70 : 0 : A B (0 ) (30 ) 50 1
1 4 1.1................................................ 5 1. A............................................... 6 1.3 B............................................... 7 8.1 A............................................... 9 3 ( 1) 11 3.1............................................. 1 3. A............................................... 13 3.3 B............................................... 14 3.4 A............................................... 15 3.5 B............................................... 16 4 ( ) 17 4.1............................................. 18 4. A............................................... 1 4.3 B............................................... 4.4 A............................................... 3 4.5 B............................................... 4 5 5 5.1 A............................................... 8 5. B............................................... 9 5.3 A............................................... 30 5.4 B............................................... 31 6 3 6.1 A............................................... 34 6. B............................................... 35 6.3 A............................................... 36 6.4 B............................................... 37 6.5 A............................................... 38 6.6 B............................................... 39 6.7 A............................................... 40 6.8 B............................................... 41 7 4 7.1............................................. 43 7. A............................................... 44 7.3 B............................................... 45 7.4 A............................................... 46 7.5 B............................................... 47 7.6 A............................................... 48
7.7 B............................................... 49 7.8 A............................................... 50 7.9 B............................................... 51 8 3 5 8.1 A............................................... 53 8. B............................................... 54 8.3 A............................................... 55 8.4 B............................................... 56 8.5 A............................................... 57 8.6 B............................................... 58 9 59 9.1 A............................................... 60 9. B............................................... 61 9.3 A............................................... 6 9.4 B............................................... 63 10 64 10.1 A............................................... 65 10. B............................................... 66 10.3 A............................................... 67 10.4 B............................................... 68 10.5 A............................................... 69 10.6 B............................................... 70 10.7 A............................................... 71 10.8 B............................................... 7 10.9 A............................................... 73 10.10 B............................................... 74 10.11 A............................................... 75 10.1 B............................................... 76 11 77 11.1 A............................................... 78 11. B............................................... 79 1 80 1.1.............................................. 80 1............................................... 80 1.3........................................ 81 1.4.......................................... 81 3
1 (1) α( ) θ( ) θ = π 180 α α = 180 π θ (1) ( ) () P O X O OP OP OX n : α + 360 n () : θ + πn (3) (3) S l r θ( ) l S : l = rθ (4) : S = 1 r θ = 1 rl (5) r ( ):θ 4
1.1 (1) π π 360 360 π α( ) θ( ) α : θ = 360 : π π θ = 180 α α = 180 π θ (3) ( ) r θ l S r πr r 360 360 π ( ) = ( ) l : πr = θ : π l = rθ S l r r πr ( ) = ( ) S : πr = θ : π S = 1 r θ 5
1. A A1-1 (1) 15 () 45 (3) π 8 (4) 5 1 π A1- OX OP (1) 30 () 60 (3) 780 (4) 410 A1-3 OP OP α + 360 n(n ) 0 α 360 (1) 480 () 315 (3) 810 (4) 930 A1-4 (1) 4 π () 1 150 4 6
1.3 B B1-1 (1) 30 () 60 (3) 7 (4) 40 (5) π 3 (6) 11 6 π (7) 7 1 π (8) B1- OX OP (1) 10 () 110 (3) 90 (4) 760 B1-3 OP OP α + 360 n(n ) 0 α 360 (1) 550 () 1190 (3) 400 (4) 1370 B1-4 (1) 5 5 π () 0 30 6 7
(1) y r 0 θ < 180 -r x O r x sin θ = y r cos θ = x r (6) (7) y tan θ = y x (8) -r () 1 (3) θ sin θ cos θ 1 sin θ 1 (9) 1 cos θ 1 (10) tan θ sin cos! (4) θ 0 0 < θ < π sin θ cos θ tan θ π π < θ < π π θ π < θ < 3 π 3 sin θ cos θ tan θ π 3 π < θ < π π 8
.1 A A-1 ( ( ) ) α 0 30 45 60 90 10 135 150 180 θ sin θ cos θ tan θ 9
A- ( ( ) ) α 180 10 5 40 70 300 315 330 360 θ sin θ cos θ tan θ 10
3 ( 1) 3 tan θ = sin θ cos θ (11) sin θ + cos θ = 1 (1) 1 + tan θ = 1 cos θ (13) (1) sin θ = 1 cos θ cos θ = 1 sin θ 11
3.1 (11) sin θ = y r cos θ = x r tan θ = y x x, y ( ) = sin θ cos θ = *1 tan θ x, y ( ) = tan θ = * ( ) ( ) (1) sin θ = y r, cos θ = x r ( ) x, y, r ( ) = *3 r, x, y *4 ( ) = *5 (13) (1) *6 *7 *1 y * x y x *3 x +y r *4 x + y = r *5 1 *6 cos θ *7 sin θ cos θ + 1 = 1 cos θ 1
3. A A3-1 (1) θ 3 sin θ = 3 5 cos θ tan θ () π < θ < π tan θ = 1 sin θ cos θ A3- (1) tan θ + 1 tan θ = 1 sin θ cos θ () (sin θ + cos θ) + (sin θ cos θ) = (3) tan θ + (1 tan 4 θ) cos θ = 1 13
3.3 B B3-1 (1) θ 4 cos θ = 5 sin θ tan θ 13 () 0 < θ < π tan θ = sin θ cos θ B3- (1) (tan θ + cos θ) (tan θ cos θ) = 4 sin θ cos θ () 1 + sin θ + cos θ 1 sin θ = cos θ 14
3.4 A A3-3 (1) sin θ + cos θ = 1 sin θ cos θ sin3 θ + cos 3 θ () sin θ cos θ = 1 sin θ cos θ sin3 θ cos 3 θ A3-4 θ (1) y = cos θ + sin θ 0 θ < π () y = tan θ + 4 tan θ + 5 π < θ < π 15
3.5 B B3-3 (1) sin θ + cos θ = 1 5 sin θ cos θ sin3 θ + cos 3 θ () sin θ cos θ = 1 5 sin θ cos θ sin3 θ cos 3 θ B3-4 θ (1) y = sin θ + 4 cos θ 0 θ < π () y = tan θ + tan θ + π < θ < π 16
4 ( ) Point (1) θ + nπ n θ + nπ θ 3 sin(θ + nπ) = sin θ (14) cos(θ + nπ) = cos θ (15) tan(θ + nπ) = tan θ (16) () θ sin( θ) = sin θ (17) cos( θ) = cos θ (18) tan( θ) = tan θ (19) (3) θ + π sin(θ + π) = sin θ (0) cos(θ + π) = cos θ (1) tan(θ + π) = tan θ () (4) θ + π sin(θ + π ) = cos θ (3) cos(θ + π ) = sin θ (4) tan(θ + π ) = 1 tan θ (5) 17
4.1 θ θ θ P, Q P, Q θ y ( P ( Q (a) (c),, (b) (d) ) ) -1 O 1 P Q -1 1 x P Q x (e) P Q x P y 1 Q y (f) tan( θ) sin( θ) cos( θ) tan( θ) = (g) (h) (e)(f) tan( θ) = (i) (j) = (k) 18
θ + π θ θ + π P, Q P, Q θ ( ) (l) (m) P, y 1 P(a,b) Q ( (n), (o) ) -1 O 1 x P, Q P (a, b) Q a, b Q -1 Q( (p), (q) ) ( (cos θ, sin θ) = (r), (s) ) (t) (u) tan(θ + π) tan(θ + π) = (v) (w) = (x) 19
θ + π θ θ + π P, Q P, Q θ ( ) (y) (z) P, Q y 1 P(a,b) Q ( (aa), (bb) ) -1 O 1 x OQ OP π P (a,b) Q ( (cc), (dd) ) (ee) -1 (ff) tan(θ + π ) tan(θ + π ) = (gg) (hh) = (ii) 1 0
4. A A4-1 sin 8 3 π cos 9 17 π tan 4 π A4- ( sin π ) 6 ( cos π ) ( tan π ) 4 3 A4-3 sin 7 6 π cos 7 6 π tan 7 6 π A4-4 sin 19 ( 6 π cos 15 ) 4 π tan 0 3 π 1
4.3 B B4-1 sin 7 4 14 13 π cos π tan 3 6 π B4- sin ( 34 ) π cos ( 56 ) π tan ( 3 ) π B4-3 sin 5 4 π cos 5 4 π tan 5 4 π B4-4 sin 11 ( 3 π cos 31 ) ( 6 π tan 5 ) 6 π
4.4 A A4-5 sin 17 13 π + cos 18 18 π + sin 7 9 π sin π 18 A4-6 ( (1) sin θ + π ) ( π ) cos (θ + π) + sin ( θ) cos ( θ () tan θ + tan θ + π ) ( π ) + tan θ + tan(π θ) 3
4.5 B B4-5 sin 13 6 π + tan 7 ( 6 π sin π ) + cos 5 3 4 π B4-6 ( π ) ( ) 3 (1) cos + θ sin(3π θ) sin π + θ cos(π θ) ( () cos θ + cos θ + π ) + cos(θ + π) + cos (θ + 3 ) π 4
5 (1) y = sin θ θ sin θ ( y = sin θ ) y π π 1 y 1 0 () y = cos θ θ cos θ ( y = cos θ ) y π y π 1 y 1 0 5
(3) y = tan θ θ tan θ ( y = tan θ ) y y π π : y θ 1 +α (θ = π 4 ) 6
(1) y = A sin θ y = sin θ θ y (a) ( A ) y 0 () y = sin(θ p) y = sin θ (b) (c) (d) y = (x p) y = x (3) y = sin aθ y = a sin y = sin θ y θ (e) y = cos θ y = tan θ 7
5.1 A A5-1 (1) y = sin θ () y = 1 cos θ (3) y = 3 tan θ A5- ( (1) y = sin θ π ) ( () y = cos θ π ) 6 ( (3) y = tan θ + π ) 6 8
5. B B5-1 (1) y = 3 sin θ () y = 3 cos θ (3) y = 1 3 tan θ B5- ( (1) y = sin θ π ) ( () y = cos θ + π ) 3 ( (3) y = tan θ + π ) 3 9
5.3 A A5-3 (1) y = sin 3θ () y = cos 1 θ (3) y = tan θ A5-4 ( (1) y = sin x + π ) ( () y = sin x + 1 (3) y = cos x (4) y = 3 sin x π ) + 3 3 3 30
5.4 B B5-3 (1) y = sin θ () y = cos 3 θ (3) y = tan 3θ B5-4 (1) y = cos (3θ + 3 ) ( π () y = cos x + 1 (3) y = sin x (4) y = sin θ π ) + 1 3 31
6! θ 0 θ 90 ABC ABC = θ, ACB = 90, AB = r, BC = x, AC = y ( ) sin α = y r (6) ( ) cos α = x r (7) -r P(x,y) x r y r y O r x ( ) tan α = y x (8) 0 θ < π θ r = 1 (6) (7) r = 1 y 1 sin θ = y 1 = y = (y ) (9) cos θ = x = x = (x ) (30) 1 (9) (30) sinθ y -1 O 1 x cosθ x -1 3
P (x, y) x θ y tan θ = y x (31) x x = 1 tan θ = (y ) (3) P O 1 x x = 1 y 33
6.1 A A6-1 0 θ < π θ 3 (1) sin θ = () cos θ = 1 A6-0 θ < π θ (1) sin θ < 1 () cos θ > 1 (3) cos θ 1 (4) sin θ 1 34
6. B B6-1 0 θ < π θ (1) cos θ = 1 () sin θ = 1 (3) cos θ = 3 () 4 sin θ = B6-0 θ < π θ 3 (1) sin θ () cos θ < 1 (3) sin θ 1 (4) cos θ > 0 35
6.3 A A6-3 0 θ < π θ ( (1) cos θ + π ) 3 = () sin θ = 1 ( (3) cos θ + π ) 3 = 4 3 A6-4 0 θ < π θ ( (1) sin θ π ) < 1 () cos θ ( 6 1 (3) sin θ + π ) > 1 6 36
6.4 B B6-3 0 θ < π θ ( (1) sin θ π ) = 1 () cos θ 6 = 1 ( (3) sin θ + π ) = 1 6 B6-4 0 θ < π θ ( (1) cos θ + π ) 3 < () sin θ 1 ( (3) cos θ + π ) 3 > 4 3 37
6.5 A A6-5 π < θ π (1) tan θ = 3 () tan θ = 1 (3) tan θ 3 (4) ( 3 < tan θ < 1 (5) tan θ + π ) > 3 4 38
6.6 B B6-5 π < θ π (1) tan θ = 1 () tan θ = 1 (3) 3 tan θ > 1 (4) 1 < tan θ < 3 ( 3 (5) tan θ π ) > 1 6 39
6.7 A A6-6 0 θ < π (1) sin θ + cos θ = 0 () sin θ 3 cos θ = 0 A6-7 0 θ < π (1) cos θ + 7 sin θ () sin θ 4 < 5 cos θ (3) sin θ < tan θ 40
6.8 B B6-6 0 θ < π (1) cos θ 3 sin θ 3 = 0 () *8 tan θ = sin θ B6-7 0 θ < π (1) sin θ 3 sin θ < 0 () cos θ sin θ + 1 (3) 3 tan θ + ( 3 + 1) tan θ + 1 0 *8 tan θ = sin θ cos θ 41
7 (1) sin(α + β) = sin α cos β + cos α sin β (33) sin(α β) = sin α cos β cos α sin β (34) cos(α + β) = cos α cos β sin α sin β (35) cos(α β) = cos α cos β + sin α sin β (36) () tan α + tan β tan(α + β) = 1 tan α tan β tan α tan β tan(α β) = 1 + tan α tan β (37) (38) 4
7.1 ( ) 43
7. A A7-1 (1) sin 75 () sin 105 (3) cos 15 A7-0 < α < π, 0 < β < π sin α = 3 5, cos β = 5 13 (1) sin(α + β) () cos(α + β) (3) sin(α β) (4) cos(α β) 44
7.3 B B7-1 (1)cos 75 ()sin 195 (3) cos 165 B7- π < α < π, 3 π < β < π sin α = 5 13, cos β = 4 5 (1) sin(α + β) () cos(α + β) (3) sin(α β) (4) cos(α β) 45
7.4 A A7-3 (1) tan 75 () tan 105 (3) tan 15 A7-4 (1) y = x + 1 y = 1 x 4 θ 3 () y = x 3 y = x + 1 θ tan θ 46
7.5 B B7-3 (1)tan 165 ()tan 195 (3) tan 55 B7-4 (1) y = 3x + 1 y = x + 3 θ () y = x 1 y = 3x + θ tan θ 47
7.6 A A7-5 α β γ tan α = 1 tan β = tan γ = 3 α + β + γ A7-6 sin x + cos y = 1 cos x + sin y = 1 4 sin(x + y) 48
7.7 B B7-5 A B C tan A = tan B = 4 tan C = 13 A + B + C B7-6 sin x sin y = 1 cos x + cos y = 1 3 cos(x + y) 49
7.8 A A7-7 ( π ) ( π ) (1) tan 4 + θ tan 4 θ () tan α + tan β sin(α + β) = tan α tan β sin(α β) = 4 tan θ 1 tan θ 50
7.9 B B7-7 (1) cos(x + y) sin(x y) = sin x cos x sin y cos y () cos(x + y) cos(x y) = cos x sin y 51
8 3 3 sin α = sin α cos α (39) cos α = cos α sin α = 1 sin α = cos α 1 (40) tan α = tan α 1 tan α (41) sin 3α = 3 sin α 4 sin 3 α (4) cos 3α = 3 cos α + 4 cos 3 α (43) 5
8.1 A A8-1 0 < α < π sin α = 3 5 sin α cos α tan α A8-0 θ < π θ (1) sin θ = cos θ () cos θ + 3 cos θ + = 0 53
8. B B8-1 π < α < π cos α = 1 sin α cos α tan α 3 B8-0 θ < π θ (1) sin θ = sin θ () cos θ = cos θ 1 54
8.3 A A8-3 (1) sin α = 1 sin 3α 3 () cos α = cos 3α HA8-4 θ = 18 (1) cos 3θ = sin θ () sin θ 55
8.4 B B8-3 (1) sin α = 1 sin 3α 3 () cos α = 1 cos 3α 3 HB8-4 θ = 36 (1) sin θ = sin 3θ () cos θ 56
8.5 A A8-5 0 x < π x y = cos x sin x 57
8.6 B B8-5 0 x < π x y = cos x + 4 cos x + 1 58
9 sin α = 1 cos α cos α = 1 + cos α tan α = 1 cos α 1 + cos α (44) (45) (46) t = tan θ ( θ (n + 1)π (n )) sin θ cos θ tan θ sin θ = t 1 + t (47) cos θ = 1 t 1 + t (48) tan θ = t 1 t (49) 59
9.1 A A9-1 sin.5 cos.5 tan.5 A9- π < α < π sin α = 3 5 sin α cos α 60
9. B B9-1 sin 11.5 cos 11.5 tan 11.5 B9-3 π < α < π sin α = 4 5 sin α cos α 61
9.3 A HA9-3 (1) tan θ = t sin θ cos θ tan θ t () sin θ + cos θ = tan θ 6
9.4 B HB9-3 (1) tan θ = t sin θ cos θ tan θ t () sin θ + cos θ = 1 5 tan θ 63
10 ( ) α sin α = a sin θ + b cos θ = a + b sin(θ + α) (50) b a + b cos α = a a + b ( ) α sin α = a sin θ + b cos θ = a + b cos(θ α) (51) a a + b cos α = b a + b 64
10.1 A A10-1 r sin(θ + α) r > 0 (1) sin θ + cos θ () 3 sin θ + cos θ (3) sin θ cos θ A10- r sin(θ + α) r > 0 (1) 5 sin θ + 3 cos θ () 3 sin θ + 4 cos θ (3) 5 sin θ cos θ 65
10. B B10-1 r sin(θ + α) r > 0 (1) sin θ + 3 cos θ () sin θ + cos θ (3) 3 sin θ 3 cos θ B10- r sin(θ + α) r > 0 (1) sin θ + cos θ () 4 sin θ 3 cos θ 66
10.3 A A10-3 0 x < π x (1) y = 6 sin x 3 cos x () y = 5 cos x + 1 sin x A10-4 sin θ + 3 cos θ r cos(θ α) r > 0 67
10.4 B B10-3 0 x < π x (1) y = 3 3 sin x + 3 cos x () y = cos x + 5 sin x B10-4 3 sin θ cos θ r cos(θ α) r > 0 68
10.5 A A10-5 y = 3 sin x sin x cos x + cos x (1) y sin x cos x () 0 x π y x 69
10.6 B B10-5 0 x π y = 5 cos x + 8 sin x cos x 3 sin x x 70
10.7 A A10-6 0 x π x ( y = sin x + sin x + π ) + sin (x + 3 ) 3 π 71
10.8 B B10-6 0 x π x ( (1) y = sin x cos x () y = cos x + sin x + π ) 6 7
10.9 A A10-7 y = (sin x + cos x) sin x cos x 1 y x 73
10.10 B B10-7 y = sin x cos x (sin x + cos x) y x 74
10.11 A A10-8 0 θ < π θ (1) sin θ cos θ = 1 () cos θ 3 sin θ + 1 = 0 A10-9 0 θ < π θ (1) 3 sin x + cos x < () sin x + cos x 1 75
10.1 B B10-8 0 θ < π θ (1) sin θ + cos θ = 1 () 3 sin θ cos θ 1 = 0 B10-9 0 θ < π θ (1) sin θ cos θ > 1 () sin θ + 3 cos θ + 3 0 76
11 sin α cos β = 1 {sin(α + β) + sin(α β)} (5) cos α sin β = 1 {sin(α + β) sin(α β)} (53) cos α cos β = 1 {cos(α + β) + cos(α β)} (54) sin α sin β = 1 {cos(α + β) cos(α β)} (55) sin A + sin B = sin A + B sin A sin B = cos A + B cos A + cos B = cos A + B cos A cos B = sin A + B cos A B sin A B cos A B sin A B (56) (57) (58) (59) 77
11.1 A A11-1 (1) sin 75 cos 15 () cos.5 cos 67.5 (3) sin 11.5 sin.5 A11- (1) sin 105 sin 15 () cos 75 + cos 15 78
11. B B11-1 (1) sin 105 cos 75 () cos 105 sin 15 (3) cos 8.5 cos 37.5 (4) sin 17.5 sin 5.5 B11- (1) sin 75 +sin 15 () sin 165 sin 105 (3) cos 105 +cos 165 (4) cos 195 cos 75 79
1 1.1 Point (1) sin(α + β) = sin α cos β + cos α sin β (60) sin(α β) = sin α cos β cos α sin β (61) cos(α + β) = cos α cos β sin α sin β (6) cos(α β) = cos α cos β + sin α sin β (63) () tan α + tan β tan(α + β) = 1 tan α tan β tan α tan β tan(α β) = 1 + tan α tan β (64) (65) 1. Point (3) sin α = sin α cos α (66) cos α = cos α sin α = 1 sin α = cos α 1 (67) tan α = tan α 1 tan α (68) (4) sin α = 1 cos α cos α = 1 + cos α tan α = 1 cos α 1 + cos α (69) (70) (71) (5)3 sin 3α = 3 sin α 4 sin 3 α (7) cos 3α = 3 cos α + 4 cos 3 α (73) 80
1.3 Point (1) sin α cos β = 1 {sin(α + β) + sin(α β)} (74) cos α sin β = 1 {sin(α + β) sin(α β)} (75) cos α cos β = 1 {cos(α + β) + cos(α β)} (76) sin α sin β = 1 {cos(α + β) cos(α β)} (77) () sin A + sin B = sin A + B sin A sin B = cos A + B cos A + cos B = cos A + B cos A cos B = sin A + B cos A B sin A B cos A B sin A B (78) (79) (80) (81) 1.4 Point (1) ( ) α sin α = a sin θ + b cos θ = a + b sin(θ + α) (8) b a + b cos α = a a + b () ( ) (1) α sin α = a sin θ + b cos θ = a + b cos(θ α) (83) a a + b cos α = b a + b 81