Question

Sin ( 5x - pi/3) = sin x помогите пожалуйста

Answer 1

Sin(5x-π/3)-sinx=0
$2sin \frac{5x-x- \pi /3}{2} *cos \frac{5x- \pi /3+x}{2} =0$$2sin \frac{4x- \pi /3}{2} *cos \frac{6x- \pi /3}{2} =0$
2sin(2x-π/6)*cos(3x-π/6)=0
sin(2x-π/6)=0
2x-π/6=πn
2x=πn+π/6
x=πn/2+π/12
cos(3x-π/6)=0
3x-π/6=πn/2
3x=πn/2+π/6
x=πn/6+π/18

Answer 2

Sin(5x-pi/3)-sinx=0 2sin(2x-pi/6)cos(3x-pi/6)=0 Sin(2x-pi/6)=0 2x-pi/6=pin , n e z 2x=pin+pi/6 , n e z x= pin/2+pi/12 , n e z Cos(3x-pi/6)=0 3x-pi/6=pi/2 +pin, n e z 3x=4pi/6+ pin , n e z x=4pi/18 + pin/3 , n e z

Answer 3

Solving the Equation sin(5x - pi/3) = sin(x)

To solve the equation sin(5x - pi/3) = sin(x), we can use trigonometric identities and properties.

First, let's simplify the equation using the identity sin(a - b) = sin(a)cos(b) - cos(a)sin(b):

sin(5x - pi/3) = sin(x) sin(5x)cos(pi/3) - cos(5x)sin(pi/3) = sin(x)

Since cos(pi/3) = 1/2 and sin(pi/3) = sqrt(3)/2, we can substitute these values into the equation:

sin(5x)(1/2) - cos(5x)(sqrt(3)/2) = sin(x) (1/2)sin(5x) - (sqrt(3)/2)cos(5x) = sin(x)

Next, we can use the identity sin(a) = cos(π/2 - a) to rewrite the equation:

(1/2)sin(5x) - (sqrt(3)/2)cos(5x) = sin(x) (1/2)sin(5x) - (sqrt(3)/2)sin(π/2 - 5x) = sin(x)

Now, we can simplify the equation further:

(1/2)sin(5x) - (sqrt(3)/2)sin(π/2 - 5x) = sin(x) (1/2)sin(5x) - (sqrt(3)/2)sin(π/2)cos(5x) + (sqrt(3)/2)cos(π/2)sin(5x) = sin(x) (1/2)sin(5x) - (sqrt(3)/2)cos(5x) = sin(x)

We can now solve for x by isolating the terms with sin(x) on one side of the equation:

(1/2)sin(5x) - (sqrt(3)/2)cos(5x) - sin(x) = 0

To solve this trigonometric equation, we can use various methods such as factoring, the quadratic formula, or graphing. However, in this case, it is not possible to find an exact solution algebraically. We can use numerical methods or graphing to approximate the solutions.

Please let me know if you need further assistance or have any other questions!