Colografx cm
COBISS.SR-ID
a1 = a,am+1 = am · a,m ∈ N, a0 =1,a =0, a m = 1 am ,a =0,m ∈ N. (a,b ∈ R \{0},m,n ∈ Z)
am · an = am+n , (am )n = amn , (ab)m = am bm , a b m = am bm . (a =0,b =0,a + b =0) 3 4 0 ; (√ 2)0 2 ; 3 π 0 ; a 3 0 ; a0 + b0 ; Æ (a + b)0 a0 . 1 2 2 ; 1 2 2 ; 1 2 3 ; 1 2 3 ; 8 · 2 3 ; Æ 8 2 3 ; 8 1 2 1 ; 2 · 4 1 ;
1 3 2 23 0
;
+0 5
. ( 2)55 :( 2)51 ; 1 2 3 : 1 4 1 ; ( 3)3 56 :(34 )42 ; 274 :(23 · 270 ); (23 )7 (27 )3 + 1 3 34 + 1 9 17 2 5 + 3 7 · 12 5 2 ; 4 5 : 3 4 + 3 2 4
; 1 4+ 3
:
. ( 27) 15 · ( 9)20 ( 3) 7 · ( 3)2 . (a =0,b =0) a 2 a 5 ; a 2 : a 5 ; a5 : a 2 ; a 2 · a 3 · a5 ; (2a) 2 · 3a3 ; Æ 5a 3 · 6a 2 15a 5 ; 2a 3 9a 2 · 3 4 a 4 · a8 ; (ab) 2 · (ab)4 ; (a2 b) 3 · (ab2 )3 ;
∈ Z 2n · 2=2n+1 ; 21 n 2 n =2 n ; 2 3n +3n =3n+1 ; 2 n +2 n+1 =3 2 n . (n,x ∈ N,a,b ∈ R) ax +2ax a2x 3ax +2a2x ; ( 1)2n +(+1)2n (+1)2n+1 ( 1)2n+1 ; 2a(a b)10 a(
(3a3 b 2 )2 x2 c 1 (x 1 y ) 3 9x 2 y 3 ; a5 z 2 b 3 2 · x 2 c 2y 5 2 ;
(a2 · b3 ) 3 : a2 b 1 c x 1 y 2 z 3 5
(a =0,b =0,m,n ∈ Z)
am a n a2n m ; am n am+n an m ; am+n : a2m 3n ; (a + b)m (a b)m ,a = b;
(a2m 2am +1) · (am 1) 1 ,a =1,a = 1; Æ a b 2n 1 · b a 2n 1 ;
(a + b) 2m · (a + b)2 3m · 0 5 (a + b)2m ,a = b; (a2n b2m ):(an bm ),an = bm .
0 5a 3 b3 · 5a4 b 2 , a = 0 125,b =16; x2 y 2x 5 y 2 x = 1 4 ,y = 8. N N = 3n+1 5n +3n 5n+2 +6 3n 5n 22n+1 · 3n +3n+1 · 4n +2n+1 · 6n+1 ,n ∈ N. c d a + b 3 · (a + b) 2 a + b c d 4 ,a = b,c = d. a,b,m,n,p,q,x,y
(x y )3 (x+y )3 (x2 y 2 ) 3 , |x| = |y |; am (an +a) an (am +a); ( p)8 : p3 4 ; 35a2n x bn :7an x bn 1 ;
(3ab2 ) 3 1 3 a2 b2 2 :3(ab2 )4 ; Æ (a 5 b7 ) 1 : a 4 b4 ; x5 + x12 x 5 + x 12 , x = 1; y 29 y 21 y 21 y 29 , y = ±1; a5 + a6 + a7 a 5 + a 6 + a 7 ;
;(a + b =0,a b =0,b =0);
x + y 1 y x y
(y =0,x = y,x y =0); a 2 x 2 x 2 a 2 · a x 1 + x a 1 1 (a =0,x =0); (a x) 1 (a + x) 1 · (a x) 1 +(a + x) 1
(a =0,a x = 0,a + x =0);
1+ x 1 x
(1+ x =0, 1 x =0) 5x 1 6=0; 3+10x 1 =0; 9x 1 8(2x +1) 1 =0; 6(y 3) 1 4(y 2) 1 =0; (5 x 1 ) 1 =2 2 ; Æ (9 x 1 ) 1 =3 2 . 239 1043=2 ·
·
a =191274 31; b =0 00023; c =1007 1957; d =0 23 · 105 . r = a · 10m , 1 ≤ a< 10 m ∈ Z 1
20000 ;
Æ 1 5 ; 3 4 10 2 + 1 4 10 3 .
∈ N,a ∈ R. n √a (I) n a, n ∈ N a =0, n =2k +1,k ∈ N,a ∈ R, (II) n a, n =2k,k ∈ N a> 0.
≥ 0,n ∈ N ( n √a )n = a.
∈ R,n ∈ N
≥ 0,n ∈ N
√an = a,n |a|,n
√ab = n √a n √b. a ≥ 0,b> 0,n ∈ N n a b = n √a n √b a ≥ 0,m,n ∈ N ( n √a )m = n √am . a ≥ 0,m,n ∈ N n √a = mn√am . a ≥ 0,m,n ∈ N n m √a = nm√a. a,b ≥ 0,n ∈ N a · n √b = n √an b. √ 16; √ 9 · 0; √ 49 · 100; 4 · ( 25); ( 7)2 ; ( 5)3 ; ( 2)6 ? a √8a ; √ 9a; √ 36a2 ; √6a2 ; √ 5a3 ; √5 a; √27+ a ? √36x2 ; √a4 ; √a6 ; (1 a2 ); (x 3)4 ; Æ (2+ y )10 .
√9 · 16; √25 · 81; √7 · 28; √6 · 24; √5 · 45; Æ √5 · 3 · 60; √6 · 8 · 3; √3 · 53 · 35 · 5; √23 · 53 · 10
9 16 ; 25 81 ; 1 9
;
144 ; 4+ 9 4 ; Æ 1 144 169 ; 6+ 1 4 ; 1 1 25 ; 1 1 8 25 2 . 4 √81; 4 √81; 5 √ 32; 9 √1; 4 1 625 ; Æ 3 1000 27 ; √0 36; √0 81.
(a ≥ 0,b ≥ 0,c ≥ 0)
25 · 7 · 9; √0 09 · 64 · 49; √18 · 25; √16a2 b4 ; √48a5 b12 c3 ; Æ √63ab21 c2n (n ∈ N); √8a7 c4n (n ∈ N) (a,b> 0) 5√7; 0 5√ 10; 1 3 √27; 7 5 14 ; a 1 a ; Æ 2a b 4a ; b3 √b4 ; ab2 a b3 ; (x 1) 1 x2 1 (x2 > 1); x + y a b a2 2ab + b2 xy + y 2 (x + y> 0,a b =0,y> 0). (x 2) 3 x 2 ,x> 2; (x +3) 2 x +3 ,x> 3; (x +2) x x +2 ,x< 2; (x 1) 5x x 1 ,x> 1.
x a a2 x 2x 1 x 3a x a2 ,x> 0,a =0;
10 ax2 y 2 4 y ax2 y 4 + 2 x ax4 y 2 ,a> 0,x> 0,y> 0; ax2 y 2 + bx2 y 2 +3x ay 2 + by 2 4y √ax2 + bx2 ,x< 0,y> 0, a + b> 0
√ab 3 √a2 b,a> 0,b> 0; √x 3 √x2 4 √x3 ,x> 0;
5 √ax3 · 6 √a2 x · 15√a7 x4 ,a> 0,x> 0; 3 x4 y 2 : 3 xy 2 ,y =0;
3 √56a2 : 3 √7a 1 ,a =0; Æ 4 √a2 b3 x3 : 6 √a3 b2 x4 ,a> 0,b> 0, x> 0.
( 4 √x 2 12√x ) · 4 √x,x> 0;
(√x +2 3 x2 y +3 4 xy 2 ) · √xy,x> 0,y> 0; x √ax+1 : 2x √ax+1 ,a> 0,x ∈ N; √xy · 3 x2 y 3 · 6 x9 y 7 ,x> 0,y> 0;
(√a7 · 3 √a2 ):( 4 √a5 · 12√a11 ),a> 0; Æ
( 3 √ x2 · 4 √x3 ):(x√x · 3 √x ),x> 0;
( 6 √x3 · 3 √x4 ):( 6 √x5 · √x ),x> 0;
(√a · 3 √a2 · 4 √a3 ):(a 3 √a · √a ),a> 0; x y 2 z · 3 x2 y z · 4 xy z 3 ,x> 0,y> 0,z> 0;
5 x4 y 4 z : 10 x9 y 2 z 8 : 15 x 4 y 6 z 9 ,x> 0,y> 0,z> 0;
( 3 xy 2 )5 · 3 x2 y 4 ,x> 0,y> 0; n a b2 2 · 2n b2 a 4 ,a> 0,b> 0,n ∈ N;
( 4n x2 y 2 )3 : n x y 2 ,x> 0,y> 0,n ∈ N
4 a 3 a√a,a> 0; x 3 x 4 √x,x> 0;
3 a b a b ,a> 0,b> 0; 5 x 3 √x : 4 x 3 √x,x> 0; ( 3 √x2 )2 :(√x3 )4 · x√x,x> 0; Æ ( x√x )3 : 3 √x√x,x> 0; x a x a x √a,a> 0,x ∈ N;
4 a 3 √a2 a3 4 √a : 4 3 √a18 ,a> 0; 5 x 6 √x5 : 4 x3 3 √x2 ,x> 0; 3 a a√a 2 4 a2 √a3 ,a> 0
a + ib, a b i
a + ib = c + id ⇔ a = c ∧ b = d
(a + ib)+(c + id)=(a + c)+ i(b + d). (a + ib) · (c + id)=(ac bd)+ i(ad + bc). a z = a + ib, b a =Re(z ),b =Im(z ) i2 = 1 a + ib c + id = ac + bd c2 + d2 + i bc ad c2 + d2 ,c 2 + d2 > 0 z = a + ib z = a ib. z = a + ib |z | = √a2 + b2 . x y x + yi =2+3i; (x 3)+(y +2)i =1; (x y )+(x + y )i =5i; (3x +5y ) (2x y )i =1+ i; x +2yi =(5+2i) (4+9i) a b
(1+ i)a +(2+3i)b = i; (1+3i)a (1 2i)b =5; (2+ i)(a + ib)=5 5i. i3 ; i4 ; i23 ; 1 i ; i 50 . (1+ i)2 ; (1 i)2 ; (1+ i)4 ; (1 i) 4 . i2 + i3 + i4 + i25 ; i5 + i 4 + i125
z1 =3,z2 =4+2i; z1 =2 3i,z2 =2+3i;
z1 =5+4i,z2 =9 5i; z1 = 2 3 ,z2 = 5i.
(3+5i)(2 4i); (5+7i)(5 7i);
(1+ i)(2 i)+ i; (1 i)(2+ i) i;
(2 i)(3+ i)(2i +1) (2 3i)(3+4i)+(2+ i)2 +(1+ i)4 i.
(3+2i)3 ; (1+ i)3 ; (1 i)3 ; (2+ i)4 . z1 z2 x+yi
(x,y ∈ R)
z1 =5+6i,z2 =2+ i;
z1 =1 i,z1 =1+ i;
z1 =5+3i,z2 =2; z1 =3 2i,z2 =4+3i; z1 =5+ i,z2 =(2 3i)(1+ i).
(1 i)2 1+ i ; 1 1+ i + 1 1 i ; 1+ i √3 1 i √3 + 1 i √3 1+ i √3 ; 7+4i 5 i 3+ i 2+3i ; 2+ i15 i3 i12 z1 = 3+ i z 3 + z 2 + az + b =0, a b. z1 =3+4i z = 2 z +1 z 7 2 .
)(15
p(z )= z 3 +3z 2 + z +2 p(z0 ) p(z0 ), z0 =2 i. z = a + ib.
|z |2 ; z + |z |; z 2 ; (z ); |z |; Æ z z ,z =0. z1 z 2 + z 1 z2 ≤ 2|z1 ||z2 |.
|z1 + z2 |≤|z1 | + |z2 |; |z1 z2 |≥ |z1 |−|z2 | |z | < 1 2 , |(1+ i)z 3 + iz | < 3 4 z + z =4, |z | =1, z z z 2 + |z | =0; z 2 + |z | =0; |z |2 + z =0; |z |− z =1+2i. z |z 2i| = |z |, |z i| = |z 1|; |z 2| = |z +2i|, |z i| = |z +2|; |z +1| = |z +4| = |z i|. z z 2 = 2; z 2 = i; z 2 = i; z 2 = 3+4i. 1+ z + z 2 =0, (az 2 + bz )(bz 2 + az )= a2 ab + b2 ; (a + b)(a + bz )(a + bz 2 )= a3 + b3 . z 2n + z n +1,z ∈ N.
x
ax 2 + bx + c =0,a,b,c ∈ R,a =0.
x1,2 = b ± √b2 4ac 2a , b2 4ac ≥ 0,
x1,2 = b ± i |b2 4ac| 2a , b2 4ac< 0.
ax2 + bx + c =0,a,b,c ∈ R,a =0
D = b2 4ac.
D> 0, D =0, D< 0,
x2 9=0; 4x2 20=0;
8 x2 =0; 25 4x2 =0; x2 +4x =0; Æ x2 2x +1=0; x2 5x +6=0; x2 3x 4=0; 2x2 3x +1=0
2x2 =0; 1 3 x2 =0; x2 =6x; x2 = 7x;
3x2 = 12x; Æ 5x2 3x =0; 3x2 +4x =0;
0 1x2 0 3x =0; x2 25=0; 3x2 300=0; x2 7=0; 4x2 25=0; 169x2 =1; 3x2 +2=0; 2x2 5=0; x2 + π 2 =0. x2 8x +15=0; x2 +7x +10=0; x2 +95x 96=0;
15x2 22x +8=0; x2 6x +9=0; Æ x2 x + 1 4 =0;
73x2 40x 33=0; 3x2 5x 1=0; x2 + x +1=0; x2 6x +15=0. (x +1)2 +(x +2)2 2(x 3)2 =5x2 ; (x 1)(x +4) 5(x +2)(x 3)=(x +1)2 (x 3)2 ; (x 3)2 +2(x 1)2 =4x +3; (x +5)(x 4)+3(x +2)(x 3)=(x +3)2 +13; 5(x +0 2)2 7 2=0. (a,b ∈ R)
ax2 =0; ax2 + bx =0; 2x2 3ax 2a2 =0;
ax2 (a b)x b =0; x2 +2ax + a2 +4b2 =0;
Æ abx2 (a2 b2 )x ab =0. x 1 x +1 x x 1 2=0; x +2 2x 3 + 2x 3 x +2 =2; 2x 1 x +1 = x +1 x 2 ; x +1 x 1 + x 1 x +1 = 25 12 ; 1 x 2 + 1 x 5 + 3x2 4(x 2)(x 5) =0;
Æ 6 x2 1 2 x 1 =2 x +4 x +1 (a,b ∈ R)
1 a + 1 b + 1 x = 1 a + b + x (a =0,b =0);
x a x b + x b x a +2=0(a = b); x2 (a + b)2 4abx (a + b)2 (a b)2 =0; x2 (a b)x ab =0; x + 1 x =2cos a;
Æ x + a x a + x a x + a = a(3x +2a) x2 a2 ; a2 x2 + bx = b2 x2 + ax (a = b,a = b); x2 +1
a2 x 2a 1 2 ax = x a .
x 2 +7x 8=0 8 x 1 =1,x 2 = 8
9x 2 6x +1=0 6 x 1 = x 2 = 1 3
3x 2 + x +2=0 23 x 1,2 = 1 ± i√ 23 6
x 2 8x +15=0
3x 2 +2x 5=0 x 2 x +2=0
x 2 + x + 1 4 =0
2x 2 +1=0
x 2 x =0
25x 2 +10x +1=0
2(1+ i)z 2 4(2 i)z 5 3i =0; (2+ i)z 2 (5 i)z +2 2i =0; z 2 (2+ i)z + i =0 x2 + |x 1| =1;
(|x| +1)2 =4|x| +9; x2 + x 3|x +1| +3=0; 2x2 −|5x 2| =0; x2 + |x|− 2=0. |x2 +4x +2| = 1 3 (5x +16); |x2 2x 1| = 1 3 (5x +1). α 6cos 2 α +(2π +3√5)sin α 6 π √5=0.
m x 2 2mx + m 2 1=0 ( 2, 4) x 2 2ax +4=0,a ∈ R x2 + px + q =0, (p,q ∈ R) a ∈ R x2 +(p +2a)x + q + ap =0; x2 +(p + a)x + q p2 4 =0 Æ (a 2 + b2 )x 2 +2acx + c 2 b2 =0 2ay 2 +2cy + b =0(a,b,c ∈ R). Æ x2 + px + q =0 x2 + p1 x + q1 =0(p,q,p1 ,q1 ∈ R) pp1 =2(q + q1 ), k (k +3)x 2 2(k +1)x + k 5=0 (a 2 + b2 )x 2 +2acx + c 2 b2 =0 ax 2 + cx + b 2 =0(a,b,c ∈ R) (p 2 + q 2 + r 2 )x 2 +2(p + q + r )x +3=0, (p,q,r ∈ R)
b2 x 2 +(b2 + c 2 a 2 )x + c 2 =0 a x2 ax +1=0 x2 x + a =0; x2 + x + a =0 x2 + ax +1=0; 2x2 (3a +2)x +12=0 4x2 (9a 2)x +36=0 a,b,c x1 x2 (x1 <x2 ) x 2 2(a + b + c)x +3(ab + bc + ca)=0
x1 <a + b + c<x2 .
km/h
B 80km . A B
km/h 96km
a =9cm ,b =7cm c =10cm
98cm 2 ? 4km
240km a x2 2ax (a +3)=0 x 2 +(2n +1)x +(2n 1)=0,n ∈ N Æ x1 x2 ax2 + bx + c =0, a =0 x1 + x2 = b a ,x1 x2 = c a , x1 x2 ax2 + bx + c =0,
ax 2 + bx + c = a(x x1 )(x x2 ) (a,b ∈ R) x1 = 1 a + b ,x2 = 1 a b ,a = b,a = b; x1 = a +1,x2 = 1 a 1 ,a =1; x1 =2+ i√3,x2 =2 i√3; x1 = a b ,x2 = b a ,a,b =0. 5x2 2x 3=0
ax 2 + bx + c =0, (ac =0,a,b,c ∈ R). 5x 2 2(5k +3)x +5k 2 +6k +1=0 k. k x 2 (2k +1)x + k 2 +2=0 x1 ,x2 3x 2 +2x 7=0. 3x 2 1 + x1 x2 +3x 2 2 ; x 2 1 x2 + x
x 2 2 + x 3 1 + x 3 2 ; 2x 2 1 + x1 x2 +2x 2 2 x 2 1 + x 2 2 .
x2 (a +1)x +2(a 1)=0 1 x1 + 1 x2 = 5 6 ; 1 x 2 1 + 1 x 2 2 = 5 4 x1 x2 x2 + kx +1=0, k x1 x2 2 + x2 x1 2 > 2
4a2 +4a +1; a2 a 6; a2 +4a +3;
2x2 5x +2; 1 2 y 2 3yz +4z 2 ;
Æ x3 2x2 8x.
a2 a 6
a2 4 ,a = 2,a =2;
a2 +3a +2
a2 +6a +5 ,a = 5,a = 1;
x2 7x +12
x2 6x +9 ,x =3; x2 3x 10
x2 + x 2 ,x =1,x = 2; 3x2 +2x 8 12x2 7x 12 ,x = 4 3 ,x = 3 4 . x1 x2
x2 ax + a 1=0; x2 ax + a 3=0. a x 2 1 + x 2 2 a 6x 2 +6(a 1)x 5a +2a 2 =0 p q x2 + px + q =0 p q. r x 2 +(4+2r )x +5+4r =0 x1 = x2 ; x1 = x2
x1 x2 5x 2 (k +5)x + k =0,k ∈ R,k =0. z1 =1 4 x1 z2 =1 4 x2 ; k x 2 (k +3)x +4k +2=0 k k (x)=(p 2)x 2 2px +2p 3,p ∈ R. p x k (x) < 0 p k (x)=0 x 2 + ax + b +1=0(a,b ∈ R) a2 + b2 x1 x2 x2 + px + q =0, p q x1 +1 x2 +1 x2 p2 x + pq =0. x4 50x2 +49=0; x2 + 16 x2 =17;
x4 7x2 8=0;
(x +2)2 1 (x +2)2 = 80 9 ;
x4 +13x2 +36=0;
Æ a2 b2 x4 (a4 + b4 )x2 + a2 b2 =0(a,b ∈ R \{0});
x4 2(a2 + b2 )x2 +4a2 b2 =0(a,b ∈ R)
x6 7x3 8=0;
x6 9x3 +8=0;
x3 +3 x3 7 x3 7 x3 +3 = 5 6 ;
x8 17x4 +16=0;
x4 (x2 8)= 9(2x4 9) x2 +8 ;
Æ x8 a2 (2+ √2)+ b2 (2 √2) a2 (2 √2)+ b2 (2+ √2) x4 + (a2 b2 )4 =0,a,b ∈ R;
x8 (a4 + b4 )x4 + a4 b4 =0,a,b ∈ R.
(x 1)(x 2)(x 3)(x 4) (x +1)(x +2)(x +3)(x +4) =1;
x2 +2x +1 x2 +2x +2 + x2 +2x +2 x2 +2x +3 = 7 6 ;
(x 4 5)4 +(x 5 5)4 =1; (4x +1)(12x 1)(3x +2)(x +1)=4;
(x2 4x +3)2 8(x2 4x) 9=0;
Æ (x2 5x +7)2 (x 2)(x 3)=1.
2x4 +3x3 16x2 +3x +2=0;
x4 5x3 +8x2 5x +1=0;
x5 7x4 +16x3 16x2 +7x 1=0;
x7 +3x6 +2x5 +4x4 +4x3 +2x2 +3x +1=0;
x4 15 3x3 +54 52x2 15 3x +1=0;
Æ (x +1)4 =2(1+ x4 );
x6 = 257x2 68 68x2 257 . (m ∈ R)
2mx4 (2m2 +5m +2)x3 +(5m2 +4m +5)x2 (2m2 +5m +2)x +2m =0,m =0; x4 +(5 3m)x3 +2(m2 2m 2)x2 +(5 3m)x +1=0
x4 2x2 400x =9999;
x4 + x2 +6x 8=0; x4 4x3 19x2 +106x 120=0. 2(x 4 2x 2 +3)(y 4 3y 2 +4)=7. ax4 + bx2 + c =0(a =0).
90cm 2 .
15cm , 13cm . 15π,
y = ax 2 + bx + c,a,b,c ∈ R,a =0. y = a x + b 2a 2 + 4ac b2 4a T b 2a , 4ac b2 4a a> 0 4ac b2 4a x = b 2a . a< 0 4ac b2 4a x = b 2a . y = x2 +3; y = x2 4; y =(x +3)2 ; y =(x 4)2 : y =2x2 ; Æ y = 1 2 x2 . y = x2 ; y = 1 4 x2 ; y = 3x2 ; y = 2x2 +4; y =(x 1)2 ; Æ y =(x +2)2 . y = x2 6x +5; y = x2 3x; y = x2 + x +1; y =2x2 + x 1; y = 2x2 x +6. y = x ·|x|; y = x2 , |x| > 1, 1, |x|≤ 1; y = x2 ·|x| 2x ; y = |x2 + x 2|;
y = |x 1|· (x 2); Æ y = x3 x2 2 |x 1 | ;
= x4 1 |x2 1 | ; y =(x 1)(2 −|x|);
=(x 3)(2 −|x|); y =(|x|− 1)(2 x). f (x)= 2x2 + bx + c
T (2, 2).
= ax2 +6x 4
= x2 mx + m +1 y =2x2 2mx +2m +4 A(3, 1) B (5, 3). y = f (x), A
y = x2 +2px +13
y = x2 + kx +4
y = x(x 1) x(x 1)+2 ;
y = x4 + x2 +5 (x2 +1)2 ; y = x 1+ 1 x 3 , x> 3. y = 2x2 4x +9 x2 2x +4 ; y =3x +4√ 1 x2 . f (x)=3x 2 5x + k 2 3k + 49 12 ,k ∈ R
f (x) x. k y = x2 2(m +1)x +2m(m +2),m ∈ R; y = x2 +(2m +1)x + m2 1,m ∈ R. y = x 2 +(m 3)x +1 2m,m ∈ R. m y
f (x)= x2 +(k +2)x +2k (k ∈ R). k f (x)=0 k f (x k ) 2x =0 x1 =0 x2 =7 Æ k f (x k ) 2x.
f (x)=(m +1)x 2 2(m 1)x + m 5,m ∈ R.
y = ax2 + bx + c,a,b,c ∈ R,a =0
f (x)= x 2 +(m 3)x +1 2m,m ∈ R. m
f (x) < 0 x.
y = f (x)
y = f (x)
f (x)=(m +1)x 2 +2(m +1)x +2m,m ∈ R. m
f (x) < 0 x ∈ R
y = f (x) y = x 2 +2(m +1)x +3m 1,m ∈ R A. y = x2 +3x 4
ax2 + bx + c =0,a,b,c ∈ R. c(a + b + c) < 0,
x2 2(m 1)x + m +5=0,m ∈ R, ( 2, 3). a f (x)= x2 6ax +2 2a +9a2 a x2 + ax + a 1=0 m (2m 5)x2 +6(m 3)x +3(m 5)=0 a x1 x2 f (x)=3x2 4x 4a
3 <x1 < x2 < 1. m (1 m)x2 4(2 m)x +2(2 m)=0 Æ 1
(m 3)x2 +2(m 4)x + m 5=0 k f (x)=(k 2)x 2 2(k +3)x +4k, x2 3x> 0; x2 +4x ≤ 0; 2x2 5x +3 > 0; x2 +12x 36 ≥ 0; x2 < 3x; Æ x2 +4x 3 > 0; x2 2x +4 > 0; x2 4x +4 > 0. x2 + x +1 4 x2 > 0; x3 3x2 +2x> 0;
x 9 x2 5x +6 < 1; 2x2 +3x 1 x2 + x +1 < 0; 15 4x x2 x 12 < 4; Æ 4 1+ x + 2 1 x < 1;
(x +2)(x2 2x +1) 4+3x x2 ≥ 0; 2x x2 9 ≥ 1 x +2 ; 1+ 12 x2 < 7 x ; 1 < 3x2 7x +8 x2 +1 ≤ 2.
|2x 3 | + x x2 3x +2 < 1; |x2 5x +5| < 1; |x 6| >x2 5x +9; |x 6| > |x2 5x +9|; x2 3x 1 x2 + x +1 < 3;
3x x2 4 ≤ 1; x2 5x +4 x2 4 ≤ 1. p
f (x)=(p2 +2p 3)x2 4px + p;
f (x)= px2 +(1 p)x 2p;
f (x)=(1 p2 )x2 +2(p 1)x + p. x. a x2 + ax 2 x2 x +1 < 2; x + a x2 + x +1 < x x2 +2x +3 ; 3x2 5x + a2 3a + 37 12 > 0; x ∈ R
(m +1)x2 (m 1)x + m =0 x 2 1 + x 2 2 ≥ 1; x2 +(2m +2)x + m =0 1 x 2 1 + 1 x 2 2 > 8? (x 1)(x 2)(x 3)(x 4)+1 001 > 0, x ∈ R.
2 4(2n 1)x +15
2
n 4 > 0, x k (k 2)x 2 +8x + k +4 > 0
x2 (a +1)x + a +4=0; 3x2 +5x + a =0 a x2 +(a 3)x + a2 =0; 4x2 4(a 2)x + a =0, x2 +(y +1)x + y 2 y +1=0; (y 2)x2 +(4y 6)x +5y 6=0; x2 (4y 2)x +(5y 2 5y 1)=0
x y =1,x2 xy + y 2 2x +3y 2=0; 2x2 3y 2 + x y +42=0, 2y 3x =2;
x2 + y 2 =2(xy +2),x + y =6;
x2 + xy + y 2 =13,x + y =4;
x2 xy + y 2 =7,x y =1.
x2 3xy +2y 2 =0,x2 3x y +3=0; 4x2 7xy +3y 2 =0, 5x2 +2xy 6y 2 =1.
x2 +3xy y 2 =3, 2x2 xy + y 2 =4;
x2 +2xy +5y 2 =113,y (x + y )=28; 3x2 2xy +5y 2 35=0, 5x2 10y 2 5=0;
x2 xy + y 2 =21,y 2 2xy +15=0;
xy +3y 2 x +4y 7=0, 2xy + y 2 2x 2y +1=0.
x2 + y 2 =25,x2 +2y 2 =41; x + xy + y =11,x2 y + y 2 x =30; (x2 y 2 )xy =180,x2 xy y 2 = 11. y + x2 =3,x + y =3; y = x2 5x +4, 2x = y +6;
x2 = y +4x 3,y + x2 =5x 7;
y = |x +1|,x2 + y =3; y 2 =2(x 1),x +1=2y ;
Æ y = a x2 a ,x + y = a,a> 0.
a
y = x2 ,y = a;
y = x2 4,x + y = a;
y x = a,y = |x2 x|;
y = |x2 x 2|,y = x + a.
x3 y 3 =19(x y ),x3 + y 3 =7(x + y );
x3 + y 3 =7,xy (x + y )= 2;
x3 + y 3 =35,x + y =5;
x3 y 3 =7,x y =1.
(x2 + y 2 )xy =78,x4 + y 4 =97; 3xy x2 y 2 =5, 7x2 y 2 x4 y 4 =155;
x4 + x2 y 2 + y 4 =481,x2 + xy + y 2 =37;
(x +1)2 (y +1)2 =27xy, (x2 +1)(y 2 +1)=10xy. 5cm, 24cm 2 . 42cm 30cm; 40cm 60cm 2 ; R =25cm r =10cm .
a(x)= b(x)
a(x)= b2 (x) ∧ b(x) ≥ 0
Æ
a(x) ≤ b(x) 0 ≤ a(x) ≤ b2 (x) ∧ b(x) ≥ 0, a(x) ≥ b(x) a(x) ≥ b2 (x)) ∧ b(x) ≥ 0
a(x) ≥ 0 ∧ b(x) ≤ 0
√x +7= x +1;
√5x 6= x;
x + √4+ x2 =8;
6+ √2+ x =3; y = √12 y ;
√7 x = x 1;
1 √1+5x = x;
4√y +6= y +1;
21+ √2y 7= y.
1 x = √3x2 7x +3; √2x2 x 5=1 x;
√6 4x x2 = x +4; √z 2 +8=2z +1; x2 4x +6= √2x2 8x +12; Æ 2x2 +3x 5√2x2 +3x +9+3=0
√2x +1+ √x 3=4;
√2x 4 √x +5=1;
√2x 1+ √x 2= √x +1;
√4x +1+ √x +2= √x +3;
√2x +14 √x 7= √x +5; Æ √x +6 √x 7=5; √x 2+ √4 x = √6 x; √x +3+ √x +4= √x +2+ √x +7
√4x2 +9x +5 √2x2 + x 1= √x2 1; √4x2 + x 5 √2x2 3x +1= √x2 1;
√3x2 +5x 8 √3x2 +5x +1=1; y 2 +4y +8+ y 2 +4y +4= 2(y 2 +4y +6);
√x2 +8x √x +1 + √x +7= 7 √x +1 ; Æ √ x2 6x √x 1 + √x 5= 5 √x 1 ; √2x2 3x 2+ √x2 5x +6=2√x2 2x; √2x2 +8x +6+ √x2 1=2x +2.
3 √x +1+ 3 √3x +1= 3 √x 1;
3 √x 1+ 3 √x 2= 3 √2x 3;
3 √x +1+ 3 √x +2 3 √2x +3=0;
3 x +3 5x +2 + 3 5x +2 x +3 = 13 6 ;
3 (1+ x)2 +4 3 (1 x)2 =5 3 √ 1 x2 ;
3 (a + x)2 2 3 (a x)2 = 3 √a2 x2 ,a ∈ R
y = ax ,a> 0,a =1
a> 1
y = ax x ∈ R x ∈ R, 0 <a< 1, y = ax x ∈ R x ∈ R. y = 1 2 x ; y = 1 2 x ; y =0 3x ; y = 10 3 x ; y =5x y =3x +1; y =2x 1; y = 1 2 x 1 2 y = 5x ; y =1 5x ; y = 2 5x
y =2x+1 ; y = 1 2 x+2 ; y =3x 2 . y =2|x| ; y = 1 2 |x| ; y =2x−|x| ; y =2 x2 |x| . x y 7x > 7y ; (√2)x > (√2)y ; π 3 x > π 3 y ; (√2 1)x > (√2 1)y ;
3 2 x <
3 2 y .
2 3 a 4 3 ,
1; 0 <a< 1? [ 2, 0] y =2x2 +x 2 ; y =3x2 +2x 1 ; y = 1 3 x2 +x 2 . ϕ : R → R,x ∈ R ⇒ a x = y ∈ R,a> 0,a =1 1 1 R R+ ,
3x · 2y = 1 9 , y x =2; 2y 5x =20, x + y =1; 1 3 x · 5y =75, x + y =1; 7x+1 · 2y =4, y x =3.
23 6x > 1; 16x > 0 125; 1 3 x > 1 9 ;
0 14x2 2x 2 ≤ 0 12x 3 ; 1 3 √x+2 > 3 x 1 3 x+ 1 2 2 x > 1 √27 ;
0 32x2 3x+6 < 0 00243;
0 2 x2 +2 x2 1 > 25;
(x 3)2x2 7x > 1; (x2 8x +16)x 6 < 1.
52x+1 > 5x +4; 2x+2 2x+3 2x+4 > 5x+1 5x+2 ; 2x +2 x+1 3 < 0; 4 1 x 1 2 1 x 2 3 ≤ 0;
25x < 6 5x 5. x =log a b ax = b (a> 0,a =1,b> 0).
alog a b = b, a> 0,a =1,b> 0, log a (xy )=log a x +log a y, x> 0,y> 0,a> 0,a =1,
log a xs = s log a x, a> 0,a =1,x> 0,s ∈ R,
log a x y =log a x log a y, x> 0,y> 0,a> 0,a =1,
log a 1=0, a> 0,a =1,
log a a =1, a =0,a =1,
log b a = 1 loga b , a> 0,b> 0,a =1,b =1,
log a b = logc b logc a , a> 0,b> 0,c> 0,a =1,c =1,
log as x = 1 s log a x, x> 0,a> 0,a =1,s =0 lg . log .
log 2 (8 16 32); log 3 9 27 ; log 1 8 1 4 ; log 1 4 1 8 . x
log x 16=2; log x 27= 3 4 ; log x 1000= 3;
log x 7= 1 3 ; log 4 x = 1 2 ; Æ log x 27= 1 3 ;
log 100 x =0 2; log 1 3 x =8; log 125 x = 2 3 . log b 7 √3 b =3, 1 9 , 1 27 . (a,b,c> 0)
x =3ab; x = 1 2 ab c ; x = a2 bc3 ; x = a2 b 1 3 c2 ; x =2(a b)(a>b); Æ x = a b2 c2 (b>c> 0); x = 4a√ab 5b √a2 b ; x = a 3 √b2 ; x = a√b c3
x,
log x =log5 log2+log4; log x =log7+log9 log3;
log x =2log3+3log5; log x =3log2 2log3;
log x = 1 2 log3+ 2 3 log5 1 3 log2; Æ log x = 1 3 log125 (log3+2log √2);
log x = 1 4 (2log2 4log3+5log4).
log( 2)( 3)=log( 2)+log( 3);
log( 3)2 =2log( 3);
log( 2)4 =2log( 2)2 ;
log 2 3 =log2 log3.
a,b,M,N a =1,b =1, log a N log b M =log b N log a M.
log 3 12=log 3 7 · log 7 5 · log 5 4+1; log 3 2 · log 4 3 · log 5 4 · log 6 5 · log 7 6 · log 8 7= 1 3 . x z1 =log(2
3) z1 = z 2
log ab a =3, log ab 3 √a b =3(a> 0,b> 0,ab = 1)
log 12 2= a, log 6 16; log 6 2= a, log 6 5= b, log 3 5
log 24 54 log 2 3= a; log 5 6, log 10 2= a, log 10 3= b; log 10 122 5 log 10 5= a, log 10 7= b.
log 5 9 8 lg2= a, lg7= b;
log 9 40 lg15= a, log 20 50= b.
log 12 18= a log 24 54= b, ab +5(a b)=1.
log 12 24= a log 18 54= b, 5(a + b) 3ab =8.
f (x)=log 1 x 1+ x . a b f (a)+ f (b)= f a + b 1+ ab ?
log 3 4 log 4 3;
log 1 3 0 4 log 1 4 0 4;
log 3 108 log 5 375
log 1 4 (log 2 3 log 3 4);
5 log5 log25 ; log 2 log100;
log 3 64 · log 2 1 27 ; (log 3 4+log 2 9)2 (log 3 4 log 2 9)2 ;
y =log a x,a> 0,a = 1,x> 0 a> 1, y =log a x x> 0, x> 1 0 <x< 1. 0 <a< 1, y =log a x x> 0, x> 1 0 <x< 1 y =log 1 2 x; y =log 1 3 ( x); log 2 x; y =log 3 |x|; y = | log 10 x| y =1+log 2 x; y = 1+log 1 2 x; y = |1+log 2 x|; y = 1 2 log 1 2 (x 1) y = 1 2 log 3 (x +1)2 ; y = log 2 x | log2 x| ; y =2log 2 x . x log 1 2 (x +1); log 3 2x 1 2 ; log 2 x 2 x +2 ; log 10 x +3 x +1 ; log 1 2 (x2 5x +6); Æ log 3 ( 4x2 +4x +3); log 2x x2 5; y =log |x|−4 2
y =log x2 3x +2 x +1 ; y =log 2 log 3 log 4 x; y =log |4 x2 |; y =log(3x 3 x ).
(log 2 3) 1 +(log 5 3) 1 > 2; 1 log2 π + 1 log5 π > 2 0 <a< 1, 0 <b< 1 log a 2ab a + b · log b 2ab a + b ≥ 1.
log 3 4 > log 4 5; log 4 5+log 5 6+log 6 7+log 7 8 > 4 4. f : R+ → R, R+ x → log a x = y ∈ R,a> 0,a =1 1 1 R+ R, a> 0,a =1
log a f (x)=log a g (x) f (x)= g (x),f (x) > 0,g (x) > 0.
lg(3x 1)+lg(12 x)=2; log 2 (x 1)+log 2 x =1;
log 3 x +log 3 (x +2)=1;
log 2 (x +1)+log 2 x =1;
lg x +ln(x 3)=1
log 2 x log 16 x =3;
7log 25 x log 5 x =5.
log 3 1+log 3 (2x 7) =1;
log 4 2log 3 (1+log 2 (1+3log 3 x)) =0 5;
log x 1 3=2;
log 4 log 3 log 2 x =0
log 5 x +log 25 x =log 1 5 √3;
log x 2 · log x 16 2=log x 64 2;
log x 2 log 2x 2=log 4x 2.
x1+log 3 x =3x;
3log x 3 · xlog 3 x =9; x2log 2 x =2x2 ;
1+log 2 (x 1)=log x 1 4;
51+log 4 x +5log 1/4 x 1 = 26 5 ;
91+log 3 x +31+log 3 x =210;
(lg x)2 lg x3 +2=0;
log 3x 3 x +log 2 3 x =1.
log 2 (9 2x )=10lg(3 x) ;
log 4 24x =2log 2 4 ;
9log 1/3 (x+1) =5log 1/5 (2x2 +1) ;
3log 2 3 x + xlog 3 x =162.
log 2 (9x 1 +7)=2+log 2 (3x 1 +1);
log 7 (6+7 x )=1+ x;
7lg x 5lg x+1 =3 · 5lg x 1 13 · 7lg x 1 ;
22lg4x 1 7lg4x =7lg4x 1 3 4lg4x ;
4log 16 x 3log 16 x 1 2 =3log 16 x+ 1 2 22log 16 x 1 . (a,b,c ∈ R)
log a2 x +log x2 a =1;
log x 2 log x2 a +1=0;
log √x a · log a2 a2 4 2a x =1;
log a x +log b x +log c x =log abc x.
log 2 (x2 +2x 7)= 1 log9 6x+x2 4 ;
log 3x+7 (9+12x +4x2 )+log 2x+3 (6x2 +23x +21)=4.
x +log 2 y = y log 2 3+log 2 x,
x log 2 72+log 2 x =2y +log 2 y ;
x log 2 3+log 2 y = y +log 2 y,
x log 3 12+log 3 x = y +log 3 4.
y =log a x (a> 0,a =1,x> 0)
0 <a< 1, a> 1.
log a f (x) < log a g (x)
0 <a< 1 f (x) >g (x), g (x) > 0, a> 1 f (x) <g (x), f (x) > 0.
log 5 (26 3x ) > 2;
log 3 (13 4x ) > 2;
log 4 (19 3x ) > 2;
log 2 (9 2x ) > 3.
log 1 2 x 1 2 +log 1 2 (x 1) ≥ 1;
log 1 2 x + 1 2 +log 1 2 x ≥ 1.
log 2 2 (2 x) 8log 1 4 (2 x) ≥ 5;
log 2 5 (6 x)+2log 1 √5 (6 x)+log 3 27 ≥ 0;
log 2 3 (5 x)+5log 1 3 (5 x)+6 ≥ 0;
log 2 2 (3 x)+log 3 √2 (3 x) ≥ 4
log 1 2 (x2 4x +3) ≥−3;
log 1,5 2x 8 x 2 < 0;
log 0,3 (x2 5x +7) > 0;
log 0,3 log 6 x2 + x x +4 < 0;
log 2 log 3 x +1 x 1 < log 1 2 log 1 3 x 1 x +2 .
2log 2 0,5 x + xlog 0,5 x > 2 5;
3log x+2 < 3log x2 +5 2;
5log 3 x 2 x < 1;
2 5 log 0,25 (x2 5x+8) ≤ 2 5.
log x (x3 x2 2x) < 3;
log x 5 (x2 8x +16) ≥ 0;
log x 3x 1 x2 +1 > 0;
log 2x (x2 5x +6) < 1.
r =5cm
sin15◦ , cos15 ◦ = 1 2 2+ √3.
cos22 ◦ 30 , sin22◦ 30 = 1 2 2 √2.
sin cos α, 1 6 5 6 ; 4 √65 7 √65 ?
sin3 x +cos 3 x sin3 x cos 3 x , tg x =2.
3sin α cos α sin α +2cos α =1, tg α. tg α +ctg α =3,
tg α ctg α; tg 2 α ctg 2 α; tg 3 α ctg 3 α. tg α +ctg α = p, tg 2 α +ctg 2 α.
sin x +cos x = s sin x cos x = p, p = 1 2 (s2 1). sin α +tg α cos α +ctg α α.
sin α cos α, 3sin α +4cos α =5. sin2 α + sin2 β =1?
2 cosec2 α tg α 1 cosec 2 α +1; cos α 1+sin α +tg α;
sin3 x +cos3 x sin x +cos x + sin3 x cos3 x sin x cos x + sin4 x cos4 x sin2 x cos2 x .
sin α 1 cos α = 1+cos α sin α ; 1 2cos2 α
sin α cos α =tg α ctg α;
3(sin4 α +cos 4 α) 2(sin6 α +cos 6 α)=1;
sin3 α(1+ctg α)+cos 3 α(1+tg α)=sin α +cos α; 2sin α cos α ctg α sin α cos α =2tg 2 α.
sin2 x sin x cos x cos2 x(sin x +cos x) sin2 x cos2 x =sin x +cos x; (tg α sin α)2 +(1 cos α)2 = 1 cos α 1 2 ; tg α + 1 cos3 α 1 sec α tg α = sin2 α cos3 α ; 1+sin α +cos α +tg α =(1+cos α)(1+tg α).
sin2 α cos2 α +cos4 α cos2 α sin2 α +sin4 α ; tg α +ctg β ctg α +tg β tg α · ctg β ; sin α cos α ctg α 1 (sin α +cos α)2 ; 1+ctg 2 α 1 sin2 α ; tg α +ctg α 1 sin α cos α ; Æ tg α(cos α cos3 α) cos( α)cos α sin α sin( α)=1.
ctg 5π 3 ; cos315 ◦ ; sin300 ◦ ; tg 7π 4 .
sin 9π 4 ; cos(6π +1); sin(8π 1); sin8 5π. (a2 + b2 )tg π 2 + α cos(3π α) (a2 b2 )ctg(2π α) sin 5π 2 α .
(α ∈ R)
sin α 2sin(π α)
cos(π + α) cos α = 1 2 tg α; sin(π α)
cos π 2 α +cos(2π α) sin π 2 + α sin π 2 α cos π 2 α = tg 2 α +1 tg 2 α 1 ; cos 3π 2 α ctg π 2 + α cos( α)
cos(2π + α)tg(π α) = sin α.
sin750◦ · cos390◦ · tg1140◦ ctg405◦ · sin1860◦ · cos780◦ ; cos 17π 6 · sin 7π 3 · tg 17π 4 ctg 10π 3 cos 7π 4 sin 8π 3 .
sin(α + β )=sin α cos β +cos α sin β, sin(α β )=sin α cos β cos α sin β, cos(α + β )=cos α cos β sin α sin β,
cos(α β )=cos α cos β +sin α sin β,
tg(α + β )= tg α +tg β 1 tg α tg β ,
tg(α β )= tg α tg β 1+tg α tg β ,
ctg(α + β )= ctg α ctg β 1 ctg α +ctg β ,
ctg(α β )= ctg α ctg β +1 ctg α ctg β .
α β tg α = 1 2 tg β = 1 3 , α + β = π 4 . tg α = √2+1 √2 1 , tg β = 1 √ 2 α,β ∈ 0, π 2 , α β = π 4 .
sin(α + β )sin(α β )=sin2 α sin2 β ;
cos π 3 α = 1 2 (cos α + √3sin α);
sin π 4 + α = √2 2 (sin α +cos α). (2+3tg 2 y )tg(x y )=tg y, 2tg x 3tg y =0
cos(x y ) cos(x + y ) = ctg y +tg x ctg y tg x ;
cos(α + β )
sin(α + β ) cos(α β ) = tg α +tg β 1+tg α tg β ;
cos(α β ) = 1 tg α tg β 1+tg α tg β ; cos α +sin α cos α sin α =tg π 4 + α .
sin2α =2sin α cos α.
cos2α =cos2 α sin2 α.
tg2α = 2tg α 1 tg 2 α .
ctg2α = ctg 2 α 1 2ctg α
sin2α = 2tg α 1+tg 2 α ; cos2α = 1 tg 2 α 1+tg 2 α ;
sin3α =3sin α 4sin3 α; cos3α =4cos 3 α 3cos α.
sin3 α = 3sin α sin3α 4 ; cos3 α = cos3α +3cos α 4 .
sin2α, cos2α tg2α
cos α = 4 5 α ∈ 3π 2 , 2π ; sin α = 3 5 α ∈ π 2 ,π
sin2α, cos2α tg2α
cos α = 5 13 sin α> 0; sin α =0 6 α ∈ 0, π 2 .
2tg15◦ 1 tg 2 15◦ ; 1 2sin2 α +cos2α. (cos5+sin5)2 =1+sin10.
sin15◦ cos15 ◦ = 1 4 ; 1 4sin2 α cos 2 α =cos 2 2α.
tg x =2 √3,
sin2x; cos2x; tg2x; ctg2x. tg π 3 = √3, sin 2π 3 ; cos 2π 3 ; tg 2π 3 ; ctg 2π 3
sin4x =4sin x cos x(1 2sin 2 x);
cos4x =8cos 4 x 8cos 2 x +1.
2sin2 α +cos2α =1; 1+cos2α 1 cos2α =ctg 2 α;
sin2α sin α 1 cos α +cos2α =tg α; cos4 α +sin4 α =1 0 5sin2 2α.
tg(α +45◦ )+tg(α 45◦ )=2tg2α;
cos2α sin2α tg2α = cos4α cos2α ;
1+2cos2α = sin3α sin α .
sin x = 1 4 ,
sin3x; cos3x; tg3x.
cos3 x cos3x cos x + sin3 x +sin3x sin x =3;
tg3x tg2x tg x =tg3x tg2x tg x. tg3x =tg π 3 x tg π 3 + x .
sin α 2 = 1 cos α 2 .
cos α 2 = 1+cos α 2
tg α 2 = 1 cos α 1+cos α .
ctg α 2 = 1+cos α 1 cos α .
tg α 2 = 1 cos α sin α , α = π (2k +1) k ∈ Z;
tg α 2 = sin α 1+cos α α = π (2k +1) k ∈ Z. A = sin x +2cos x tg x ctg x , tg x 2 =2.
sin5x sin3x; cos x 2 cos x 3 cos x 4 ; 3sin4x cos5x 7 ;
cos7x cos5x; sin(α β )cos(α + β ); Æ cos(α + β )cos(2α + β ).
sin2 α cos α = 1 4 (cos α cos3α);
sin π 4 + α +sin π 4 α = √2cos α;
sin π 6 + α +sin π 6 α =cos α.
1+sin2α sin α +cos α ; sin α sin β sin α +sin β . tg π 4 + α tg π 4 α , tg2α =3.
sin x +sin2x +sin3x; sin20◦ +sin34◦ +sin24◦ +sin30◦ (cos α +cos β )2 +(sin α +sin β )2 =4cos 2 α β 2 ;
sin α 2sin2α +sin3α cos α 2cos2α +cos3α =tg2α. cos x cos α cos x cos β = sin2 α cos β sin2 β cos α tg 2 x 2 =tg 2 α 2 tg 2 β 2 . sin x +sin y +sin z, x + y + z = π. a sin α + b cos α, a,b ∈ R a2 + b2 =0
f (x)=sin2x; f (x)=cos x 7 ;
f (x)=sin2x cos5x; f (x)=sin 3x 7 +cos x 3 +tg 2x 5
f (x)= 1 1 cos x . f (0),f (π/3) f ( π/4).
f (x)=sin4x √3cos4x;
f (x)=sin x +cos x; f (x)=sin x cos x;
y =sin x +2; y =5 cos x.
y = sin x; f (x)= 3 2 sin x;
f (x)= cos2x; y =sin2x; y =sin x 2 .
f (x)=cosec x.
f (x)=sec x.
f (x)=2sin 4 3 x + π 3 ; f (x)= 3 2 sin 2x π 3 ;
f (x)= 4 3 sin 2x + π 4 .
f (x)= 2sin x 2 + π 6 .
f (x)=2cos x 2 π 3 .
sin x = a, |a|≤ 1
cos x = a, |a|≤ 1
tg x = a
x =( 1)n arcsin a + nπ,n ∈ Z = arcsin a +2kπ, arcsin a + π (2k +1), k ∈ Z
x = ± arccos a +2nπ,n ∈ Z = arccos a +2nπ, arccos a +2nπ, n ∈ Z
ctg x = a x =arctg a + nπ,n ∈ Z x =arcctg a + nπ,n ∈ Z
2sin x √3=0; sin2x 1=0;
2cos2x 1=0; cos πx = √3 2 .
sin2 x +2sin x =0; 2sin x cos x sin x =0.
cos3x +cos5x =0; sin x π 2 +sin 3π 2 + x = √2
sin x +cos x =1+sin x cos x; cos2x √2sin x cos2x =0
tg3x cos x =0
3tg 3 x +tg x =0.
tg(2x +1)ctg(x +1)=1.
sin2x = 1 2 .
cos(sin x)= √3 2
cos2 x = 1 2 .
sin2 x = 3 4 ;tg 2 x = 1 3 ; cos x2 = 1 2
4sin2x cos2x +1=0, (0,π ).
2sin x +3sin2x =0; 2cos2x 3cos x +2=0;
3cos x +2sin2x =0; 2sin x +3cos2x 3=0.
cos x 2 =1+cos x; 2cos x 4 cos x 2 =1;
1 2sin x 6 =cos x 3 .
sin x 1+cos x =sin x 2 .
cos3x +2cos x =0.
sin3x sin x + cos3x cos x = 5 2 +cos4x.
2cos 2 x +3cos x 2=0; 2sin2 x +sin x 1=0.
sin3x +cos2x =1.
sin4 x +cos 4 x = 7 2 sin x cos x.
2sin2 x +cos2 x = 3 2 sin2x; 8cos 2 x +6sin x 3=0. cos x sin5x = 1 2 sin4x; cos2x cos3x =cos5x.
sin3x sin2x =sin11x sin10x;
cos x cos3x =cos5x cos7x.
sin6x +sin4x =0; sin x =cos2x.
sin x +sin2x +sin3x =0; cos x =cos3x +2sin2x.
√3sin x +cos x =1; sin x √3cos x =2;
sin x + √3cos x = √2; √2sin x √2cos x =1.
sin13x +cos13x = √2sin17x.
12cos x 5sin x = 13.
sin x +cos x =1
2sin x √3 > 0; 2cos x +1 < 0; tg x √3 ≤ 0; ctg x √3 > 0; 2sin x +1 > 0; Æ 2sin x √3 ≤ 0
sin x cos x> 0; sin x cos x< 0.
sin x +sin2 x +sin3 x> 0.
cos2x sin2x ≥ 0.
sin x +cos2x> 1
| sin x| > 1 2 .
sin2x> cos x; sin 3π 2 x > √3 2
2sin x cos x ≥ √2 2
sin(3x 1) < 1 2 . a sin x + b cos x + c,a =0,b =0,c =0
a =31,α =54◦ 15 ,β =76◦ 20 ;
b =6,α =37◦ 25 ,γ =102 ◦ 45 .
b =18,c =13,α =44◦ 30 ;
a =13 48,c =7 02,β =138◦ 27 .
a =10,b =18,α =28◦ 35 ;
a =9,c =16,γ =81◦ 20 .
a =10,b =18,c =9;
a =2,b =3,c =4. A B C D A B Æ CD =2570 m ,<BCD =79◦ 34 ,<ACD =32◦ 31 , <BDC =33◦ 34 ,<ADC =78◦ 45 M A,B C AB = c,AC = b,<CAB = α, <AMN = β,<AMC = ε, CABM MA,MB
MC ?
c =123 3 m ,b =282 6 m,α = 78◦ 35 ,β =28◦ 44,ε =41◦ 16 ?
R,β,γ ; b + c,a,β γ ; b + c,a,α.
s =24 5,α =18◦ ,γ =12◦ 15 ; R =3 5,b + c =8,α =51◦ 10 ; S =86,a =12,b2 + c2 =574; R =26,a =42,hb =31.
a =2√2,α =45◦ ,β =120 ◦ ; a =3+ √3,b =3√2,α =75◦ ; a = √6,b =2√3,c =3 √3; b = √6,c =3+ √3,α =45◦ 200 m α. 1:2 a =15,c =7, b =9 d =6. Æ
100cm 2 , 21cm . d α. P = 1 2 d2 sin2α.
ABCD AB =32, BC =34,DA =20 AC =17.
α : β =1:2 a : b =1: √3.
a,b,c, α,β,γ s = a + b + c 2 , P R, P = 1 2 b · c · sin α
P = s(s a)(s b)(s c)
h =250km ,α =46◦ 30 ,t =30s,R =6370km .
1 2 ; 8; Æ 1 4 ; 1 2 ; 34. ( 2)55 51 =( 2)4 =16; 23 :4=2; ( 3)168 :3168 =1; 2 5 + 3 7 12 5 2 = 14+36 35 2 =0 49; 4 5 · 4 3 + 24 34 1 = 26 34 1 = 34 26 = 81 64 ; 1 64 . a 7 ; a3 ; a7 ; 3 4 a; Æ 1 6 a3 ; (ab)2 ; a 3 · b3 ; a 3 · b5 ; 3 25 a5 b.
a2x ; a(a b)10 ; 10an
a6 cx3 b4 ; 4 · a10 b6 x4 c2 z 4 y 10 ; a4 c5 x5 z 15 b14 y 10 . an ; am+n ; a4n m ; (a2 b2 )m ; am 1; Æ ( 1)2n 1 = 1; 0 5(a + b)2m ; an + bm
0 5a 3 b3 · 5a4 b 2 =2 5ab. a = 0 125 b =16, 5. 1 2 x3 y 3 ; x = 1 4 y = 8 4. N =
)
x2 a2 ax ; x a ;
5 x =6. x1 = 5 6 ; x1 = 10 3 ; x1
√9 · √16=3 · 4=12; √7 · 4 · 7= √72 · 22 =14; Æ 3 4 ; 5 9 ; 5 4 ; 13 12 ; 5 2 ; Æ 5 13 ; 5 4 ; 3 10 . 3; 2; 1 5 ; Æ 10 3 ; 0 9 2 15√ 7; 15√ 2; 4ab2 ; 4a2 b6 c√3ac ; Æ 3b10 cn √7ab ; 2a3 c2n √2a. 5√7= √52 · 7= √175; √2 5; √3; 35 2 ; √a; Æ 4a2 b 4a = √ab; √b10 ; a2 b4 a b3 = √a3 b; (x 1)2 (x 1)(x +1) = ⎧ ⎪ ⎨ ⎪ ⎩ x 1 x +1 ,x> 1 x 1 x +1 ,x< 1; a>b (x + y )2 (a b)2 (a b)2 (x + y )y = x + y y , a<b (x + y )2 (a b)2 (a b)2 (x + y )y = x + y y . 3(x 2); 2(x +3); x(x +2); 5x(x 1) . x a a2 x 2x 1 x 3a x a2 = |
| a √x 2√x 3 a |a| = √x = |a| a 3 a |a| 2 √x = 0, a< 0, 4√x, a> 0; (10|xy |− 4y +2x|y |√a =8xy √a; (|xy | +3x|y |− 4|x|y )√ a + b =6xy √a + b. 6 √a7 b5 = a 6 √ab5 ; 12√x23 = 12√x12 · 12√x11 = x 12√x11 ; 30√a30 x31 = 30√a30 x30 · 30√x = ax 30√x; x; 2a; Æ 12√b5 x.
(1+ i)2 =1+2i + i2 =2i; 2i; (1+ i)4 = (1+ i)2 2 =(2i)2 = 4; 1 4 . 2i +1. z1 + z2 =7+2i,z1 z2 = 1 2i; z1 + z2 =4,z1 z2 = 6i; z1 + z2 =14 i,z1 z2 = 4+9i; z1 + z2 = 2 3 5i,z1 + z2 = 2 3 +5i. 26 2i; 52 +35i 35i +72 =74; 3+ i; 3 2i; 9+13i. 17+2i. 33 +3 · 32 · 2i +3 · 3 · 4i2 +8i3 =27+54i 36 8i = 9+46i; 2+2i; 2 2i; (3+4i)2 = 7+24i. 5+6i 2+ i = (5+6i)(2 i) (2+ i)(2 i) = 10 5i +12i +6 4+1 = 16 5 + 7 5 i; i; 5 2 + 3 2 i; 6 25 17 25 i; 12 13 + i 5 13 . 1 i; 1; 1 2 + 41 26 i; 1 2 + 3 2 i. a = 20,b = 50. f (2+ i)=0. z = (2i)50 ( 2i)48 i · (2i)49 = 4 3 . |1+ i| = √12 +12 = √2; √2; √5; √2 2 ; z1 = a + ib,z2 = c + id. z1 · z2 =(ac bd)+ i(ad + bc), |z1 · z2 | = (ac bd)2 +(ad + bc)2 = a2 c2 + b2 d2 +
= (a2 + b2 )(c2 + d2 )= a2 + b2 · c2 + d2 = |z1 |·|z2 | Æ z1 = z2 5 34 ; √13 3 ; 5√ 5 z1 = a + ib,z2 = c + id.
z1 + z2 = a + ib + c + id = a + c + i(b + d)= a + c (b + d)i = a bi + c di = z 1 + z 2 . 2 3 i. p(z0 )=11 2i,p(z 0 )=11+2i.
|z |2 = a2 + b2 ; a + √a2 + b2 + ib; a2 b2 2abi; a ib = a + ib = z ; |a ib| = √a2 + b2 = |z |; Æ a2 b2 a2 + b2 + i 2ab a2 + b2 z1 = a + ib,z2 = c + id. z1 z 2 + z 1 z2 =(a + ib)(c id)+(a ib)(c + id)=2(ac + bd) ≤ 2 (ac bd)2 +(ad + bc)2 =2|
i, i; i, i; 1; 3 2 2i. z =1+ i; z = 3 2 + i 3 2 ; z = 5 2 + 5 2 i. z = x + iy. x2 y 2 +2xyi = 2, x2 y 2 = 2 2xy =0. x =0 y =0. x2 = 2, x x =0, y 2 =2, y = ±√2. z1 = i√2 z2 = i√2; z1 = 1+ i √2 ,z2 = 1+ i √2 ; z1 = 1 i √2 ,z2 = 1+ i √ 2 ; z1 =1+2i,z2 = 1 2i. z 3 =1. z 2n + z n +1= 3,n =3k 0,n =2k ± 1, k ∈ Z. (x 3)(x +3)=0,x1 =3,x2 = 3; x1,2 = ±√5; x1,2 = ±2√2; x1,2 = ± 5 2 ; x(x +4)=0,x1 =0,x2 = 4; Æ (x 1)2 =0,x1,2 =1; (x 2)(x 3)=0,x1 =2,x2 =3; (x +1)(x 4)=0,x1 = 1,x2 =4; 2(x 1) x 1 2 =0,x1 = 1 2 ,x2 =1. x1 = x2 =0; x1 = x2 =0; x1 =0,x2 =6; x1 =0, x2 = 7; x1 =0,x2 =4; Æ x1 =0,x2 = 3 5 ; x1 =0,x2 = 4 3 ; x1 =0,x2 =3; x1 = 5,x2 =5; x1 = 10,x2 =10; x1 = √7,x2 = √7; x1 = 5 2 ,x2 = 5 2 ; x1 = 1 13 ,x2 = 1 13 ; x1 = i 2 3 ,x2 = i 2 3 ; x1 = i 5 2 ,x2 = i 5 2 , x1 = iπ,x2 = iπ.
a =0, a = 2, x1 = 1 3 , a =1 x1 = 1; x1 = 1 x2 = a +1 a 1 . z1,2 = 4(2 i) ± 16(2 i)2 +8(1+ i)(5+3i) 4(1+ i) , z1 = 4 i 1+ i = 3 5i 2 ,z2 = i 1+ i = 1+ i 2 ; D = 2i =(1 i)2 ,z1 = 2(2 i) 5 ,z2 =1 i. z1 =1+ √3 2 + i 2 ,z2 =1
3 2 + i 2 . x ≥ 1, x2 + x 2= 0,x ≥ 1, x1 =1. x< 1, x2 x =0,x< 1, x2 =0; |x|2 = |x2 |. x1 =4,x2 = 4; x1 =2,x2 =0; x1 = 1 2 ,x2 =2,x3 =
,x
; x1 = 1, x2 =1. x2 +4x +2 ≥ 0, x2 +4x +2 ≥ 0, 3x2 +7x 10=0, x1 =1 x2 +4x +2 < 0, x2 +4x +2 < 0, 3x2 +17x +22=0, x2 = 2; x1 =1,x2 =4 sin α =
3 sin
x1 = m 1,x
,
<m 1 < 4 2 <m +1 < 4, 1 <m< 3. |a| > 2 |a| =2 (x1 = x2 = a) |a| < 2
D =(p +2a)2 4(q + ap)= p2 4q +4a2 ≥ 0;
D = p2 4q +(p + a)2 ≥ 0.
D1 =4b2 (a2 + b2 c2 ) D2 =4(c2 2ab). D1 + b2 D2 =4b2 (a b)2 ≥ 0,
D1 < 0 D2 < 0
D + D1 =(p p1 )2 ≥ 0, D< 0
D1 < 0 k = 3 k = 3
D =16k +64, k ≥−4.
D1 =4a2 c2 4(a2 + b2 )(c2 b2 )=4b2 (b2 + a2 c2 ),D2 =4c2 2ab.
D1 +4b2 D2 =4b2 (b a)2 ≥ 0 D1 ≥ 0 D2 ≥ 0
D =4(p + q + r )2 12(p2 + q 2 + r 2 )= 4 (p q )2 +(q r )2 +(r p)2 ≤
0
D =(b2 + c2 a2 )2 4b2 c2 =(b2 + c2 a2 +2cb)(b2 + c2 a2 2cb) = (b + c)2 a2 (b c)2 a2 =(b + c + a)(b + c a)(b c + a)(b c a ). a,b,c b + c + a> 0, b c a< 0,b + c a> 0,b c + a> 0, D< 0. x0 x 2 0 x0 + a =0 x 2 0 ax0 +1=0. x0 (1 a)= (1 a). a =1 x0 = 1. x0 = 1, a = 2 a =1 a = 2. a1 = 2,a2 =1; a =3.
D =4(a2 + b2 + c2 ab bc ca)=2 (a b)2 +(b c)2 + (c a)2 > 0, x1 = a + b + c 1 2 √D, x2 = a + b + c + 1 2 √D, x1 <a + b + c<x2 x2 +(7 x)2 =10x +7 x +4
x km/h
80 x +4 + 80 x 4 =8 1 3 , x =20 96 x 12 96 x = 2 3 . 48km/h , 36km/h . 10x +5=(x +3)(x 13), (x 3) 720 x +40 =720, x 1 x 10 + 1 x = 1 12 . x1 =30 x2 =4,
1cm . 16km/h . x1,2 = a ± √a2 + a +3, a2 + a +3= k 2 , k ∈ N. a1,2 = 1 ± √4k 2 11 2 . 4k 2 11= m2 ,m ∈ N 4k 2 m2 =11 (2k m)(2k + m)=11, 2k m =11 2k + m =1 2k m =1 2k + m =11. k =3 m =5, a1 = 3, x1 =0,x2 = 6 a2 =2 x1 = 1,x2 =5 D =4n2 4n +5. D =4n2 4n +5= k 2 ,k ∈ N. k 2 (2n 1)2 =4, (k 2n +1)(k +2n 1)=4=1 · 4=2 · 2. k n (a2 b2 ) x2 2ax +1=0; (a 1)x2 a2 x + a +1=0; x2 4x +7=0; x2 a2 + b2 ab x +1=0.
y +256=0
(k 2 2) 2 2= k 2 (k 2 4)+2, x1 + x
=
1 x
=1. k 2 4 > 0, k< 2 k> 2 (2a +1)2 ; (a 3)(a +2); (a +1)(a +3); 2 x 1 2 (x 2); 1 2 (y 4z )(y 2z ); Æ x(x2 2x 8)= x(x 4)(x +2). (a 3)(a +2) (a 2)(a +2) = a 3 a +2 ; a +2 a +5 ; x 4 x 3 ; x 5 x 1 ; x +2 4x +3 . x 2 1 + x 2 2 =(x1 + x2 )2 2x1 x2 =(a 1)2 +1 ≥ 1. a =1; a =1. x 3 1 + x 3 2 =(x1 + x2 )3 3x1 x2 (x1 + x2 )=(x1 + x2 ) (x1 + x2 )2 3x1 x2 = 1 2 a2 1 2 a +1= 1 2 a + 1 2 2 + 9 8 . x 3 1 + x 3 2 a = 1 2 . p + q = p,pq = q. (1, 2) (0, 0). r = ±1; r = 2
kz 2 +(2k +20)z +60 3k =0; k1 = 5 3 ,k2 =15. x1 =2x2 x1 + x2 = k +3,x1 x2 =4k +2, x1 =2 k +3 3 ,x2 = k +3 3 , 2 k +3 3 2 =4k +2, k1 =0,k2 =12. x2 3x +2=0, x1 =2,x2 =1, x2 15x +50=0, x1 =10 x2 =5. p< 1; p1 =1,p2 = 15 4 . a2 + b2 = (x1 + x2 ) 2 +(x1 x2 1)2 =(x 2 1 +1)(x 2 2 +1). p = 2,q = 1 p =1,q x1,2 = ±7,x3,4 = ±1; x1,2 = ±1,x3,4 = ±4; x1,2 = ±2√2,x3,4 = ±i; x1 =1,x2 = 5,x3,4 = 2 ± 1 3 i; x1,2 = ±2i,x3,4 = ±3i; Æ
b =8,c = 10. T b 6 , b2 +84 12 ,ymin =4, x =1, b =6. a = 1. a = 9 7 4(m +1) m2 4 = 16(m +2) 4m2 8 m1 =6,m2 = 2. y = x2 +6x 8
p ∈{−4, 3, 3, 4}. k = ±4. x 12 x P (x)= x(12 x). P (x)= x2 +12x x =6. 11 11=121 f (x)= x + x2 , x x1 = 1 2 f (x)= x2 +(64 x)2 =2x2 128x +642 x1 =32 y = (x2 x +2) 2 x2 x +2 =1 2 x2 x +2 =1 2 x 1 2 2 + 7 4 . x = 1 2 y (1/2)= 1 7 ;
y = 5 (x2 +1)2 1 x2 +1 +1= 1 5 5z 1 2 2 + 19 20 ,z = 1 x2 +1 . 19 20 z = 1 10 , x = ±3; y =(x 3)+ 1 x 3 +2 ≥ 2+2=4, a + 1 a ≥ 2 a> 0. x =4. y =2 · (2x2 4x +8)+1 2x2 4x +8 =2 1+ 1 2(x 1)2 +6 . 7 3 x =1; 1 x2 ≥ 0, 1 ≤ x< 1. y = f (x)=3x +4√ 1 x2 . x> 0 f (x)=3x + 4√1 x2 > 3 · ( x)+4 1 ( x)2 , f (x) >f ( x) x< 0 x ≥ 0 y 2 =9x2 +16 16x2 +24x√ 1 x2 = (4x 3√1 x2 )2 +25 ≤ 25. y = √25=5 4x 3√1 x2 =0, x = 3 5 . x = 5 6 . f 5 6 = k 2 3k +2=0 k1 =1,k2 =2. T (m +1,m2 + 2m 1). x = m +1,y = m2 +2m 1, m,m = x 1, y = x2 2. M (t,t2 2),t ∈ R y = x2 2tx +2(t2 1), (m = t 1) y = x2 2; y = x 3 4 . D = m2 +2m +5=(m +1)2 +4 > 0; m1 = 1,m2 = 11 3 ; y = x2 +4x 5 D =(k 2)2 ≥ 0; f (x k ) 2x = x2 kx, k =7, ymin = 49 4 x = 7 2 . M (1, 2)
1◦ a b c x = b 2a y = 4ac b2 4a a = b 2x , y = 4 · b 2x · c b2 4 · b 2x = bx 2 + c.
2◦ b a c y = ax2 + c. 3◦ c a b, x = b 2a . x2 f (x) < 0 x; x = 3 m 2 , m =3 2x, y = 4(1 2m) (m 3)2 4 = x 2 +4x 5; m(x 2)+ x2 3x +1 y =0. x =2 y = 1 M (2, 1). D =4 (m +1)2 2m(m +1) =4(m +1)(1 m). |m| > 1 f (x) < 0 x, m +1 < 0, m< 1. m = 1 f (x)= 2 < 0, m ≤−1; x = 1,y = 4(m +1)(m 1) 4(m +1) = m 1, m +1 =0. x = 1 ( 1, 2); m(x2 +2x +2)+ x2 +2x y =0 m, x2 +2x +2=0 x2 +2x y =0, f (x)= ax2 + bx + c. f (0)= c, f (1)= a + b + c. c(a + b + c) < 0, f (0) f (1) x0 , 0 <x0 < 1. c(a + b + c) < 0, b2 b2 4c(a + b + c) > 0, (b2 4ac) (b +2c)2 > 0, b2 4ac> 0,
D =4(m 1)2 4(m +5)=4(m +1)(m 4) > 0 f (x)= x2 2(m 1)x + m +5, f (3)=5(4 m) f ( 2)=
5(m +1), f ( 2) · f (3)=25(m +1)(4 m) < 0,
. a> 11 9 . f (x)= x2 + ax + a 1, D> 0,f (2) > 0 x1 + x2 2 > 4. (a 2)2 > 0,a> 1 a< 4, a m< 1 ∨ 2 <m< 5 2 ∨ m> 38 11 .
,
<a< 1 4 . m< 4 7 ∨ 13 7 <m< 2 ∨ 3 <m. 3 <m< 11 3 .
(k +3)2 4k (k 2) > 0, 1 3 (7 2√ 19) <k< 1 3 (7+2√19) (∗) k =2 g (x)= x2 2(k +3) k 2 x + 4k k 2 . Æ x1 x2 g (x), g (2) < 0 g (3) < 0, 4 2(k +3) k 2 2+ 4k k 2 < 0 9 2(k +3) k 2 3+ 4k k 2 < 0. 2 <k< 5. (∗), k (2, 5).
x< 0 ∨ x> 3; x ∈ [ 4, 0]; x< 1 ∨ x> 3 2 ; x =6; 3 <x< 0; Æ 1 <x< 3; x ∈ (−∞, +∞); x =2. x2 + x +1 > 0 x, 4 x2 > 0, ( 2, 2); x(x2 3x +2) > 0. (0, 1) ∪ (2, +∞); x< 2 ∨ x> 3; x< 1 2 ∨ x> 1; x< 3 ∨ 3 √7 2 <x< 3 √7 2 ∨ x> 4; Æ x< 1 ∨ x> 1; x ≤−2 ∨ x =1 ∨−1 <x< 4; x< 3 ∨−2 <x< 3; 3 <x< 4; 1 ≤ x ≤ 6. |2x 3 |− x2 +4x 2 x2 3x +2 < 0 ⇔ x ≥ 3 2 ∨ x2 +6x 5 x2 3x +2 < 0 ∨ x< 3 2 ∧ x2 +2x +1 x2 3x +2 < 0 ⇔ x ≥ 3 2 ∧ x 5 x 2 > 0
∨ x< 3 2 ∧ x (1 √2) x 1 > 0 ⇔ x< 1 √2 ∨ 1 <x< 2 ∨ x> 5; 3 <x< 4 ∨ 1 <x< 2; 1 <x< 3; 1 <x< 3
x2 + x +1 > 0 3(x 2 + x +1) <x2 3x 1 < 3(x 2 + x +1), x< 2 ∨ x> 1; Æ x ≤−4 ∨−1 ≤ x ≤ 1 ∨ x ≥ 4; 0 ≤ x ≤ 8 5 ∨ x ≥ 5 2 . p> 3; p< 1 2 √2 7 ; √2 1 <p ≤ 1. 6 <a< 2; a ≤−1; a< 3 √5 2 ∨ a> 3+ √5 2 . x 2 1 + x 2 2 = 1 4m m2 (m +1)2 ≥ 1 m ∈ [ 3, 0], m = 1; 1 x 2 1 + 1 x 2 2 = 4(m +1)2 2m m2 > 8 m ∈ ( 1/2, 2) m =0. x2 5x +5= t. (x 1)(x 2)(x 3)(x 4)+1 001 =(t 1)(t +1)+1 001= t2 1+1 001= t2 +0 001 > 0. f (x)=(y 2 +1) 2 x 2 4y (y 2 1)x +4y 2 (y 2 +1)2 > 0, f (x) ≥ 0 f (x) D =16y 2 (y 2 1)2 16y 2 (y 2 +1) 2 = 64y 4 ≤ 0 n =3. x k 2 > 0 D = 4(k +6)(k 4) < 0, k> 2 k< 6 k> 4. k =5. D ≥ 0,x1 + x2 < 0,x1 x2 > 0
a 2 2a 15 ≥ 0,a +1 < 0,a +4 > 0
4 <a ≤−3; 0 <a< 25 12 . 3 ≤ a ≤ 1; a ∈ [4, +∞) x, D =(y +1)2 4(y 2 y +1)= 3(y 1)2 . D ≥ 0, y =1, x = 1. Æ ( 1, 1).
D =4 (2y 3)2 (y 2)(5y 6) = 4(y 1)(y 3) ≥ 0, y =1 y =2 y =3. ( 1, 1), ( 2, 2) (3, 3); ( 3, 1) (3, 2) x =1+ y. (1+ y )2 (1+ y )y + y 2 2(1+ y )+3y 2=0, y 2 +2y 3=0, y1 =1,y2 = 3. x =1+ y x1 =2,x2 = 2, (2, 1) ( 2, 3); (2, 4), ( 4, 5); (4, 2), (2, 4); (3, 1), (1, 3); (3, 2), ( 2, 3). y 2 , x y 2 3 x y +2=0, x y =2 x y =1. x =2y,x 2 3x y +3=0 x = y,x 2 3x y +3=0. (2, 1) 3 2 , 3 4 , (1, 1) (3, 3); (1, 1), ( 1, 1), i √3 3 , 4i √3 9 , i √3 3 , 4i √3 9 .
4, 2x2 15xy +7y 2 =0, x =7y y =2x. x =7y,x 2 +3xy y 2 =3 y =2x,x 2 +3xy y 2 =3. 7 √23 , √23 23 , 7 √23 , √23 23 , (1, 2), ( 1, 2); 4 (x y )2 =1. 9 2 , 7 2 , ( 3, 4), (3, 4) 9 2 , 7 2 ; (3, 2), 25 √113 , 16 √113 , ( 3, 2), 25 √113 , 16 √113 ; ( 4, 5), ( 3√3, √3), (3√3, √3), (4, 5); (2, 3) (c, 1),c ∈ R x2 = u y 2 = v u + v =25, u +2v =41 (9, 16), x = ±3,y = ±4, (3, 4), (3, 4), ( 3, 4) ( 3, 4). xy = u,x + y = v. (5, 1), (1, 5), (2, 3), (3, 2). xy = u,x2 y 2 = v. u1 =9,v1 = 20 u2 = 20,v2 =9. ± 10+ √181, ∓ 9 10+ √181 , ±i 10+ √181 , ± √181 10 , (±5, ±4), (±4i, ∓5i) (0, 3) (1, 2) A(5, 4) B (2, 2)
(1, 2) 1 √17 2 , 3+ √17 2 ;
(3, 2) Æ (a, 0), (0,a), a< 0 a =0 (0, 0), a> 0 ( √a,a) (√a,a) a< 17 4 a = 17 4 1 2 , 15 4 . a> 17 4 ,
a< 1 a = 1 (1, 0),
1 <a ≤ 0 ( √a,a + √ a ) (1+ √1+ a,a +1+ √1+ a ), a> 0 (1 √1+ a,a +1 √1+ a )
(1+ √1+ a,a +1+ √1+ a ).
(x y )(x 2 + xy + y 2 19)=0, (x + y )(x 2 xy + y 2 7)=0
1◦ x y =0 x = y (0, 0), (√ 7, √7), ( √7, √7).
2◦ x + y =0. y = x 2x3 = 38x, (0, 0), (√19, √19), ( √19, √19).
3◦ x y =0 x + y =0. x2 + xy + y 2 =19,x2 xy + y 2 =7 x2 + y 2 =13, xy =6, (2, 3), ( 2, 3), (3, 2), ( 3, 2). ( 1, 2), (2, 1), 1 ± i√ 3, 1 2 ∓ i √3 2 , 1 2 ± i √3 2 , 1 ∓ i√ 3 ; (2, 3), 3, 2); ( 1, 2), (2, 1) x2 + y 2 = 78 xy , 97+2(xy )2 = 782 (xy )2 . ( 3, 2), ( 2, 3), (2, 3), (3, 2); ( 3, 2), ( 2, 3), (2, 3), (3, 2); (x 2 + y 2 + xy )(x 2 + y 2 xy )=481. (±4, ±3), (±3, ±4); x + 1 x +2 y + 1 y +2 =27, x + 1 x y + 1 y =10. x + 1 x = u,y + 1 y = v. (2, 2 ± √3), 1 2 , 2 ± √3 , (2 ± √3, 2), 2 ± √3, 1 2 .
6cm 8cm
24cm 18cm; 8cm, 15cm , 17cm; 30cm , 40cm , 50cm . (x2 +2x +1= x +7 ∧ x ≥−1) ⇔ (x2 + x 6=0 ∧ x ≥−1) ⇔ (x =2 ∨ x = 3) ∧ x ≥−1 ⇔ x ≥−1 ⇔ x =2 x1 =2; x1 =2,x2 =3; x1 = 15 4 ; y1 =3; Æ x1 =3; x1 =0; y1 =19; y1 =28. 1 x = √3x2 7x +3 ⇔ 1 x ≥ 0 ∧ (1 x)2 =3x2 7x +3 ⇔ x ≤ 1 ∧ 2x2 5x +2=0
x ≤ 1 ∧ x =2 ∨ x = 1 2 ⇔ x = 1 2 . x1 = 1 2 ; x1 = 3; x1 = 1; z1 =1; y = x2 4x +6. x1 =2; Æ x1 = 9 2 ,x2 =3 √2x2 +3x +9= y ) x1 =4; x1 =20; x1 =2; x1 = 2 9 ; x1 =11; Æ x1 = 12 5 ,x2 =4; x2 = 47 24 . √x +1(√4x +5 √2x 1)= √x +1 √x 1. x1 = 1.
4x +5
2x 1= √x 1 7x2 26x 45=0, 9 7 . x2 =5 9 7 x1 =1,x2 = 3+2 √53 7 ; √3x2 +5x 8 < √3x2 +5x +1 y 2 +4y +6= z. y1 = 2;
a> 1, ax >ay x>y, 0 <a< 1, ax >ay x<y. x>y ; x>y ; x>y ; x<y ; x>y. a 2 3 <a 3 4 ; a 2 3 >a 3 4 . f (x)= x2 + x 2 [ 2, 0] f ( 1/2)= 9/4, f ( 2)=0, min y =2 9/4 x = 1 2 , max y =20 =1, x = 2; max y = 1 3 , min y = 1 9 ; max y (x)= 1 3 9/4 =39/4 , min y = 1 3 0 =1. 24(x 0,5) =25(14 x) . 4(x 0 5)=5(14 x) 4x 2= 70 5x, x1 =8; x1 =2; x1 = 15; x1 =6. x1 = 3, x1 = 4 3 ; x1 =1,x
x
=1,x
= 4. 5|4x 6|
, |4x 6| =6x 8. x ≥ 3 2 x< 3 2 x1 = 7 5 ; x1 = 8 7 ; x1 = 3 5 ; x1 = 2 7 x1 =0,x2 = 1 2 ; x1 = 1,x2 =6; x1 =0,x2 = 3 2 ; x1 =1,x2 =2;
2x = y y 2 6y +8=0, y1 =4,y2 =2, x1 =2,x2 =1. 2√x = t, x1 =4; Æ 32x+4 = t, t2 12t +27=0, t1 =9,t2 =3, 32x1 +4 =9, x1 = 1, 32x2 +4 =3, x2 = 3 2 ; 2x = t. x1 = 2,x2 =3; 32(x2 1) 4 · 3x2 1 +3=0. 3x2 1 = t t2 4t +3=0, t1 =1,t2 =3, 3x2 1 =30 , 3x2 1 =31 . x2 1=0 x2 1=1, x1,2 = ±1, x3,4 = ±√2; 2 x 3 2 = y, 2x =8y 2 , 8y 2 2=15y, y1 = 1 8 ,y2 =2 y> 0, x1 =5;
5x 2 = y. x1 =2. 36= 10+4 x 2 2x 2 (2x )2 +10 · 2x 144=0. y 2 +10y 144=0 y1 = 8 y2 = 18, 2x =8, 2x = 18 2x > 0, x1 =3; x1 =2. 2 · 22x 5 · 2x · 3x +3 · 32x =0, 32x ,
.
·
3 x = y, 2y 2 5y +3=0, y1 =1,y2 = 3 2 , x1 =0,x2 = 1
( 2, 0); ( 2, 3); ( 1, 2); ( 1, 2)
23 6x > 20 , 3 6x> 0, x< 1 2 ; x> 3 4 1 3 x > 1 3 2 , x< 2; x ∈ −∞, 1 2 ∪ (1, +∞); x> 2. x ∈ (−∞, 1) ∪ (0, 2); x< 1 2 ,x> 1; 1 <x< 0, 0 <x< 1 0 <x 3 < 1 x 3 > 1. 3 <x< 3 5,x> 4; x< 3, 5 <x< 6 x> 0; x> 0; x ∈ (0, 1); x< 0,x ≥ 1 2 ; 0 <x< 1.
log 2 8+log 2 16+log 2 32=3+4+5=12;
log 3 9 log 3 27=2 3= 1; 2 3 ; 3 2 . 1 81 ; 1 10 ; 1 2 ; Æ 273 ; 5 √100; 1 6561 ; x = 1 25 . log 3 7 √3= 1 7 , log 81 7 √3= 1 28 , log 1 9 7 √3= 1 14 , log 1 27 7 √3= 1 21 .
log x =log3+log a +log b; log x =log 1 2 +log a +log b log c;
log x =2log a +log b +3log c; log x =2log a + 1 3 log b 2log c; log x =log2+log(a b); Æ log x =log a log(b + c) log(b c);
log x =log4 1 2 log5+log a 1 4 log b;
log x = 1 2 log a + 1 3 log b; log x = √b log a +3log c.
log x =log 5 4 2 =log10, x =10; x =21; x =32 · 53 =1125; x = 8 9 ; x = √3 · 3 25 2 ; Æ x = 5 6 ; x = 8 3 . M =1 N =1, 0=0. M =1 N =1, log a N log b N = log a M log b M , log N b log N a = log M b log M a ,
log a b =log a b, x =0
log ab 3 √a b =log ab 3 √a log ab b = 1 3 log ab a log ab ab a = 1 3 · 3 log ab ab +log ab a =1 1+3=3.
log 6 16=4log 6 2= 4 log2 6 = 4 log2 12 log2 2 = 4 1 a 1 = 4a 1 a ;
log 3 5= log 6 5 log 6 3 = b log 6 6 log6 2 = b 1 a 3a +1 a +3 ; log 5 6= log10 6 log10 5 = log10 2+log 10 3 log10 10 log10 2 = a + b 1 a ;
log 10 122 5=log 10 352 log 10 10=2(log 10 5+log 10 7) 1 =2(a + b) 1
log 5 9 8= log9 8 log5 = log 98 10 log 10 2 = log2+2log7 log10 log10 log2 = a +2b 1 1 a ;
log 9 40= log40 log9 = 1+2log2 2log3 . log 100 2 log20 = b, 2 log2 1+log2 = b, log2= 2 b 2+ b , a =log3+log 10 2 ,
log3 log2= a 1, log3=log2+ a 1 log 9 40= 1+2log2 2log3 = 5 b 2ab +2a 4b +2 . log 3 2= 2 a 2a 1 log 3 2= 3 b 3b a . a ∈ ( 1, 1),b ∈ ( 1, 1). log 3 108=3+log 3 4 > 4, log 5 375=3+log 5 3 < 4 1 2 ; √5; 18; Æ 23 1 3 log 2 124 · 8 1 3 =124 · 2=248. x(y + z x) log x = y (z + x y ) log y = z (x + y z ) log z = 1 t . log x = tx(y + z x, log y = ty (z + x y ), log z = tz (x + y z ), y log x + x log y =2txyz,y log z + z log y =2txyz, 2log x + x log z =2txyz. y log x + x log y = y log z + z log y = z log x + x log z, log xy y x =log z y y z =log xz z x , xy y z = z y y z = x z z x . log a (n +1). log 2 log 2 4 √2= log 2 log 2 8 √2= log 2 log 2 2 1 8 = log 2 1 8 log 2 2 = log 2 1 8 = log 2 2 3 =3 · log 2 2=3; n.
log 6 5 √25=25 1 log 6 5 =25log 5 6 =52log 5 6 =5log 5 36 =36 log 8 7 √49=49 1 log 8 7 =49log 7 8 =72log 7 8 =7log 7 64 =64, √36+64=10. a loglog a log a =(alog a 10 ) loglog a =10loglog a =log a. 5=10log5 , 5log20 =10log5 log20 , 20=10log20 , 20log5 =10log5 log20 , 5log20 =20log5 log x = 1 10 log144= 1 10 · 2 15836, 1441/10 = 1 6438; log x = 1 2 log2+ 1 2 log2 ,x =1 6818; log x = 6 5 log754; log x = 1 2 log2+ 1 3 log3+ 1 4 log4 ; I = (A +15) 0 354 3 1 54 4 1/5 , A =34 51/4 301/2 . A. log A = 1 4 · 1 53782+ 1 2 · 1 47712=1 12302 A =13 275. I = K, K =(28 275 · 0 354 3 · 1 54 4 ) 1/5 , log K = 1 5 (log28 275 3log0 354 4log1 45) = 1 5 (1 45141 3 · (0 54900 1) 4 · 0 16137)=0 43179, K =2 7026 I = 2 7026; Æ (x y )2/5 , x =1536 3, y =3040 9
x +1 > 0, x> 1; x> 1 4 ; x ∈ (−∞, 2) ∪ (2, +∞);
x ∈ (−∞, 3) ∪ ( 1, +∞); x ∈ (−∞, 2) ∪ (3, +∞); Æ x ∈ 1 2 , 3 2 ; 2x x2 > 0 2x x2 =1, x ∈ (0, 1) ∪ (1, 2); x ∈ (−∞, 5) ∪ ( 5, 4) ∪ (4, 5) ∪ (5, +∞). x2 3x +2 x +1 > 0, 1 <x< 1 2 <x< +∞, x ∈ ( 1, 1) ∪ (2, +∞); x> 4; x = ±2; x> 0. 1 log 2 3 + 1 log 5 3 =log 3 2+log
. log (3x 1) · (12 x) =log100, (3x 1)(12 x)=100, x1 = 16 3 ,x2 =7, x1 =2; x1 =1; x1 =1; x1 =5; x1 =16; x1 =25. x1 =4; x1 =3; x1 =1+ √3; x1 =8.
(9
x1 =2; x1 =0,x2 =2; 3log 2 3 x =(3log 3 x )log 3 x = xlog 3 x . 9x 1 +
7=4(3x 1 +1) 3x 1 = y. x1 =2, x2 =1; x1 =0; x1 =100; x1 =25; x1 =64. a> 0,a =1,x1 = a, 1 log2 x · log2 a 2log2 x +1=0, log 2 2 x = log2 a 2 , x1 = 2 1 2 log 2 a , 0 <a< 1; a ≤ 0,a =1,a =2 0 <a< 1, 1 <a< 2 a =3 x1 = a +2, 2 <a< 3 a> 3 x1,2 = a ± 2; 0 <a,b,c =1,abc =1 bc =1 ca =1 ab =1, x> 0. x1 =1. 9 6x + x2 > 0 x =3, 9 6x + x2 =1 x =2,x =4 x2 +2x 7= |x 3|. x1 = 5; x1 = 1 4 .
x(4+2log 2 3)= y (2+log 2 3), y =2x. x +log 2 (2x)=2x log 2 3+log 2 x x = 1 2log2 3 1 , 1 2log 2 3 1 , 2 2log 2 3 1 ; 1 2 log2 3 , 2 2 log2 3 .
26 3x > 25, 3x < 1, −∞ <x< 0; x< 1; x< 1; x< 0. x> 1 x 1 2 (x 1) ≤ 1 2 , x2 3 2 x ≤ 0, 0 ≤ x ≤ 3 2 . 1 <x ≤ 3 2 ; 0 <x ≤ 1 2 −∞ <x< 2. log 1 4 (2 x)= 1 2 log 2 (2 x), x< 2 log 2 2 (2 x)+4log 2 (2 x) 5 ≥ 0. t2 +4t 5 ≥ 0 t ≤−5 t ≥ 1. x< 2 log 2 (2 x) ≤−5 log 2 (2 x) ≥ 1. x ∈ (−∞, 0] ∪ 63 32 , 2 ;
−∞ <x ≤−119, 1 ≤ x< 6; −∞ <x ≤−22, 4 ≤ x< 5;
−∞ <x ≤ 1, 47 16 ≤ x< 3. log 1 2 (x2 4x +3) ≥ log 1 2 1 2 3 , 0 <x2 4x +3 ≤ 8. 1 ≤ x< 1 3 <x ≤ 5; 4 <x< 6; 2 <x< 3; 4 <x< 3,x> 8; x> 2
0 <x< 1 2 ,x> 2; x> 0 01; x> 2; 1 ≤ x ≤ 4. 0 <x< 1 x> 1. x> 2; 3 <x< 4, 4 <x< 5,x> 5; 1 3 <x< 1, 1 <x< 2; 0 <x< 1 2 , 1 <x< 2, 3 <x< 6.
95 · 0 017453rad=1 65804rad
46 0 000291rad=0 01339rad
54 · 0 000005rad=0 00027rad 95◦ 46 54 =1 67170rad; 1 08621rad; 1 37564rad .
1rad= 180◦ π ≈ 57 29578 ◦ =57◦ 17 44 806 , 1rad= 180 · 60
≈ 3438 , 1rad= 180 60 60 π ≈ 206265 . 0 2755rad=0 2755 · 57 29578 ◦ =15 78499 ◦ , 0 2755rad=0 2755 · 3438 =947 169 , 0 2755rad=0 2755 · 206265 =56826 . 0 2755rad=15◦ 47 6 ; 62◦ 14 5 ; 39◦ 0 31 . 1 27991rad
cos22 ◦ 30 = 1 2
a,b ∈ R, x ∈ R. tg x =2 sin x =cos x,
3sin α +4cos α =5, sin2 α +cos2 α =1,
sin α = 3 5 cos α = 4 5 . sin2 α +sin2 β =1 sin2 α =1 sin2 β,
sin2 α =cos 2 β. sin α> 0 cos β> 0, sin α = cos β. sin α =sin(90◦ β ), α + β =90◦ ,
2 cosec2 α tg α 1 cosec 2 α +1= 1 (cosec2 α 1) 1 ctg α 1 (cosec 2 α 1)
= 1 ctg 2 α 1 ctg α ctg α ctg 2 α = ctg α(1 ctg α)(1+ctg α) 1 ctg α ctg 2 α =ctg α + ctg 2 α ctg 2 α =ctg α,α = π 4 + kπ ∧ α = π 2 + nπ, k,n ∈ Z.
cos α 1+sin α + sin α cos α = cos2 α +sin α +sin2 α (1+sin α)cos α = 1+sin α (1+sin α)cos α = 1 cos α
=sec α α = π 2 + kπ,k ∈ Z. x = π 4 + kπ 2 ,k ∈ Z. sin α 1 cos α = sin α 1 cos α 1+cos α 1+cos α = sin α(1+cos α) sin2 α = 1+cos α sin α ,α = kπ, k ∈ Z; 3(sin4 α +cos 4 α) 2(sin6 α +cos 6 α)
=3(sin4 α +cos4 α) 2(sin2 α +cos 2 α)(sin 4 α sin2 α cos 2 α +cos 4 α)
=3sin4 α +3cos 4 α 2sin4 α +2sin2 α cos 2 α 2cos 4 α =sin4 α +2sin2 α cos 2 α +cos 4 α =(sin2 α +cos 2 α) 2 =1; (sin α +cos α)2 1 ctg α sin α cos α = 2sin α cos α cos α sin α sin α cos α = 2sin α 1 sin 2 α sin α =2tg 2 α.
cos2 x(sin x +cos x) sin2 x cos 2 x = cos2 x sin x cos x .
tg x tg y tg(x + y )= tg x +tg y 1 tg x tg y Æ tg(x + y )= 1
sin(α + β )sin(α β )
=(sin α cos β +sin β cos α)(sin α cos β sin β cos α)
2 α cos2
=sin2 α sin2 β. 2+3tg 2 y = tg x tg y 1+tg x tg y =tg y, tg x = 3 2 tg y tg y. ctg y +tg x ctg y tg x = cos y sin y + sin x cos x cos y sin y sin x cos x = cos(x y ) cos(x + y ) x = π 2 + nπ,k ∈ Z; y = nπ,n ∈ Z;
, α = π 4 + kπ,k ∈ Z.
sin3α =sin(2α + α)=sin2α cos α +cos2α sin α =2sin α cos α cos α +(1 2sin2 α)sin α =2sin α cos 2 α +sin α 2sin3 α =2sin α(1 sin2 α)+sin α 2sin3 α =2sin α 2sin3 α +sin α 2sin3 α =3sin α 4sin3 α. sin α = 1 4 5 2 = 3
tg α = sin α cos α = 3 4 sin2α = 24 25 , cos2α = 7 25 , tg2α = 24 7 .
tg 2 x 2 = 1 cos x 1+cos x tg 2 x 2 = 1 cos α +cos β 1+cos α cos β 1+ cos α +cos β 1+cos α cos β = 1+cos α cos β cos α cos β 1+cos α cos β +cos α +cos β = (1 cos α)(1 cos β ) (1+cos α)(1+cos β ) =tg 2 α 2 ctg 2 β 2 , 1+cos α cos β =0 x =2kπ,α =2 π
,m ∈ Z. z = π (x + y ), sin z =sin π (x + y ) . sin x +sin y +sin x =sin x +sin y +sin(x + y ) =2sin x + y 2 cos x y 2 +2sin x + y 2 cos x + y 2 =2sin x + y 2 cos x
1◦ x ∈ (−∞, +∞). 2◦ T = π.
3◦ y =0 sin2x =0, x = kπ 2 ,k ∈ Z. x =0,x = π 2 ,x = π.
5◦ y> 0 x ∈ kπ, π 2 + kπ ,k ∈ Z, y< 0 x ∈ π 2 + nπ,π (n +1) ,n ∈ Z. x ∈ (0,π/2) x ∈ (π/2,π ). 6◦ x ∈ π 4 + kπ, π 4 + kπ , x ∈ π 4 + kπ, 3π 4 + kπ ,k ∈ Z.
2sin x √3=0 ⇔ sin x = √ 3 2 x ∈ π 3 +2kπ ∨ x = 2π 3 +2nπ, k,n ∈ Z;
sin2x 1=0 ⇔ sin2x =1 2x = π 2 +2kπ ⇔ x = π 4 + kπ, k ∈ Z;
2cos x =1=0 ⇔ cos2x = 1 2 2x = π 3 +2kπ ∨ 2x = π 3 +2nπ,k,n ∈ Z x = ±π 6+ mπ,m ∈ Z; cos πx = √3 2 ⇔ πx = ± π 6 +2kπ ⇔ x = ± 1 6 +2k,k ∈ Z. sin2 x +2sin x =0 ⇔ sin x(sin x +2)=0 ⇔ sin x =0 ∨ sin x = 2 1 ≤ sin x ≤ 1, x = kπ,k ∈ Z;
2sin x cos x sin x =0 ⇔ sin x(2cos x 1)=0 ⇔ sin x =0 ∨ cos x = 1 2 , x = kπ ∨ x = ± π 3 +2nπ,k,n ∈ Z. cos3x = cos5x cos(3x + π )= cos5x 3x + π = ±5x +2kπ. x = π 8 (2k +1) x = π 2 (2 +1) k, ∈ Z; sin x π 2 +sin 3π 2 + x = √2 ⇔− sin π 2 x cos x = √2 ⇔ cos x +cos x = √2 cos x = √2 2 , x = ± π 4 +2kπ,k ∈ Z.
sin x +cos x =1+sin x cos x
⇔ sin x sin x cos x +cos x 1=0
⇔ sin(1 cos x) (1 cos x)=0
⇔ (1 cos x)(sin x 1)=0
⇔ 1 cos x =0 ∨ sin x 1=0
⇔ x =2kπ ∨ x = π 2 +2nπ k,n ∈ Z;
cos2x √2sin x cos2x =0 ⇔ cos2x(1 √2sin x)=0
⇔ cos2x =0 ∨ 1 √2sin x =0
⇔ x = π 4 +2kπ ∨ x = 3π 4 +2 π, k, ∈ Z 1 √2sin x =0
cos2x =0, x = π 4 (2k +1),k ∈ Z cos3x =0 cos x =0, x = π 6 + kπ 3 ,x = π 2 (2m +1),k,m ∈ Z. x =
∈
⇔ tg x =0
∈
= kπ,k ∈
,n ∈ N.
(3tg
x
cos(sin x)= √3 2 , sin x =2kπ ± π 6 2kπ ± π 6 ≤ 1, k =0. sin x = ± π 6 , x =( 1)k arcsin π 6 + kπ,k ∈ Z. x = ± π 4 + kπ,k ∈ Z. x = ± π 3 + kπ,k ∈ Z; x = ± π 6 + nπ,n ∈ Z.
k> 1,n> 2 k n, (0,π ). 2sin x +3sin2x =0 ⇔ 2sin x +6sin x cos x =0 ⇔ 2sin x(1+3cos x)=0 ⇔ sin x =0 ∨ cos x = 1 3 ⇔ x = kπ ∨ x = ± arccos 1 3 +2nπ, k,n ∈ Z; x = π 2 + kπ
x = ± arccos 3
+2nπ,k,n ∈ Z; x = π 2 + kπ
x =( 1)n arcsin 3 4 + nπ, k,n ∈ Z; x = kπ ∨ x =( 1)n arcsin 1 3 + nπ, k,n ∈ Z.
cos x 2 =1+cos x ⇔ cos x 2 =2cos 2 x 2 ⇔ cos x 2 2cos x 2 1 =0
⇔ cos x 2 =0 ∨ cos x 2 = 1 2
⇔ x = π +2kπ ∨ x = ± 2π 3 +4nπ, k,n ∈ Z;
x =2π +4mπ ∨ x =8 π,m, ∈ Z;
x =6nπ ∨ x =3π +12kπ, n,k ∈ Z
1+cos x =2cos 2 x 2 , cos x 2 =0. sin x 1+cos x =sin x 2 ⇔ sin x 2cos 2 x 2 sin x 2 =0
⇔ 2sin x 2 1 cos x 2 =0
⇔ sin x 2 =0 ∨ cos x 2 =1
⇔ sin x 2 =0 x =2kπ,k ∈ Z cos3x, Æ cos x(4cos 2 x 1)=0 cos x =0 ∨ 4cos 2 x 1=0. cos x =0 x = π 2 (2k +1),k ∈ Z. 4cos 2 1=0, 2(1+cos2x) 1=0, cos2x = 1 2 , x = kπ ± π 3 ,k ∈ Z.
sin3x sin x + cos3x cos x = 5 2 +cos4x ⇔ sin4x 1 2 sin2x = 3 2 +2cos 2 2x
⇔ 4cos2x = 3 2 +2cos 2 x ∧ sin2x =0
⇔ cos2x = 1 2 ⇔ x = ± π 6 + kπ,k ∈ Z.
cos x = t t, 2t2 +3t 2=0 t1 = 2,t2 = 1 2 . cos x = 2 cos x = 1 2 , x = π 3 +2kπ ∨ x =2nπ π 3 ,k,n ∈ Z; x = π 2 +2kπ ∨ x =( 1)n π 6 + nπ, k,n ∈ Z.
cos2x =1 2sin2 x sin3x =sin x(3 4sin2 x),
sin x(3 4sin2 x)+1 2sin2 x =1, 4sin 3 x +2sin 2 x 3sin x =0. sin x = t, 1 ≤ t ≤ 1, 4t3 +2t2 3t =0, t1 =0,t2,3 = 1 ± √ 13 4 1 √ 13 4 < 1, x = kπ ∨ x =( 1)n arcsin √3 1 4 + π,k, ∈ Z.
sin4 x +cos 4 x = 7 2 sin x cos x
⇔ sin4 x +2sin2 x cos 2 x +cos 4 x 2sin2 x cos2 x = 7 2 sin x cos x
⇔ (sin2 x +cos2 x)2 2sin2 x cos 2 x = 7 2 sin x cos x
⇔ 1 1 2 sin2 2x = 7 4 sin2x.
sin2x = t 2t2 +7t 4=0 t1 = 1 2 t2 = 4.
sin2x = 1 2 x =( 1)n π 12 + kπ 2 ,k ∈ Z.
2sin2 x +cos 2 x =3sin x cos x.
cos 2 x =0, 2tg 2 x 3tg x +1=0, x = π 4 + kπ x =arctg 1 2 + nπ, k,n ∈ Z; x = 7π 6 +2kπ ∨ x =2nπ π 6 k,n ∈ Z
cos x sin5x = 1 2 sin4x
⇔ 1 2 sin(5x + x)+sin(5x x) = 1 2 sin4x
⇔ 1 2 sin6x + 1 2 sin4x = 1 2 sin4x
⇔ sin6x =0 ⇔ x = kπ 6 ,k ∈ Z; x = kπ 3 ∨ x = nπ 2 k,n ∈ Z.
sin3x sin2x =sin11x sin10x ⇔ 1 2 cos(3x 2x) cos(3x +2x) = 1 2 cos(11x 10x) cos(11x +10x) ⇔ cos21x =cos5x ⇔ x = kπ 8 ∨ x = nπ 13 k,n ∈ Z; x = mπ 8 ,m ∈ Z. x = kπ 5 ∨ x = π 2 + nπ k,n ∈ Z; sin x =cos2x ⇔ sin x cos2x =0 ⇔ sin x sin π 2 2x =0 ⇔ 2sin 1 2 3x π 2 cos 1 2 π 2 x =0 ⇔ sin 1 2 3x π 2 =0 ∨ cos 1 2 π 2 x =0 ⇔ x = π 6 + 2kπ 3 ∨ x = π 2 2 π, k, ∈ Z, ⇔ x = π 6 + 2kπ 3 , k ∈ Z
sin x +sin2x +sin3x =0
⇔ 2sin2x cos x +sin2x =0
⇔ sin2x(2cos x +1)=0
x ∈ 2kπ + π 4 ,π (2k +1)+ π 4 k ∈ Z; x ∈ π (2k 1)+ π 4 , 2kπ + π 4 , k ∈ Z. sin x +sin2 x +sin3 x> 0 ⇔ sin x(1+sin x +sin2 x) > 0 1+sin x +sin2 x> 0 x, sin x> 0, 2kπ<x<π +2kπ, k ∈ Z. 3π 8 + nπ ≤ x ≤ π 8 + nπ,n ∈ Z cos2x =1 2sin2 x Æ sin x = t, t(1 2t) > 0, 0 <t< 1 2 , 0 < sin x< 1 2 . sin x> 0 sin x< 1 2 x ∈ (0,π ), sin x) x ∈ 2kπ, 2kπ + π 6
1◦ β1 =59◦ 27 ,γ1 =91◦ 58 ,c1 =20 89 2◦ β2 =120◦ 33 ,γ2 =30◦ 52 ,c2 =10 72; α =33◦ 47 ,β =64◦ 53 ,b =14 65
α =arccos b2 + c2 a2 2bc =19◦ 43 ,β =arccos a2 + c2 b2 2ac =142◦ 36 , γ =180◦ α β =17◦ 41 ; α =28◦ 57 ,β =46◦ 34 ,γ =104◦ 29 BCD
<BCD, BD =2748 5 m , ACD
AD =1482 4 m . ABD
AD,BD <ADB, AB =2002 m .
ψ
MA = c sin ϕ sin δ = b sin ψ sin ε
MB = c sin(ϕ + δ ) sin δ
MC = b sin(ψ + ε) sin ε .
sin ϕ sin ψ = b sin δ c sin ε sin 360◦ (α + δ + ε) ψ sin ψ = b sin δ c sin ε , tg ψ = c sin ε sin(α + δ + ε) b sin δ + c cos(α + δ + ε)sin ε ϕ =360 ◦ (α + δ + ε + ψ ). ψ =114◦ 15 ,ϕ =97◦ 10 ,MA =231 95,MB =206 08, MC =114 72
α =180◦ β γ,a =2R sin α,b =2R sin β,c =2R sin γ. sin α 2 = a cos β γ 2 b + c β γ β + γ =180◦ α,β γ = ε,b c tg β γ 2 < a b + c < 1; a s = 2sin α sin α +sin β +sin γ ,a =14 773,b =24 084,c = 10 143,β =149 ◦ 45 ;
a =5 4528,β =115◦ 6 ,γ =13◦ 44 ,b =6 3387,c =1 6613; α =38◦ 40 ,b =19 183,c =14 353,β =93◦ ,γ =48◦ 20; α1 =53◦ 52 ,γ1 =47◦ 34 ,c1 =38 38,β1 =78◦ 34 ,b1 =50 968 α2 =126◦ 8 ,γ2 =47◦ 34 ,c2 =38 381,β2 =1◦ 8 ,b2 =5 706. b =2√3,c =2 2 √3; sin75◦ =sin(45 ◦ +30◦ ),β =60◦ ,γ =45◦ ,c =2√3; α =30◦ ,β =135◦ ,γ =15◦ ; a =2√3,β =30◦ ,γ =105◦ . 200tg α =(200 d)tg2α, d =100(1+tg 2 α)= 100 cos2 α , h h =25tg α h =8tg2α, 8tg2α =25tg α.
tg α = 3 5 h =15 m .
=78◦ 35 β =40◦ 48 α =26◦ 58 . ABC AC = m, cos <ACB = b2 + m2 a2 2mb
<ACB =53◦ 22, ACD <ACD =13◦ 54
<BCD = <ACB + <ACD =67◦ 16 , BCD BD =31 735. Æ BCD <CBD =31◦ 33 ϕ
ϕ =180 ◦ <ACB <CBD =95◦ 5 , Æ
3/4 × cm