さくらの個別指導 ( さくら教育研究所 ) A 2 2 Q ABC 2 1 BC AB, AC AB, BC AC 1 B BC AB = QR PQ = 1 2 AC AB = PR 3 PQ = 2 BC AC = QR PR = 1

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1 Q,, = QR PQ = = PR PQ = = QR PR = P 0 0 R 5

2 6 θ r xy r y y r, x r, y x θ x θ θ (sine) (cosine) (tangent) sin θ, cos θ, tan θ. θ sin θ = = 5 cos θ = = 4 5 tan θ = = 4 θ 5 4 sin θ = y r cos θ = x r tan θ = y x θ r y θ r x θ x y. θ sin θ = = 5 cos θ = = 5 tan θ = = 5 θ

3 .. 7. sin θcos θtan θ () θ 0 () 5 θ sin 0 = cos 45 = 60 tan 60 = = () cos 0 tan 0 () sin 45 tan 45 () sin 60 cos 60

4 8 sin θcos θtan θ θ θ 84 sin θ cos θ tan θ θ sin 5 = 0.46, cos 8 = tan = () sin () cos 48 () tan 75. θ cos θ = 4 5 = 0.8 cos θ 0.8 θ θ 7 θ 5 4 a b a b.4 θ () () θ 5 θ

5 .. 9 D sin θ = y r r θ y = r sin θ x y x = r cos θ y = x tan θ θ r y = r sin θ y r θ x x = r cos θ θ x y = x tan θ y.4 = sin 6 = tan sincostan = 6 = 54 = 54

6 m m m = sin 0 00m = = 4. 4m.6. m m. 0m.6m = tan = =.89.8 D D = =5.4 () 5.4m E.6m.7 0m 40.6m 0m D

7 .... sin θcos θtan θ x = r cos θ, y = r sin θ r y tan θ = y x = sin θ cos θ x + y = r θ x (cos θ) + (sin θ) = ( x r ) + ( y r ) = (cos θ) ( ) sin θ + = sin θ cos θ (cos θ) cos θ = tan θ tan θ = sin θ cos θ sin θ + cos θ = + tan θ = cos θ (sin θ) (cos θ) (tan θ) sin θcos θtan θ

8 . sin θ = cos θ tan θ θ sin θ + cos θ = cos θ = sin θ = ( ) = 5 9 cos θ > cos θ = 9 = tan θ = sin θ cos θ = 5 = = 5 5 sin θ = θ.8 cos θ = sin θ tan θ θ

9 ... tan θ = cos θ sin θ θ + tan θ = cos θ tan θ = cos θ = + tan θ = + = 5 cos θ > 0 cos θ = 5 = 5 sin θ = tan θ cos θ = = 5 5 θ.9 tan θ = cos θ sin θ θ

10 4 90 θ sin α = x r = cos θ α r cos α = y y r = sin θ tan α tan θ = x y y θ x = x α = 90 θ θ 90 θ sin(90 θ) = cos θ cos(90 θ) = sin θ tan(90 θ) = tan θ.5 () 5 = 90 7 sin 5 = cos = 90 () 80 = 90 0 cos 80 = sin = 90 () 75 = 90 5 tan 75 = tan = () sin 64 () cos 78 () tan 8. () sin 6 = cos(90 ) () cos 78 = sin(90 )

11 < θ < 90 θ θ 0 θ 80 y r y P(x, y) O r r x θ 0 θ 80 θ r O x r x OP = θ P P (x, y) y θ 0 < θ < 90 r 6 P(x, y) y r sin θ = y r, cos θ = x r, tan θ = y θ x r x O r x θ P 0 (r, 0) sin 0 = 0 cos 0 = tan 0 = 0 90 (0, r) sin 90 = cos 90 = 0 80 ( r, 0) sin 80 = 0 cos 80 = tan 80 = 0 θ = 90 x = 0 tan θ.6 0 OP = 0 r = P (, ) x = y = P y r 0 sin 0 = y r = cos 0 = x r = = tan 0 = y x = = O 60 x

12 6. () 5 () 50 r = r = P y P y P r 5 P r 50 O x O x θ sin θ θ 0 < θ < 90 cos θ tan θ θ 90 < θ < 80

13 θ y r OP = θ P(x, y) P(x, y) OP = 80 θ r y P Q Q ( x, y) O 80 θ 80 θ y θ x sin(80 θ) = y r = sin θ cos(80 θ) = x r = cos θ tan(80 θ) = y x = tan θ P(x, y) r O Q( x, y) r x 80 θ 0 θ θ sin(80 θ) = sin θ cos(80 θ) = cos θ tan(80 θ) = tan θ + = 80 sin = sin cos = cos tan = tan.7 () sin 0 = sin = 80 () cos 5 = cos = 80 () tan 50 = tan = 80. () sin 40 () cos 56 () tan 00

14 8 θ θ θ.8 () 0 θ 80 sin θ = θ y θ () OP OQ θ = 0, = 80 0 () 0 θ 80 cos θ = θ x θ () OP θ = 5 5 = () () y y P Q P O x O x Q P 0 0 O P 45 O

15 .4 0 θ 80 θ.. 9 () sin θ = () cos θ =.9 0 θ 80 tan θ = θ y = P θ OP 0 θ = 0 60 O x.5 0 θ 80 θ () tan θ = () tan θ =

16 40 D y OP = θ P(x, y) 5 r = xy P y x = cos θ, y = sin θ x O x x + y = 0 θ 80 θ tan θ = sin θ cos θ sin θ + cos θ = + tan θ = cos θ.4 0 θ 80 cos θ = sin θ tan θ sin θ + cos θ = sin θ = cos θ = sin θ > 0 8 sin θ = 9 = tan θ = sin θ cos θ = = ( ) = 8 9 ( ) ( ) = θ

17 θ 80 sin θcos θtan θ () cos θ = 4 5 () tan θ = θ sin θ cos θ 0 tan θ 0 0

18 4..4 θ 8cm θ 8cm 0 DE () E E H D () D H

19 .. 4 sin θ = 4 cos θtan θ () 0 < θ < 90 () 90 < θ < 80 () 6. () 5.4 () cos θ = 5 4 tan θ = 5 () cos θ = 5 4 tan θ = 5

20 44... abc c abc sin sin sin a b R 0 < < 90 D D D = = D = 90 D = R R sin = D = a R a = < < 80 90

21 .. 45 = 90 R a = sin 90 = sin = a R R 90 < < 80 D D D = D 0 < D < 90 a D D sin D = R D D +D = +D = 80 ( sin D = sin ) = sin sin = a R 80

22 46 R a = R sin b sin = R, c sin = R θ sin θ = R R a sin = b sin = c sin = R a : sin = b : sin = c : sin a : b : c = sin : sin : sin.0 0 R 0 sin 60 = R 60 0 R = 0 sin 60 = 0 0 = 0 = 5.77

23 R () a = 5 = 45 () b = = 0.8 c = 0 R = 0 a sin = b sin, b sin = c sin, a sin = c sin.5 = 45 = 60 b = 6 a a sin = b sin 6 a a sin 60 = 6 sin 45 a = 6 6 = a =

24 48.9 () a = = 0 = 45 b () b = 4 = 45 = 60 c () c = = 0 = 0 a

25 D [] [] b a b a c D c D [][] = D + D D = (b sin ) D = (c b cos ) [] D = b cos c a a = (b sin ) + (c b cos ) = b sin + c bc cos + b cos = b (sin + cos ) + c bc cos sin + cos = = b + c bc cos.0 [] = D + D, D = (b sin ), D = (c b cos ) [] D c b a

26 50 a = b + c bc cos b = c + a ca cos c = a + b ab cos θ = + cos θ.6 b = c = 5 = 0 a a = b + c bc cos = cos 0 ( = ) = 49 0 a 5 a > 0 a = 7

27 .. 5. () b = 4c = 5 = 60 a () a = c = = 45 b () a = b = = 50 c

28 5 cos = b + c a bc, cos = c + a b, cos = a + b c ca ab.7 a = b = c = 7 cos cos = a + b c ab = + ( 7 ) = 6 = cos = = () a = 7b = c = 8 cos

29 .. 5 () a = b = 5c = cos. cos () a = 9b = 4 c = 7 () a = 7b = 6c = () a = 0b = 4c = 4

30 54... a = b = + = 60 c = 80 ( + ) c = + ( + ) ( + ) cos 60 = 4 + ( + + ) 4( + ) = 6 c > 0 c = 6 6 sin = sin 60 sin = 6 = + = 0 < 0 + c 60 =45 = 5 =80 ( ) = 75 () c = 6 = 45 = 75

31 a = c = + = 45 b + 45

32 56. sin : sin : sin = 7 : 5 : 46 abc a : b : c = sin : sin : sin a : b : c = 7 : 5 : a = 7b = 5c = cos = = 5 0 = = 0 5 a : b : c = 7 : 5 : a = 7b = 5c =.5 sin : sin : sin = 8 : 7 : 7

33 P P 4 P 70mP 50m 70m 50m 4 P cos 4 = P = P + P P P cos 4 = = = 597 > 0 = () 40m.6 P P 94 P 50mP 0m 50m 94 0m P

34 m P P = 60, P = P H P 00m PH PH = P sin 0 P P P = 80 ( ) = 45 P P sin 60 = 00 sin 45 P = 00 sin 60 sin 45 = 00 = m PH = P sin 0 = 00 6 = P H () 50 6 m 50 6 =.47.5m.7 00m P H P H = 60, H = H P 0 00m 75 0 P H PH

35 D D M M 4 H D () cos M H M () H () M = = 4 M = M = sin 60 () M = M = sin 60 = 4 = M cos M = + M M M = 4 + ( ) ( ) = 4 4 H M D ( ) 6 () sin M = = = 6 H = sin M = 4 = 4 6 H D D

36 () cos M 4 D H M () MH..4 4 D = 60, = 5, D = D 5 60 () D ()

37 a : b = 7 : = 60 sin 6 = 5, D = 4, E = D-EFGH cos HF 4 E D 5 H F G 4 () 7 () [ ]

38 6... = h h = b sin S S = c b sin b c h xy θ S x S = S xy sin θ θ y S S = bc sin, S = ca sin, b a S = ab sin. a = b = 4 = 0 S S = ab sin 4 = sin c = 4 =

39 .9 S.. 6 () b = 0c = 8 = 45 () a = 6c = 5 = 50 () 4.8 a = 7b = 8c = 9 () cos () sin () S () cos = = = 8 7 () sin > 0 ( ) 5 sin = = = () S = bc sin = = 5

40 64.0 S () a = 5b = 7c = 8 () a = b = 4c = 5... : a h h h a h a a S S S = ah, S = a h = ah S : S = :

41 : S S S : S k : k : ( ) ( ) k : k : k : k : m : n m : n. DE : E = 5 : D DE : DE = 5 : = 5 : 9 E DE : DE

42 66.9 O P P Q Q () Q O () P Q () P Q () = H H = OH tan 0 = = O 0 H Q = = 4 () P P Q : 4 = : () P Q ( ) : = : 4. D 60 () D : D D () D :

43 .. 67 O D D O : O O : O O : O OD : OD 4. () () () (4) (5) (6) :

44 68.4 D D D : h S : : D : S h S h D V D V h S V = Sh, V = S h = Sh V : V = :.4 P Q k : P Q k : k : P Q

45 .. 69 k : k : k : m : n m : n m : n.0 P 0cm Q P 0cm 0cm 0cm () P Q () P Q () P Q Q 0 0 = 0 (cm) 0 : 0 = : : = 9 : 4 () : = 7 : 8.5 P Q () P Q () P Q

46 70.. D V D H H h H = H = DH = h h H D H sin 60 = H H = sin 60 = h = H = ( ) = H H D D D S S = sin 60 = 4 V = Sh = 4 = 60.6 a a

47 .. 7 S h V V = Sh () h S S h S S 5 () 5 III

48 7.5 P 5cmcm Q P 4cm P P Q π 5 π = π 4 (cm ) Q P Q r V V = 4 πr

49 .. 7 r P Q r x P Q Q πr r πr r = πr P r V P V = πr = 4 πr P Q x r r x x r r r x r x π(r x ) πr πx = π(r x )

50 74. 4 π = 4 π π = π 4 π : π = 4 : 6 = :.8.7 cm 5cm cm 4 5cm 4 cm 4 π 4 = 44π (cm ) 4 5cm π 5 = 500 π (cm ) 500 > 44 5cm.9 P Q P 0cmQ 5cm P Q 7

51 .. 75 D r S S = 4πr. 4 4π = 6π π 4 = 6π V V = = ( ) ) = ( ) ( ) = Sr V = 4 πr S = 4πr

52 76 E.8 45 S S S : S r 4 S r S = + + = 6r 5r r 4r 5 S = 4 = 6 6r = 6 r = S = S + r( ) = 6 + = 6 S = 4πr = 4π = 4π S : S = 6 : 4π = 9 : π.4.8 V V V : V = S : S

53 S : S..4 7 () = 6D = 4 = 60 D ()

54 78 8 = D = E = D D-EFGH DE H DE E F G 9 r cm r cm h cm cm h cm 7 () () 8 7 [ cos ED = ] 5 7, sin ED = πr h cm

55 P P 45 0m P 60 P H () H () 0m H 5 D D D = 75 = () () D

56 80 D D// = = 4D = D 7D = D D H P = P P H = () PH H P D () cos P

57 = 60, a : b = :, c = () sin () b 6 b = c = = 0 7 D D = 5, = 7, D = 7, D = θ = θ () cos θ () D S

58 8 8 D O () OD V O D () r () 5 () b 6 sin 7 () D 8 () D = 4V () D r = V

59 .4. 8 () 5 ( + ) m () 5 ( + ) m [() H = x PH = x tan 60, PH = (x + 0) tan 45 ] () 5 () 5 [() () = 90 D = 45 ] 5 [ D E E cos = ] 4 () () 4 [ () PH= x P = PH + H H = H = ] () cos P = P + P P P 5 () () + 4 [() () a = b (b) = b +6 b 6 cos 60 ] 6 a = 4 = 90 = 60 a = = 0 = 0 [ ] sin = 7 () () 6 [() D D = 4 0 cos θ D D = cos θ] () () () 48 6 π 6 π [ ] () D () D 4

60 .5 θ sin θ cos θ tan θ θ sin θ cos θ tan θ

(1) θ a = 5(cm) θ c = 4(cm) b = 3(cm) (2) ABC A A BC AD 10cm BC B D C 99 (1) A B 10m O AOB 37 sin 37 = cos 37 = tan 37

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