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CHAPTER 2 Trigonometric Functions of a General Angle
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(a) For u in quadrant I [Fig. 2.14(a)] we have x sin u y r 211 6
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(b) For u in quadrant IV [Fig. 2.14(b)] we have x sin u y r 211 6
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5, y and tan u
3 4.
211, and r y x
211 5
2.13 Find the values of sin u and cos u, given tan u
Since tan u y/x is negative, u is in quadrant II (take x 3). In either case r 216 9 5. y/r y/r 3/5 and cos u
4 and y x/r
3) or in quadrant IV (take x 4/5. x/r 4/5.
4 and
(a) For u in quadrant II [Fig. 2.15(a)], sin u (b) For u in quadrant IV [Fig. 2.15(b)], sin u
3/5 and cos u
Fig. 2.15
2.14 Find sin u, given cos u
and that tan u is positive.
Since cos u x/r is negative, x is negative. Since also tan u y/x is positive, y must be negative. Then u is in quadrant III. (See Fig. 2.16.) Take x 4 and r 5; then y 3/5. 252 ( 4)2 3. Thus, sin u y/r
Fig. 2.16
CHAPTER 2 Trigonometric Functions of a General Angle
2.15 Find the values of the remaining functions of u, given sin u
Since sin u y/r is positive, y is positive. Since cos u II. (See Fig. 2.17.)
23>2 and cos u
1/2.
x/r is negative, x is negative. Thus, u is in quadrant
Fig. 2.17
Taking x
1, y
23, and r 23 1
#( 1)2
( 23)2
2, we have 1 tan u 1 sin u 1 23 2 23 23 3 2 23 3
tan u sec u
y>x 1 cos u
23 2
cot u csc u
2.16 Determine the values of cos u and tan u if sin u
Since sin u is negative, u is in quadrant III or IV. (a) In quadrant III: Take y cos u x r m, r 2n2 n n, x m2 2n2 tan u
m/n, a negative fraction.
m2; then y x m 2n2 m2 m 2n2 m2 n2 m2
(b) In quadrant IV: Take y cos u x r
m, r 2n2
n, x m2
2n2 tan u
m2; then y x m 2n2 m2 m 2n2 m2 n2 m2
2.17 Determine the values of the trigonometric functions of (a) 0 , (b) 90 , (c) 180 , (d) 270
0; when u r. Let P be any point except the origin on the terminal side of u. When u 0 , x r and y 90 , x 0 and y r; when u 180 , x r and y 0; and when u 270 , x 0 and y (a) u sin 0 cos 0 tan 0 cot 0 sec 0 csc 0 0 ;x y/r x/r y/x x/y r/x r/y r, y 0/r r/r 0/r r/r 0 1 0 1 0 (b) u sin 90 cos 90 tan 90 cot 90 sec 90 csc 90 90 ; x y/r x/r y/x x/y r/x r/y 0, y r/r 0/r 0/r r/r 1 0 0 1 r [See Fig. 2.18(a).] [See Fig. 2.18(b).]
undefined undefined
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CHAPTER 2 Trigonometric Functions of a General Angle
(c) u sin 180 cos 180 tan 180 cot 180 sec 180 csc 180 180 ; x y/r x/r y/x x/y r/x r/y 0/r r/r 0/( r) undefined r/( r) undefined 1 r, y 0 1 0 0 (d) u sin 270 cos 270 tan 270 cot 270 sec 270 csc 270 270 ; x y/r x/r y/x x/y r/x r/y 0, y r/r 0/r 0 0 1 r 1
[See Fig. 2.18(c).]
[See Fig. 2.18(d).]
undefined 0/( r) undefined r/( r)
Fig. 2.18
2.18 Evaluate: (a) sin 0 2 cos 0 3 sin 90 4 cos 90 5 sec 0 6 csc 90 (b) sin 180 2 cos 180 3 sin 270 4 cos 270 5 sec 180 6 csc 270
(a) 0 (b) 0 2(1) 2( 1) 3(1) 4(0) 5(1) 4(0) 6(1) 5( 1) 16 6( 1) 6 3( 1)
2.19 Using a protractor, construct an angle of 20 in standard position. With O as center, describe an arc of radius 10 units intersecting the terminal side at point P. From P drop a perpendicular to the x axis, meeting it in A. By actual measurement, OA 9.4, AP 3.4, and P has coordinates (9.4, 3.4). Then find the trigonometric functions of 20 (see Fig. 2.19).
Fig. 2.19
sin 20 cos 20 tan 20
3.4/10 9.4/10 3.4/9.4
0.34 0.94 0.36
cot 20 sec 20 csc 20
9.4/3.4 10/9.4 10/3.4
2.8 1.1 2.9
CHAPTER 2 Trigonometric Functions of a General Angle
2.20 Obtain the trigonometric functions of 50 as in Prob. 2.19. Refer to Fig. 2.19.
By actual measurement, P, on the terminal side at a distance 10 units from the origin, has coordinates (6.4, 7.7). Then sin 50 7.7/10 0.77 cot 50 6.4/7.7 0.83 cos 50 tan 50 6.4/10 7.7/6.4 0.64 1.2 sec 50 csc 50 10/6.4 10/7.7 1.6 1.3
SUPPLEMENTARY PROBLEMS
2.21 State the quadrant in which each angle terminates and the signs of the sine, cosine, and tangent of each angle. (a) 125 , (b) 75 , (c) 320 , (d) 212 , (e) 460 , ( ) 750 , (g) 250 , (h) 1000 Ans. (a) II; , , ; (e) II; , , ; (b) I; (f) I; , , , , ; ; (c) IV; , , ; (g) II; , , ; (d) III; , , (h) I; , , ;
In what quadrant will u terminate if (a) (b) (c) (d) sin u and cos u are both positive cos u and tan u are both positive sin u and sec u are both negative cos u and cot u are both negative (a) I, (b) I, (c) III, (e) (f) (g) (h) (d) II, tan u is positive and sec u is negative tan u is negative and sec u is positive sin u is positive and cos u is negative sec u is positive and csc u is negative (e) III, ( ) IV, (g) II, (h) IV
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