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Unformatted text preview: (6/26/07) Section 0.5 Trigonometric and inverse trigonometric functions Overview: If all you know about a triangle is its three angles, you cannot determine the lengths of its sides. You need to know how large the triangle is. You can, however, find the ratios of the lengths of the sides. These ratios for right triangles are given by the trigonometric functions .These functions are also used in analytic geometry and calculus to find ratios of distances from angles that are not in triangles. The inverse trigonometric functions are needed to find angles from ratios of distances. The definitions, basic properties, and graphs of these functions are reviewed in this section. Topics: Radian measure Trigonometric functions and the unit circle Some trigonometric identities A direct definition of the tangent function Graphs of the secant, cosecant, and cotangent functions The inverse sine, cosine, tangent, and cotangent functions The Law of Cosines and the Law of Sines More trigonometric identities Radian measure In calculus and analytic geometry an angle is determined by rotating one of its sides (the terminal side ) from its other side (the initial side ). The angle is positive if the rotation is counterclockwise (Figure 1) and negative if the rotation is clockwise (Figure 2). In calculus, angles are usually measured in radians . The radian measure of a positive angle with its vertex at the center of a circle of radius 1 (Figure 3) equals the length of the arc of the circle it subtends (marks off). The radian measure of a negative angle equals the negative of the length of the arc. Initial side Terminal side Initial side Terminal side x x 1 A positive angle A negative angle FIGURE 1 FIGURE 2 FIGURE 3 1 p. 2 (6/26/07) Section 0.5, Trigonometric and inverse trigonometric functions An angle x with 0 x 2 with its vertex at the center of a circle of radius r subtends an arc of length rx (Figure 4). Moreover, because the sector of the circle inside the angle x is the fraction x 2 of the circle and the area of the circle is r 2 , the area of the sector is parenleftbigg x 2 parenrightbigg ( r 2 ) = 1 2 xr 2 . x rx r FIGURE 4 Arclength = rx Area = 1 2 xr 2 Since the circumference of a circle of radius 1 is 2 , a half counterclockwise revolution is radians, which corresponds to 180 degrees. Consequently, degrees can be converted to radians and radians to degrees with the formulas degrees = [ degrees] bracketleftbigg radians 180 degrees bracketrightbigg = parenleftbigg 180 parenrightbigg radians (1) x radians = [ x radians] bracketleftbigg 180 degrees radians bracketrightbigg = parenleftbigg 180 parenrightbigg x degrees . (2) The trigonometric functions The trigonometric functions are the cosine, sine, tangent, secant, cosecant , and cotangent and are denoted y = cos x, = sin x,y = tan x,y = sec x,y = csc x , and y = cot x . To define them, we place the vertex of an angle x at the origin of a...
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