. :
(a b)² =a² 2ab+b²
(a b)³ =a³ 3a² b+3ab² b³
a² -b² =(a+b)(a-b)
a³ b³ =(a b)(a² ∓ab+b²),
(a+b)³ =a³ +b³ +3ab(a+b)
(a-b)³ =a³ -b³ -3ab(a-b)
xn-an=(x-a)(xn-1+axn-2+a² xn-3+...+an-1)
ax² +bx+c=a(x-x1)(x-x2)
x1 x2
ax² +bx+c=0
:
ap ag = ap+g
ap:ag=a p-g
(ap)g=a pg
ap /bp = (a/b)p
ap× bp = abp
a0=1; a1=a
a-p = 1/a
pÖ a =b => bp=a
pÖ apÖ b = pÖ ab
Ö a; a = 0
ax² +bx+c=0; (a¹ 0)
x1,2= (-b Ö D)/2a; D=b² -4ac
D>0 x1¹ x2;D=0 x1=x2
D<0, .
:
x1+x2 = -b/a
x1× x2 = c/a
. :
x² + px+q =0
x1+x2 = -p
x1× x2 = q
p=2k (p-.)
x² +2kx+q=0, x1,2 = -k Ö (k² -q)
-
Ö ((x2-x1)² -(y2-y1)²)
:
loga x = b => ab = x; a>0,a¹ 0
a loga x = x, logaa =1; loga 1 = 0
loga x = b; x = ab
loga b = 1/(log b a)
logaxy = logax + loga y
loga x/y = loga x - loga y
loga xk =k loga x (x >0)
logak x =1/k loga x
loga x = (logc x)/(logca); c>0,c¹ 1
logbx = (logax)/(logab)
an = a1 +d(n-1)
Sn = ((2a1+d(n-1))/2)n
bn = bn-1 × q
b2n = bn-1× bn+1
bn = b1× qn-1
Sn = b1 (1- qn)/(1-q)
S= b1/(1-q)
.
sin x = a/c
cos x = b/c
tg x = a/b=sinx/cos x
ctg x = b/a = cos x/sin x
sin (p -a) = sin a
sin (p /2 -a) = cos a
cos (p /2 -a) = sin a
cos (a + 2p k) = cos a
sin (a + 2p k) = sin a
tg (a + p k) = tg a
ctg (a + p k) = ctg a
sin² a + cos² a =1
ctg a = cosa / sina, a ¹ p n, nÎ Z
tga × ctga = 1, a ¹ (p n)/2, nÎ Z
1+tg² a = 1/cos² a, a ¹ p (2n+1)/2
1+ ctg² a =1/sin² a, a ¹ p n
:
sin(x+y) = sin x cos y + cos x sin y
sin (x-y) = sin x cos y - cos x sin y
cos (x+y) = cos x cos y - sin x sin y
cos (x-y) = cos x cos y + sin x sin y
tg(x+y) = (tg x + tg y)/ (1-tg x tg y)
x, y, x + y ¹ p /2 + p n
tg(x-y) = (tg x - tg y)/ (1+tg x tg y)
x, y, x - y ¹ p /2 + p n
.
sin 2a = 2sin a cos a
cos 2a = cos² a - sin² a = 2 cos² a - 1 =
= 1-2 sin² a
tg 2a = (2 tga)/ (1-tg² a)
1+ cos a = 2 cos² a /2
1-cosa = 2 sin² a /2
tga = (2 tg (a /2))/(1-tg² (a /2))
- .
sin² a /2 = (1 - cos a)/2
cos² a /2 = (1 + cosa)/2
tg a /2 = sina /(1 + cosa) = (1-cos a)/sin a
a ¹ p + 2p n, n Î Z
- .
sin x + sin y = 2 sin ((x+y)/2) cos ((x-y)/2)
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sin x - sin y = 2 cos ((x+y)/2) sin ((x-y)/2)
cos x + cos y = 2cos (x+y)/2 cos (x-y)/2
cos x - cos y = -2sin (x+y)/2 sin (x-y)/2
. .
sin x sin y = ½ (cos (x-y) - cos (x+y))
cos x cos y = ½ (cos (x-y)+ cos (x+y))
sin x cos y = ½ (sin (x-y)+ sin (x+y))
sin x = (2 tg x/2)/(1+tg2x/2)
cos x = (1-tg2 2/x)/ (1+ tg² x/2)
sin2x = (2tgx)/(1+tg2x)
sin² a = 1/(1+ctg² a) = tg² a /(1+tg² a)
cos² a = 1/(1+tg² a) = ctg² a / (1+ctg² a)
ctg2a = (ctg² a -1)/ 2ctga
sin3a = 3sina -4sin³ a = 3cos² a sina -sin³ a
cos3a = 4cos³ a -3 cosa= cos³ a -3cosa sin² a
tg3a = (3tga -tg³ a)/(1-3tg² a)
ctg3a = (ctg³ a -3ctga)/(3ctg² a -1)
sin a /2 = Ö ((1-cosa)/2)
cos a /2 = Ö ((1+cosa)/2)
tga /2 = Ö ((1-cosa)/(1+cosa))=
sina /(1+cosa)=(1-cosa)/sina
ctga /2 = Ö ((1+cosa)/(1-cosa))=
sina /(1-cosa)= (1+cosa)/sina
sin(arcsin a) = a
cos(arccos a) = a
tg (arctg a) = a
ctg (arcctg a) = a
arcsin (sina) = a; a Î [-p /2; p /2]
arccos(cos a) = a; a Î [0; p ]
arctg (tg a) = a; a Î [-p /2; p /2]
arcctg (ctg a) = a; a Î [ 0; p ]
arcsin(sina)=
1)a - 2p k; a Î [-p /2 +2p k;p /2+2p k]
2) (2k+1)p - a; a Î [p /2+2p k;3p /2+2p k]
arccos (cosa) =
1) a -2p k; a Î [2p k;(2k+1)p ]
2) 2p k-a; a Î [(2k-1)p; 2p k]
arctg(tga)= a -p k
a Î (-p /2 +p k;p /2+p k)
arcctg(ctga) = a -p k
a Î (p k; (k+1)p)
arcsina = -arcsin (-a)= p /2-arccosa =
= arctg a /Ö (1-a ²)
arccosa = p -arccos(-a)=p /2-arcsin a =
= arc ctga /Ö (1-a ²)
arctga =-arctg(-a) = p /2 -arcctga =
= arcsin a /Ö (1+a ²)
arc ctg a = p -arc cctg(-a) =
= arc cos a /Ö (1-a ²)
arctg a = arc ctg1/a =
= arcsin a /Ö (1+a ²)= arccos1/Ö (1+a ²)
arcsin a + arccos = p /2
arcctg a + arctga = p /2
sin x = m; |m| = 1
x = (-1)n arcsin m + p k, kÎ Z
sin x =1 sin x = 0
x = p /2 + 2p k x = p k
sin x = -1
x = -p /2 + 2 p k
cos x = m; |m| = 1
x = arccos m + 2p k
cos x = 1 cos x = 0
x = 2p k x = p /2+p k
cos x = -1
x = p + 2p k
tg x = m
x = arctg m + p k
ctg x = m
x = arcctg m +p k
sin x/2 = 2t/(1+t2); t - tg
cos x/2 = (1-t²)/(1+t²)
.
: af(x)>(<) a()
1) a>1, .
2) a<1, .
: :
logaf(x) >(<) log a j (x)
1. a>1, : f(x) >0
j (x)>0
f(x)>j (x)
2. 0<a<1, : f(x) >0
j (x)>0
f(x)<j (x)
3. log f(x) j (x) = a
: j (x) > 0
f(x) >0
f(x) ¹ 1
:
1. :
sin 2x - Ö 3 cos x = 0
2sin x cos x -Ö 3 cos x = 0
cos x(2 sin x - Ö 3) = 0
....
2. ....
3.sin² x - sin 2x + 3 cos² x =2
sin² x - 2 sin x cos x + 3 cos ² x = 2 sin² x + cos² x
sin x = 0, cos x = 0,
, => cos x
-:
Sin a ³ m
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2p k+a 1 = a = a 2+ 2p k
2p k+a 2 = a = (a 1+2p)+ 2p k
:
I cos (p /8+x) < Ö 3/2
p k+ 5p /6< p /8 +x< 7p /6 + 2p k
2p k+ 17p /24 < x< p /24+2p k;;;;
II sin a = 1/2
2p k +5p /6 = a = 13p /6 + 2p k
cos a ³ (=) m
2p k + a 1 < a < a 2+2 p k
2p k+a 2< a < (a 1+2p) + 2p k
cos a ³ - Ö 2/2
2p k+5p /4 = a = 11p /4 +2p k
tg a ³ (=) m
p k+ arctg m = a = arctg m + p k
ctg ³ (=) m
p k+arcctg m < a < p +p k
:
(xn) = n× xn-1
(ax) = ax× ln a
(lg ax)= 1/(x× ln a)
(sin x) = cos x
(cos x) = -sin x
(tg x) = 1/cos² x
(ctg x) = - 1/sin² x
(arcsin x) = 1/ Ö (1-x²)
(arccos x) = - 1/ Ö (1-x²)
(arctg x) = 1/ Ö (1+x²)
(arcctg x) = - 1/ Ö (1+x²)
-:
(u × v) = u× v + u× v
(u/v) = (uv - uv)/ v²
.
y = f(x0)+ f (x0)(x-x0)
x
1.
2. k = x
3. X0, f(x0), f (x0),
:
ò xn dx = xn+1/(n+1) + c
ò ax dx = ax/ln a + c
ò ex dx = ex + c
ò cos x dx = sin x + cos
ò sin x dx = - cos x + c
ò 1/x dx = ln|x| + c
ò 1/cos² x = tg x + c
ò 1/sin² x = - ctg x + c
ò 1/Ö (1-x²) dx = arcsin x +c
ò 1/Ö (1-x²) dx = - arccos x +c
ò 1/1+ x² dx = arctg x + c
ò 1/1+ x² dx = - arcctg x + c
