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Fig. 2.11
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2.7 Determine the signs of the functions sine, cosine, and tangent in each of the quadrants.
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sin u y/r. Since y is positive in quadrants I and II and negative in quadrants III and IV and r is always positive, sin u is positive in quadrants I and II and negative in quadrants III and IV.
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CHAPTER 2 Trigonometric Functions of a General Angle
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cos x/r. Since x is positive in quadrants I and IV and negative in quadrants II and III, cos is positive in quadrants I and IV and negative in quadrants II and III. tan y/x. Since x and y have the same signs in quadrants I and III and opposite signs in quadrants II and IV, tan is positive in quadrants I and III and negative in quadrants II and IV. (NOTE: The reciprocal of a trigonometric function has the same sign in each quadrant as the function.)
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2.8 Determine the values of the trigonometric functions of angle u (smallest positive angle in standard position) if P is a point on the terminal side of u and the coordinates of P are (a) P(3, 4),
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(a) r 232 y/r x/r y/x x/y r/x r/y
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(b) P( 3, 4),
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42 5 (b) r
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(c) P( 1,
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2( 3)2 4/5 3/5 4/( 3) 3/4 5/( 3) 5/4
42 5 (c) r 2( 1)2 ( 3)2 210 3 210> 10 210 >10 3 1>3 210 210>3 [See Fig. 2.12 (c).] sin u cos u 4/3 tan u cot u 5/3 sec u csc u 3> 210 1> 210 3>( 1) 1>( 3) 210>( 1) 210>( 3)
[See Fig. 2.12(a).] sin cos tan cot sec csc 4/5 3/5 4/3 3/4 5/3 5/4 sin cos tan cot sec csc
[See Fig. 2.12(b).]
Fig. 2.12
Note the reciprocal relationships. For example, in (b) sin u 1 csc u 4 5 cos u 1 sec u 3 5 tan u 1 cot u 4 3 etc.
Note in (c) the rationalizing of the denominators: sin u 3 210 1 210 3 # 210 210 210 # 210 210 210 1 3 210 10 210 10
cos u
Whenever the denominator of a fraction is an irrational number, an equivalent fraction with a rational denominator will be given as well.
2.9 In what quadrant will u terminate, if (a) sin and cos are both negative (b) sin and tan are both positive (c) sin is positive and sec is negative (d) sec is negative and tan is negative
CHAPTER 2 Trigonometric Functions of a General Angle
(a) Since sin y/r and cos third-quadrant angle.
x/r, both x and y are negative. (Recall that r is always positive.) Thus,
is a
(b) Since sin is positive, y is positive; since tan angle.
y/x is positive, x is also positive. Thus, is a first-quadrant
(c) Since sin is positive, y is positive; since sec is negative, x is negative. Thus, is a second-quadrant angle. (d) Since sec is negative, x is negative; since tan is negative, y is then positive. Thus, is a second-quadrant angle.
2.10 In what quadrants may u terminate if (a) sin is positive (b) cos is negative (c) tan is negative (d) sec is positive
is a first- or second-quadrant
(a) Since sin angle.
is positive, y is positive. Then x may be positive or negative, and
(b) Since cos is negative, x is negative. Then y may be positive or negative, and is a second- or third-quadrant angle. (c) Since tan is negative, either y is positive and x is negative, or y is negative and x is positive. Thus, be a second- or fourth-quadrant angle. (d) Since sec is positive, x is positive. Thus, may be a first- or fourth-quadrant angle.
8 17
2.11 Find the values of cos u and tan u, given sin u
and u in quadrant I.
y/r 8/17, we take y 8 and r 17. Since u is in
Let P be a point on the terminal line of u. Since sin u quadrant I, x is positive; thus x 2r2 y2 2(17)2
(8)2
To draw the figure, locate the point P(15, 8), join it to the origin, and indicate the angle u. (See Fig. 2.13.) cos u x r 15 17 and tan u y x 8 15 16,
The choice of y 8 and r 17 is one of convenience. Note that 8/17 16/34 and we might have taken y 34. Then x 30, cos u 30/34 15/17 and tan u 16/30 8/15.
(See Prob. 2.6.)
Fig. 2.13
2.12 Find the values of sin u and tan u, given cos u
Since cos u is positive, u is in quadrant I or IV. Since cos u x/r 5/6, we take x 5 and r
5 6.
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