called a vector) OP has rotated one and a half times around, while the smaller angle implies that it has undergone only half of a rotation For our purposes now, this doesn t matter But in some disciplines and situations, it does! Negative angles are encountered in trigonometry, especially in graphs of functions Multiple revolutions of objects are important in physics and engineering So if you ever hear or read about an angle such as p /2 or 5p, you can be confident that it has meaning The negative value indicates clockwise rotation An angle larger than 2p indicates more than one complete rotation counterclockwise An angle of less than 2p indicates more than one complete rotation clockwise
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