Question

Помогите пожалуйста

Answer 1

1.
  cos15°sin15° + (cos75°-sin75°)² =
= ¹/₂ * (2sin15°cos15°) + cos² 75° - 2sin75°cos75°+sin² 75° =
= ¹/₂ * sin(2*15°) + (cos² 75°+sin² 75°) - sin(2*75°)=
= ¹/₂ * sin30° + 1 - sin150° =
= ¹/₂ * ¹/₂ + 1 - sin(180°-30°)=
= ¹/₄ + 1 - sin30°=
=¹/₄ + 1 - ¹/₂ = ¹/₄ + 1 - ²/₄ = 1 - ¹/₄ = ³/₄ = 0.75

2.
log²₂ (x) - 3log₂ (x)=4
ОДЗ:  x>0

y=log₂ (x)
y² - 3y-4=0
D=(-3)² - 4*(-4)=9+16=25=5²
y₁=(3-5)/2= -1
y₂=(3+5)/2=4

При у=-1
log₂ (x)=-1
x=2⁻¹
x=¹/₂

При у=4
log₂ (x)=4
x=2⁴
x=16

Ответ: ¹/₂;  16.

3.
$\frac{log_{5}12-2log_{5}2}{log_{5}18+log_{5}0.5}= \frac{log_{5}12-log_{5}2^2}{log_{5}(18*0.5)}= \frac{log_{5}( \frac{12}{4} )}{log_{5}9} = \frac{log_{5}3}{log_{5}9}= \\ \\ =log_{9}3=log_{3^2}3 = \frac{1}{2}log_{3}3= \frac{1}{2}*1= \frac{1}{2}$

4.
$\frac{1}{7- \sqrt{39} }+ \frac{1}{7+ \sqrt{39} }= \frac{7+ \sqrt{39}+7- \sqrt{39} }{7^2-( \sqrt{39} )^2}= \frac{14}{49-39}= \frac{14}{10}=1.4$

5.
sinx - ¹/₂ =0
sinx= ¹/₂
x=(-1)ⁿ π/6 + πk, k∈Z