2009 I 2 II III 14, 15, α β α β l 0 l l l l γ (1) γ = αβ (2) α β n n cos 2k n n π sin 2k n π k=1 k=1 3. a 0, a 1,..., a n α a

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1 009 I II III 4, 5, α β α β l 0 l l l l γ ) γ αβ ) α β. n n cos k n n π sin k n π k k 3. a 0, a,..., a n α a 0 + a x + a x + + a n x n 0 ᾱ 4. [a, b] f y fx) y x 5. ) Arcsin 4) Arccos ) ) Arcsin ) 3) Arccos 3 5) Arctan 3 6) Arctan 3 ) 6. n ) tan x nπ π/, nπ + π/) f n x) f n x) fn x) Arctan x ) sin x [nπ π/, nπ + π/] g n x) cos x [nπ, n + )π] h n x) g n x) x) h n x) h x) Arcsin x g n n

2 7. ) cos Arcsin x) ) tan Arcsin x) < x < ) 3) sin Arctan x) 4) sin Arctan x) 8. Arctan tan x ) π 4 Arctancos x) ) ) 9. ) cosh x sinh x ) tanh x cosh x 3) sinhx + y) sinh x cosh y + cosh x sinh y 4) coshx + y) cosh x cosh y + sinh x sinh y 5) tanhx + y) tanh x + tanh y + tanh x tanh y 0. ) cosh x, sinh x, tanh x cosh x x 0 x 0 ) ). ) tanh x ) tan x) x 3) Arcsincos x) 4) Arctan x + x

3 009 I II III 4, 5, ,, α β 0, β, αβ β β β 0,, α 0, β, αβ 0,, β 0, α, αβ α 0 αβ β γ : 0α 0βαβ) 0β 0ααβ) γ αβ 0, β, αβ 0, α, αβ αβ 0,, α α, 0,, β β α β [0, ] / / α α rcos θ + i sin θ) α r θ

4 000 n cos k n n π + i sin k n π k k e k n πi cos k n π + i sin k n π n k cos sin e k n πi e πi) k n cos kn π + i sin kn ) π ζ e πi/n n n e k n πi ζ k ζ + ζ + + ζ n ζ ζn ζ k ζ n k e n πi) n e n n πi e πi ζ ζn ζ ζ ζ 0 n k e k n πi 0 0 n cos k n π k n k sin k n π 0 e kπi/n coskπ/n) + i sinkπ/n) kπ/n k k n n 0 0 e iθ e iφ e iθ+φ) sin cos e iθ+φ) cosθ + φ) + i sinθ + φ) cos θ cos φ sin θ sin φ + i sin θ cos φ + i cos θ sin φ cos θ + i sin θ)cos φ + i sin φ) e iθ e iφ

5 3 e πi/5 e 4πi/5 e 6πi/5 O e 0πi/5 e 8πi/5 : coskπ/n) + i sinkπ/n) n 3 α a 0 + a α + a α + + a n α n a 0 + a α + a α + + a n α n a 0 + a α + a α + + a n α n a 0 + a ᾱ + a ᾱ + + a n ᾱ n a 0 + a ᾱ + a ᾱ + + a n ᾱ n a 0 + a ᾱ + a ᾱ + + a n ᾱ n 0 ᾱ x 3x i 0 + i i 4

6 4 M m [m, M] [m, M] y fx) y x fx) m x fx) M x fa ) fa ) m, a < a ) fb ) fb ) M, b < b ) CASE : a < a < b < b fa ) m, fb ) M [m, M] y fx) y x [a, b ] a < c < a c fc) m fx) m x fc) > m m < y 0 < fc) y 0 y 0 a < c < c fc ) y 0 c c < c < a fc ) y 0 c [a, b ] fx) y 0 x 3 M y 0 m a a b b 3: CASE y fx) CASE : a < b < a < b m < y 0 < M y 0 fx) y 0 x a, b ) b, a ) a, b ) CASE 3 : b < a < a < b a < c < a c fx) fc) x [b, a ] [a, b ] fx) fc) x

7 5 CASE 4 : a < b < b < a CASE 3 m M CASE 5 : b < a < b < a CASE m M CASE 6 : b < b < a < a CASE m M CASE 5 ) π 6, ) π, 3) π 6, 4) 3 4 π, 5) π 6, 6) π 3 ) Arcsin sin θ π θ π θ Arcsin π 6 )

8 6 6 ) y 0 tan x y 0 x tan x 0 y 0 x 0 x x 0 + mπ m Arctan y 0 tan x y 0 x π/ < x < π/ tan x y 0 x Arctan y 0 x Arctan y 0 + mπ m f n x) nπ π/, nπ + π/) nπ π < f n y 0 ) < nπ + π fn y 0 ) x n x 0 + nπ Arctan y 0 + nπ y 0 fn x) Arctan x + nπ ) y 0 y 0 sin x y 0 x sin x 0 y 0, π/ x 0 π/ x 0 x mπ + ) m x 0 m cos x sin x + π/) cos x y 0 sin x + π ) y 0 cos x y 0 x x 0 x mπ + ) m x 0 π m π + ) m x 0 m sin x y 0 x π/ x π/ Arcsin y 0 x x x mπ + ) m Arcsin y 0, x m + ) m Arcsin y 0 m Arcsin y 0 nπ π g n y 0 ) nπ + π, nπ h n y 0 ) n + )π n gn y 0 ) x m n h n y 0 ) x m n + gn y 0 ) nπ + ) n Arcsin y 0, h n y 0 ) n + π + ) n+ Arcsin y 0 y 0 [, ] gn x) ) n Arcsin x + nπ, h n x) ) n+ Arcsin x + n + π

9 7 h 0 x) Arccos x Arccos x Arcsin x + π y x 80 4 y f f π Arctan x f 0 f O x f Arcsin x g 0 g g g y O g x h h y O h 0 Arccos x x h 4: f n, gn, h n 7 ) sinarcsin x) x cos θ sin θ cos θ π/ θ π/ 0 cos θ sin θ π θ π

10 8 π/ Arcsin x π θ Arcsin x cos Arcsin x) sin Arcsin x) x ) ) tan θ sin θ cos θ sin θ sin θ π θ π θ Arcsin x sin Arcsin x) x tan Arcsin x) sin Arcsin x) x 3) tan Arctan x) x sin θ tan θ π/ θ π/ cos θ 0 cos θ cos θ cos θ sin θ + cos θ cos tan θ + θ sin θ tan θ cos θ θ Arctan x sin Arctan x) tan θ tan θ + tan Arctan x) tan Arctan x) + x x + 4) sin Arctan x) sin Arctan x) cos Arctan x) sin Arctan x) 3) cos Arctan x) 3) cos θ θ Arctan x + tan θ cos Arctan x) π < θ < π x + x sin Arctan x) x + x + x x +

11 9 8 ) tan θ θ tan Arctan y) y tan θ) tan θ tan Arctan tan x )) tan x tanθ + φ) π ) tan 4 Arctan cos x) tan θ + tan φ tan θ)tan φ) tan π 4 tan Arctan cos x)) + ) tan π 4 tan Arctan cos x))) cos x + cos x sin x cos x tan x tan π Arctantan x) π Arctan cos x) + nπ 4 n n x 0 0 π Arctan + nπ 4 Arctan π/4 n 0 ) Arctantan x) + Arctancos x) π Arctan x) + x

12 0 Arctantan x) ) tan + tan x ) tan x x) + tan 4 x cos x cos 4 x sin x cos 4 x + sin 4 x cos 3 x sin x cos x cos 4 x + sin 4 x Arctancos x)) + cos x cos x) sin sin x) x + cos x 4 sin x cos x 4 sin x cos x + cos x sin x) 4 sin x cos x 4 sin x cos x + cos 4 x 4 cos x sin x + sin 4 x sin x cos x cos 4 x + sin 4 x Arctantan x) + Arctancos x) ) Arctantan x) ) + Arctancos x)) 0 Arctantan x) + Arctancos x) x 0 Arctantan 0) + Arctan cos 0) ) Arctan 0 + Arctan 0 + π 4 π ) cosh x sinh x e cosh x sinh x + e x ) e x e x ) x ex + + e x cosh x sinh x ) cosh x sinh x sinh x cosh x 0 ex + e x 4 x 0 cosh 0 sinh 0 0

13 ) tanh x ) tanh x sinh x cosh x cosh x sinh x cosh x ) cosh x 3) sinh x cosh y + cosh x sinh y ex e x e y + e y + ex + e x ex e y + e x e y e x e y e x e y 4 ex e y e x e y ex+y e x+y) e y e y + ex e y e x e y + e x e t + e x e y 4 sinhx + y) 4) 3) cosh x cosh y + sinh x sinh y ex + e x e y + e y + ex e x ex e y + e x e y + e x e y + e x e y 4 ex e y + e x e y ex+y) + e x+y) e y e y + ex e y e x e y e x e y + e x e y 4 coshx + y) 5) 3) tanh x + tanh y sinh x cosh x + sinh y cosh y sinh x cosh y + sinh y cosh x cosh x cosh y 4) sinhx + y) cosh x cosh y + tanh x tanh y + sinh x sinh y cosh x cosh y cosh x cosh y + sinh x sinh y coshx + y) cosh x cosh y cosh x cosh y tanh x + tanh y tanhx + y) + tanh x tanh y

14 cosh x : ex + e x, sinh x : ex e x, tanh x : sinh x cosh x ex e x e x + e x cosh sinh tanh cosh x e x y e x sinh x tanh x O x tanh x sinh x e x 5: cos θ + sin θ cos θ, sin θ) x + y ) cosh t sinh t

15 3 cosh t, sinh t) x y sinh t R R cosh t cosh t, sinh t) x y x > 0 9 ) 5) θ 0, 0) cos θ, sin θ) x t t 0, 0) cosh t, sinh t) x x y x ±π/4 cosh t, sinh t) t cos t, sinh t) x y x > 0 x x y x y y sinh t sin t x + y t cos t O x O cosh t t θ x cos θ, sin θ) x + y x. ABC BC, CA, AB a, b, c sin A a sin B b sin C c

16 4 sin A sinh a sin B sinh b sin C sinh c tanh t 5) cos t eit + e it, sin t eit e it i cosh t et + e t, sinh t et e t 0 ) y cosh x x y cosh x ex + e x ex ) + e x e x ) e x y + 0 e x y ± y x log y ± ) y x y ± y y cosh x y y + y y 0 y y y ) y ) y + 0 y y

17 5 x log y + ) y 0, x log y ) y 0 x y x cosh x x 0) cosh x log x + ) x [, ) [0, ) cosh x x 0) cosh x log x ) x [, ), 0] y sinh x y sinh x ex e x ex ) e x e x ) ye x 0 e x y ± y + e x 0 y y x log y + ) y + x sinh x sinh x log x + ) x + y tanh x x y tanh x ex e x e x + e x ex ) e x ) +

18 6 e x 0 y) e x ) + y e x + y y tanh x, ) + y x log y log + y y x tanh x tanh x log + x x, ) ) ) cosh x x 0) cosh x cosh x ) log x + )) x + x ) x x + x + x x x + x x + x x + x x x y cosh x 9) cosh x ) cosh y sinh y cosh y sinh y y 0 sinh y 0 sinh y cosh y x cosh x ) x cosh x x 0) ) cosh x ) log x )) x x ) x x x x x x x x x x x x x

19 7 y cosh x cosh x ) cosh y sinh y cosh y sinh y y 0 sinh y 0 sinh y cosh y x cosh x ) x sinh x ) sinh x ) log x + )) x + x + ) x + x + x + + x x + x + x + x + + x x + x + x + x + y sinh x cosh y > 0 sinh x ) sinh y cosh y cosh y cosh y sinh y sinh y + x + sinh x ) x + tanh x ) tanh x ) log + x ) ) + x x x x + x +x x x) ) + x) x) x y tanh x tanh x ) tanh y cosh y

20 8 9 ) tanh y cosh y tanh x ) tanh y x ) tanh x tanh x ) sinh x cosh x sinh x cosh x cosh x cosh x tanh x tanh x cosh x tanh x cosh x sinh x cosh 3 x sinh x fx) log fx) ) f x) fx) f x) fx)log fx) ) sinh x log cosh x log sinh x) log cosh x) cosh x sinh x sinh x cosh x cosh x sinh x ) sinh x tanh x cosh x cosh x sinh x cosh 3 x sinh x

21 9 ) tan x) x ) e x logtan x)) x logtan x)) e x logtan x) ) logtan x) + tan x) x x sin x logtan x) + x / ) cos x tan x) x tan x 3) Arcsincos x)) cos x cos x sin x sin x 4) Arctan ) x + x + x +x + x +x x ) x + x ) x + x) x) + x + x)

2009 IA 5 I 22, 23, 24, 25, 26, (1) Arcsin 1 ( 2 (4) Arccos 1 ) 2 3 (2) Arcsin( 1) (3) Arccos 2 (5) Arctan 1 (6) Arctan ( 3 ) 3 2. n (1) ta

2009 IA 5 I 22, 23, 24, 25, 26, (1) Arcsin 1 ( 2 (4) Arccos 1 ) 2 3 (2) Arcsin( 1) (3) Arccos 2 (5) Arctan 1 (6) Arctan ( 3 ) 3 2. n (1) ta 009 IA 5 I, 3, 4, 5, 6, 7 6 3. () Arcsin ( (4) Arccos ) 3 () Arcsin( ) (3) Arccos (5) Arctan (6) Arctan ( 3 ) 3. n () tan x (nπ π/, nπ + π/) f n (x) f n (x) fn (x) Arctan x () sin x [nπ π/, nπ +π/] g n

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