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CHAPTER 2 Trigonometric Functions of a General Angle
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2.3 In what quadrants may P(x, y) be located if (a) x is positive and y (b) y is negative and x 0 0 (c) y/r is positive (d) r/x is negative (e) y/x is positive
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(a) In the first quadrant when y is positive and in the fourth quadrant when y is negative (b) In the fourth quadrant when x is positive and in the third quadrant when x is negative (c) In the first and second quadrants (d) In the second and third quadrants (e) In the first quadrant when both x and y are positive and in the third quadrant when both x and y are negative
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2.4 (a) Construct the following angles in standard position and determine those which are coterminal: 125 , 210 , 150 , 385 , 930 , 370 , 955 , 870 (b) Give five other angles coterminal with 125 .
(a) The angles in standard position are shown in Fig. 2.10. The angles 125 and 955 are coterminal since 955 125 3 360 (or since 125 955 3 360 ). The angles 210 , 150 , 930 , and 870 are coterminal since 150 210 1 360 , 930 210 2 360 , and 870 210 3 360 . From Fig. 2.10, it can be seen that there is only one first-quadrant angle, 385 , and only one fourth-quadrant angle, 370 , so these angles cannot be coterminal with any of the other angles. (b) Any angle coterminal with 125 can be written in the form 125 k 360 where k is an integer. Therefore, 485 125 1 360 , 845 125 2 360 , 235 125 1 360 , 595 125 2 360 , and 2395 125 7 360 are angles coterminal with 125 .
Fig. 2.10
2.5 State a positive angle and a negative angle coterminal with each radian-measure angle: (a) /6, k 360
(a) (c) 0 (d) (e) /6 2 17 /6 10 /3 2 2 2 4 4 (b) 5 /4
(b) 5 /4, k(2 radians)
13 /6; 2 ; 0 2 7 /6; 2 /3; 3 /2; 7 /2 /6
(c) 0, (d) 17 /6, (e) 2k where k is an integer
2 2 2 17 /6 10 /3 4 2 2 /2 5 /6 4 /3 11 /6 3 /4
10 /3,
(f) 7 /2
13 /4; 5 /4
(f) 7 /2
CHAPTER 2 Trigonometric Functions of a General Angle
2.6 Show that the values of the trigonometric functions of an angle u do not depend on the choice of the point P selected on the terminal side of the angle.
On the terminal side of each of the angles of Fig. 2.11, let P and P have coordinates as indicated and denote the distances OP and OP by r and r , respectively. Drop the perpendiculars AP and A P to the x axis. In each of the diagrams in Fig. 2.11, the triangles OAP and OA P , having sides a, b, r and a , b , r , respectively, are similar; thus, using Fig. 2.11(a), (1) b/r b /r a/r a /r b/a b /a a/b a /b r/a r /a r/b r /b
Since the ratios are the trigonometric ratios for the first-quadrant angle, the values of the functions of any first-quadrant angle are independent of the choice of P. From (1) and Fig. 2.11(b) it follows that b/r b /r a/r a /r b/ a b/ a a/b a /b r/ a r/ a r/b r /b
Since these are the trigonometric ratios for the second-quadrant angle, the values of the functions of any second-quadrant angle are independent of the choice of P. From (1) and Fig. 2.11 (c), it follows that b r br rr a r ar rr b a br ar a b ar br r a rr ar r b rr br
Since the ratios are the trigonometric ratios for the third-quadrant angle, the values of the functions of any third-quadrant angle are independent of the choice of P. From (1) and Fig. 2.11 (d), it follows that b r br rr a r ar rr b a br ar a b ar br r a rr ar r b rr br
Since the ratios are the trigonometric ratios for the fourth-quadrant angle, the values of the functions of any fourth-quadrant angle are independent of the choice of P.
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