Question

а) 3sin(-4x)-3=0б)cos(4x-П/3) = 1/2в)tg x/4 = - корень из 3,делённое на 3г)sin2x*cosП/6+sinП/6*cos2x = 0д)8sin3x*cos3x = 4Ребя решите пожалуйста а то 2 выходит(

Answer 1

А)3sin(-4x)-3=0
   3sin(-4x)=3
   - sin4x=1
   sin4x= - 1
   4x= - Пи/2+2ПиK, где K принадлежит Z
   x= - Пи/8+(Пи/2)К, где к принадлежит Z
б) cos(4x-Пи/3)=1/2
   cos4x*cosПи/3+sin4x*sinПи/3=1/2
   1/2*cos4x+(корень из 3)/2*sin4x=1/2
   cos4x+(корень из 3)*sin 4x=1
   cos^2(2x)-sin^2(2x)+(корень из 3)*2sin2x*cos2x- 1=0
   cos^2(2x)-sin^2(2x)-sin^2(2x)-cos^2(2x)+(корень из 3)*2sin2x=0 
   - 2sin^2(2x)+(корень из 3)*2sin2x=0
   2sin2x(- sin2x+ корень из 3)=0
   2sin2x=0  или sin2x=корень из 3
   2x=ПиК,         2x=(-1)в степени к*arcsin(корня из 3)+ПиК, где К принадлежит Z;  
   х=(Пи/2)К       x=(-1)в степени к*arcsin(корня из 3)/2 + Пгде К принадлежит Z; 
в) tg(x/4)= - ( корень из 3)/3
   x/4=arctg((-корень из 3)/3)+ПиК,где К принадлежит Z;  
   x= - 10Пи/3+4ПиК, где К принадлежит Z;   
г) sin2x*cosПи/6+sinПи/6*сos2x=0
   sin(2x+Пи/6)=0
   2x+Пи/6=ПиК,где К принадлежит Z;  
   2х= - Пи/6+ПиК, где К принадлежит Z;  
   х= -Пи/12+Пи/2К, где К принадлежит Z;
д)8sin3x*cos3x=4
   4sin6x-4=0
   sin6x=1
   6x=Пи/2+2ПиК, где К принадлежит Z
   x=Пи/12+(Пи/3)К, где К принадлежит Z

Answer 2

а) 3sin(-4x) - 3 = 0

To solve this equation, we can start by isolating the sine term:

3sin(-4x) = 3

Next, divide both sides of the equation by 3:

sin(-4x) = 1

Now, we need to find the values of x that satisfy this equation. Remember that the sine function has a period of 2π, meaning that any value x that satisfies the equation will also satisfy the equation when 2π is added or subtracted.

Since sin(-θ) = -sin(θ), we can rewrite the equation as:

-sin(4x) = 1

Now, we can solve for x by taking the inverse sine (or arcsine) of both sides:

4x = arcsin(-1)

To find the solutions, we need to consider the range of the arcsine function, which is -π/2 to π/2. Since -1 is within this range, we can proceed:

x = arcsin(-1) / 4

Using a calculator, we can find that arcsin(-1) = -π/2.

Therefore, the solution to the equation is:

x = (-π/2) / 4 = -π/8

So, the solution to the equation 3sin(-4x) - 3 = 0 is x = -π/8.

б) cos(4x - π/3) + 1/2 = 0

To solve this equation, we can isolate the cosine term:

cos(4x - π/3) = -1/2

Next, we can take the inverse cosine (or arccos) of both sides:

4x - π/3 = arccos(-1/2)

To find the solutions, we need to consider the range of the arccos function, which is 0 to π. Since -1/2 is within this range, we can proceed:

4x - π/3 = arccos(-1/2)

Using a calculator, we can find that arccos(-1/2) = 2π/3.

Therefore, the solution to the equation is:

4x - π/3 = 2π/3

Simplifying, we have:

4x = π

x = π/4

So, the solution to the equation cos(4x - π/3) + 1/2 = 0 is x = π/4.

в) tg(x/4) = -√3/3

To solve this equation, we can take the inverse tangent (or arctan) of both sides:

x/4 = arctan(-√3/3)

To find the solutions, we need to consider the range of the arctan function, which is -π/2 to π/2. Since -√3/3 is within this range, we can proceed:

x/4 = arctan(-√3/3)

Using a calculator, we can find that arctan(-√3/3) = -π/6.

Therefore, the solution to the equation is:

x/4 = -π/6

Multiplying both sides by 4, we have:

x = -4π/6

Simplifying, we have:

x = -2π/3

So, the solution to the equation tg(x/4) = -√3/3 is x = -2π/3.

г) sin(2x)cos(π/6) + sin(π/6)cos(2x) = 0

To solve this equation, we can apply the product-to-sum formula for sine:

sin(a)cos(b) + sin(b)cos(a) = sin(a + b)

Using this formula, we can rewrite the equation:

sin(2x + π/6) = 0

Now, we can solve for x by taking the inverse sine (or arcsine) of both sides:

2x + π/6 = arcsin(0)

To find the solutions, we need to consider the range of the arcsine function, which is -π/2 to π/2. Since 0 is within this range, we can proceed:

2x + π/6 = arcsin(0)

Using a calculator, we can find that arcsin(0) = 0.

Therefore, the solution to the equation is:

2x + π/6 = 0

Simplifying, we have:

2x = -π/6

x = -π/12

So, the solution to the equation sin(2x)cos(π/6) + sin(π/6)cos(2x) = 0 is x = -π/12.

д) 8sin(3x)cos(3x) = 4

To solve this equation, we can apply the double angle formula for sine:

sin(2θ) = 2sin(θ)cos(θ)

Using this formula, we can rewrite the equation:

4sin(6x) = 4

Now, we can solve for x by dividing both sides of the equation by 4:

sin(6x) = 1

To find the solutions, we need to consider the range of the sine function, which is -π/2 to π/2. Since 1 is within this range, we can proceed:

6x = arcsin(1)

Using a calculator, we can find that arcsin(1) = π/2.

Therefore, the solution to the equation is:

6x = π/2

x = π/12

So, the solution to the equation 8sin(3x)cos(3x) = 4 is x = π/12.