121616949-math.78 - circle In the figure x is the standard...

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64 Chapter 4 Transcendental Functions To define the radian measurement system, we consider the unit circle in the xy -plane: . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ............................................................................................ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ............................................................ x (cos x, sin x ) y A B An angle, x , at the center of the circle is associated with an arc of the circle which it is said to subtend. In the figure, this arc is the portion of the circle from point (1 , 0) to point A . The length of this arc is the radian measure of the angle x ; the fact that the radian measure is an actual geometric length is largely responsible for the usefulness of radian measure. The circumference of the unit circle is 2 πr = 2 π (1) = 2 π , so the radian measure of the full circular angle (that is, of the 360 degree angle) is 2 π . While an angle with a particular measure can appear anywhere around the circle, we need a fixed, conventional location so that we can use the coordinate system to define properties of the angle. The standard convention is to place the starting radius for the angle on the positive x -axis, and to measure positive angles counterclockwise around the
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Unformatted text preview: circle. In the figure, x is the standard location of the angle π/ 6, that is, the length of the arc from (1 , 0) to A is π/ 6. The angle y in the picture is-π/ 6, because the distance from (1 , 0) to B along the circle is also π/ 6, but in a clockwise direction. Now the fundamental trigonometric definitions are: the cosine of x and the sine of x are the first and second coordinates of the point A , as indicated in the figure. The angle x shown can be viewed as an angle of a right triangle, meaning the usual triangle definitions of the sine and cosine also make sense. Since the hypotenuse of the triangle is 1, the “side opposite over hypotenuse” definition of the sine is the second coordinate of point A over 1, which is just the second coordinate; in other words, both methods give the same value for the sine. The simple triangle definitions work only for angles that can “fit” in a right triangle, namely, angles between 0 and π/ 2. The coordinate definitions, on the other hand, apply...
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