Question

1)sin5X=sinX 2)8sinX-7cosX=0 решитий

Answer 1

Sin5x=sinx
sin5x-sinx=0
2sin2xcos3x=0
sin2x=0          cos3x=0
2x=πn            3x=π/2+πn
x=π/2n            x=π/6+π/3n, n∈Z
2) 8sinx-7cosx=0| : cosx
8tgx - 7=0
tgx=7/8
x=arctg7/8+πn, n∈Z 

Answer 2

1. To solve the equation sin(5X) = sin(X), we can use the trigonometric identity:

sin(A) = sin(B) if and only if A = B + 2πn or A = π - B + 2πn

where n is an integer.

Applying this identity to our equation, we have:

5X = X + 2πn or 5X = π - X + 2πn

Simplifying each equation:

For 5X = X + 2πn: 4X = 2πn X = (2πn) / 4 X = (πn) / 2

For 5X = π - X + 2πn: 6X = π + 2πn X = (π + 2πn) / 6 X = (π/6) + (πn/3)

Therefore, the solutions for sin(5X) = sin(X) are: X = (πn) / 2 X = (π/6) + (πn/3)

where n is an integer.

2. To solve the equation 8sin(X) - 7cos(X) = 0, we can divide both sides of the equation by cos(X):

8sin(X) / cos(X) - 7cos(X) / cos(X) = 0 / cos(X)

Using the identity sin(X) / cos(X) = tan(X):

8tan(X) - 7 = 0

Adding 7 to both sides:

8tan(X) = 7

Dividing both sides by 8:

tan(X) = 7/8

To find the solution for X, we can take the inverse tangent (arctan) of both sides:

X = arctan(7/8)

Thus, the solution for 8sin(X) - 7cos(X) = 0 is: X = arctan(7/8)