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CHAPTER 2 Trigonometric Functions of a General Angle
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y-coordinate is the directed distance AP OB. A point P with x-coordinate x and y-coordinate y will be denoted by P(x, y). The axes divide the plane into four parts, called quadrants, which are numbered (in a counterclockwise direction) I, II, III, and IV. The numbered quadrants, together with the signs of the coordinates of a point in each, are shown in Fig. 2.3. The undirected distance r of any point P(x, y) from the origin, called the distance of P or the radius vector of P, is given by r 2x2 y2
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Thus, with each point in the plane, we associate three numbers: x, y, and r. (See Probs. 2.1 to 2.3.)
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2.3 Angles in Standard Position
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With respect to a rectangular coordinate system, an angle is said to be in standard position when its vertex is at the origin and its initial side coincides with the positive x axis. An angle is said to be a first-quadrant angle or to be in the first quadrant if, when in standard position, its terminal side falls in that quadrant. Similar definitions hold for the other quadrants. For example, the angles 30 , 59 , and 330 are first-quadrant angles [see Fig. 2.4(a)]; 119 is a second-quadrant angle; 119 is a third-quadrant angle; and 10 and 710 are fourth-quadrant angles [see Fig. 2.4(b)].
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CHAPTER 2 Trigonometric Functions of a General Angle
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Two angles which, when placed in standard position, have coincident terminal sides are called coterminal angles. For example, 30 and 330 , and 10 and 710 are pairs of coterminal angles. There is an unlimited number of angles coterminal with a given angle. Coterminal angles for any given angle can be found by adding integer multiples of 360 to the degree measure of the given angle. (See Probs. 2.4 to 2.5.) The angles 0 , 90 , 180 , and 270 and all the angles coterminal with them are called quadrantal angles.
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Trigonometric Functions of a General Angle
Let u be an angle (not quadrantal) in standard position and let P(x, y) be any point, distinct from the origin, on the terminal side of the angle. The six trigonometric functions of u are defined, in terms of the x-coordinate, y-coordinate, and r (the distance of P from the origin), as follows: sine u cosine u tangent u sin u cos u tan u y-coordinate distance x-coordinate distance y-coordinate x-coordinate y r x r y x cotangent u secant u cosecant u cot u sec u csc u x-coordinate y-coordinate distance x-coordinate distance y-coordinate x y r x r y
As an immediate consequence of these definitions, we have the so-called reciprocal relations: sin cos 1/csc 1/sec tan cot 1/cot 1/tan sec csc 1/cos 1/sin
Because of these reciprocal relationships, one function in each pair of reciprocal trigonometric functions has been used more frequently than the other. The more frequently used trigonometric functions are sine, cosine, and tangent. It is evident from the diagrams in Fig. 2.5 that the values of the trigonometric functions of u change as u changes. In Prob. 2.6 it is shown that the values of the functions of a given angle u are independent of the choice of the point P on its terminal side.
Fig. 2.5
CHAPTER 2 Trigonometric Functions of a General Angle
2.5 Quadrant Signs of the Functions
Since r is always positive, the signs of the functions in the various quadrants depend on the signs of x and y. To determine these signs, one may visualize the angle in standard position or use some device as shown in Fig. 2.6 in which only the functions having positive signs are listed. (See Prob. 2.7.)
Fig. 2.6
When an angle is given, its trigonometric functions are uniquely determined. When, however, the value 1 of one function of an angle is given, the angle is not uniquely determined. For example, if sin u 2, then 30 , 150 , 390 , 510 , . . . . In general, two possible positions of the terminal side are found; for example, the terminal sides of 30 and 150 in the above illustration. The exceptions to this rule occur when the angle is quadrantal. (See Probs. 2.8 to 2.16.)
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