3 () 3,,C = a, C = a, C = b, C = θ(0 < θ < π) cos θ = a + (a) b (a) = 5a b 4a b = 5a 4a cos θ b = a 5 4 cos θ a ( b > 0) C C l = a + a + a 5 4 cos θ = a(3 + 5 4 cos θ) C a l = 3 + 5 4 cos θ < cos θ < 4 < 3 + 5 4 cos θ < 6 6 < 3+ 5 4 cos θ < 4 () (3) 6 < a l < 4 log.78 = log 0 log 0.78 = log 0 0.300 = log 0 78 0 3 3.4343 3 = 0.693 n a = x, n b = y, c = n + (x + y) (n a b + c ) = {(n a ) + (n b ) + (x + y) } = { x y + (x + y) } = 4x y 4(n a )(n b ) = 4( x )( y ) = 4x y (n a b + c ) = 4(n a )(n b )
(4) P θ P 3 P O O = θ OP = a n P n OP n = a n {a n } a = θ, a n = a n (n ) {a n } θ a n = ( ) n θ P n O = a a + a 3 + ( ) n a n a a + a 3 + ( ) n a n = n ( ) k a k = k= n ( k ( ) ) k θ k= ) n = ( ( ) θ = { ( ) n } θ 3 n ( ) n 0 lim n { ( P n O = lim ) n } θ = n 3 3 θ n O
(5) I G D F K E H J C E F CD CD CD H E F G I E,F C,C J,K G = J, CJ = CH = CK, K = I, DI = DH = DG C C = J + JC K KC = J K D D = G + GD I ID = G I C C = D D D C C = D D (6) X P Q Y PY X P X QY Y P X Q = () 4,,X,Y 3,Q,Y,Q,X QX = QY, QX = QY QX QY () () Q : Q = X : Y () 3
QY P P X Q = QY P X = Q P QY : P X = Q : P QY = QX = P X QY P X QY = P X QY = QX QX = P X 4,,P,Q Q (7) θ c D a O b C CD a b O OD O CO COD DO = a, O = b OD S CD T S = a b = 8 ab T = 4S = ab CD c OD ( a ) ( b c = + ) c = a + b = l ab ( l = a + b) = l 4T T = 4 l c 4
D = θ (0 < θ < 80 ) DC = 80 θ DO = θ, DO = 80 θ OD = 90 OD O DO DO = c cos 80 θ, O = c cos θ l = c cos 80 θ + c cos θ { ( = c cos 45 cos 45 θ )} l c = cos ( ) 45 θ (8) l T = { 4 l 8 cos ( ) 45 θ = l cos ( 45 θ 8 cos ( ) 45 θ = l cos (90 θ) 8 cos ( ) 45 θ l sin θ = 8 cos ( ) 45 θ ) } I M L C E O D H F P P r C O s D O t C = r s, D = r t P O P H I PC PD P E F L M P H = a, HO = r, P E = x, P F = y CE = CL = r s, DF = DM = r t P E P L = P H P I P H P I = P F P M x(x + r s ) = y(y + r t ) = a(a + r) 5
P = P C C = (x + r s ) ( r s ) = x(x + r s ) = a(a + r) P = P D D = (y + r t ) ( r t ) = y(y + r t ) = a(a + r) P = P 6
4 ().435 a <.445,.735 b <.735 3.45 a + b < 3.47 () f(x) f(x) = px + qx + r (p, q, r p 0) 3 p = f(0) = r = c f() = p + q + r = c + f() = 4p + q + r = c + c(c + )(c + ), q = c + 3 c(c + )(c + ), r = c f(x) = c + 3 c(c + )(c + ) x c(c + )(c + ) x + c f (c + ) = c + c(c + ) c + 3 c(c + ) + c = c +
(3) C, C, C n r n O n OX,OY O XOO n+ = θ O n+ OX H n+ O n+ H n+ O n O n+ H n+ I n+ O n I n+ O n+ O n I n+ = θ (r n+ + r n ) sin θ = r n+ r n r n+ = r n + sin θ sin θ {r n } r +sin θ sin θ ( ) n + sin θ r n = r sin θ {r n } n S n (4) ( S n = r +sin θ sin θ +sin θ sin θ ) n = r {( + sin θ) n ( sin θ) n } sin θ( sin θ) n n r = sin θ( sin θ) n {( + sin θ) ( sin θ)} ( + sin θ) n k ( sin θ) k k= n ( ) n k + sin θ = r sin θ k= x 4 = a 4 b 4 3 a, b, x a a, b H H a P H= H P Q H H = P H HQ HQ = P H = a H = a b
x O.C CO OD = CO D O CD MN CD MN E,F EO = OF CO OD = EO OF EO = OF = x x P H a a a D b E O x x a x C Q F (5) G β α E β D α F H C E = CD = α, C = DF = β E G DF H DH = γ DC = DHC = CG = α, DC = β, CD = π α β, D = β γ, GD = GD = π α β+γ, C = β γ, G = β γ CG CD G = CD GE CDF E = CF F = E + EF, CE = CF + EF F = CE 3
(6) Q F H O P P Q O OP, OQ OF 4 O,P,Q,F OPFQ H OF P,Q, HP HQ P = Q Q sin(α β) = P sin β (7) Q sin α cos β Q cos α sin β = P sin β Q sin α = P tan β + Q cos α tan β Q tan β = P sin α + Q tan α Cotβ = P Cosecα + Cotα Q C F α P F C, C, C α C, C α F, F P 4
P = P F, P = P F. P, P P F + P F = P + P = = ( ) α 5