Задача 34748 ...

sova

2019-03-21 09:36:14

1.
Основное логарифмическое тождество:
[b]a[/b]^(log_( [b]a[/b])b)=b
a>0;a ≠ 1; b>0

(1-2^2)=1-4=-3
Логарифмы отрицательных чисел не существуют
Применить формулу нельзя.

2.
(8^(2/5))^(1/2)=8^((2/5)*(1/2))=8^(1/5)
∛64=∛4^3=4
8^(1/5)*4=(2^3)^(1/5)*2^2=2^(3/5)*2^(2)=2^((3/5)+2)=2^(13/5)

3.
sqrt(14)=sqrt(7)*sqrt(2)

sqrt(14)*(sqrt(7)+2-3sqrt(2))=

=sqrt(7)*sqrt(7)*sqrt(2)+2sqrt(14)-3sqrt(7)*sqrt(2)*sqrt(2)=

=7*sqrt(2) +2sqrt*(14)-6sqrt(7)

4.
a^6+b^6=(a^2)^3+(b^2)^3=
формула (m^3+n^3=(m+n)*(m^2-mn+n^2);
m=a^2; n=b^2)

=(a^2+b^2)*((a^2)^2-a^2*b^2+(b^2)^2)=(a^2+b^2)*(a^4-a^2b^2+b^4)

(a^6+b^6)/(a^4-a^2b^2+b^4)=a^2+b^2

5a)
(1+tg^230 градусов)*(1-tg^230 градусов)-ctg^260 градусов=

=(1+(sqrt(3)/3)^2)*(1-(sqrt(3)/3)^2)*-(sqrt(3)/3)^2=

=(1+(1/3))*(1-(1/3)) - (1/3)=(1^2-(1/3)^2- (1/3)=(8/9)-(6/9)=2/9

5b)
(sin(π/6) - cos(π/6))^2+tg(7π/4)=

= [b]sin^2(π/6)[/b]-2sin(π/6)*cos(π/6)+ [b]cos^2(π/6)[/b]+tg(2π-(π/4))=

= [b]1[/b]- sin(2*(π/6))- tg(π/4)=1-sin(π/3)+1=-sin(π/3)= [b]-sqrt(3)/2;[/b]

6a)
1-cos^2 α =sin^2 α

1-cos^2 α +sin^2 α =sin^2 α +sin^2 α =2sin^2 α

ctg(5π/4)=ctg(π+(π/4))=ctg(π/4)=1

(1-cos^2 α +sin^2 α)*сtg(5 π/4)=2sin^2 α *1=2sin^2 α

6b)
2sin^2 α+cos^2 α =sin^2 α +sin^2 α +cos^2 α =

=sin^2 α +(sin^2 α +cos^2 α)=sin^2 α +1

sin^4 α -1=(sin^2 α -1)*(sin^2 α +1)

(2sin^2 α+cos^2 α)*(sin^2 α -1)/(tg α*(sin^4 α -1))=

=(sin^2 α +1)*(sin^2 α -1)/(tg α*(sin^2 α -1)*(sin^2 α +1))=

=1/tg α =ctg α

При α =π/4

ctg(π/4)=1

6c

sin(- α ) =- sin α

sin(π- α )=sin α

tg((π/2)+ α )= - ctgα

sin(- α )+sin(π- α )*(tg((π/2)+ α )=

=-sin α +sin α *(cos α /sin α )=-sin α +cos α

7.
cos1230 градусов= cos( 4*360 градусов - 210 градусов)=

=cos(-210 градусов)= cos(-180 градусов - 30 градусов)=

=-cos(30 градусов)=-sqrt(3)/2:

sin(-405 градусов)=-sin405 градусов =

=- sin( 360 градусов + 45 градусов)=-sin 45 градусов= - sqrt(2)/2:

tg(-7π/3)=-tg(7π/3)=-tg(2π+(π/3))= - tg(π/3)= - sqrt(3);

ctg(29π/6)=ctg((30π/6)-(π/6))=ctg(5π- (π/6))=ctg(-π/6)=- sqrt(3).