Question

(sin 3x+sin x)/cos x=0

Answer 1

(sin 3x+sin x)/cos x=0

sinx+sin3x=0

2sin(2x)=0

2x=πn , n ∈ Z

ответ х=(πn)/2, n ∈ Z

Answer 2

To solve the trigonometric equation (sin 3x + sin x)/cos x = 0, we need to find the values of x that satisfy the equation.

Let's manipulate the given equation step by step:

(sin 3x + sin x) / cos x = 0

First, let's combine the two sine terms:

sin 3x + sin x = 0

Now, we can use the trigonometric identity for the sum of two sine functions:

2 * sin((A + B)/2) * cos((A - B)/2) = 0

In this case, A = 3x, and B = x:

2 * sin((3x + x)/2) * cos((3x - x)/2) = 0

Simplifying further:

2 * sin(2x) * cos(x) = 0

Now, there are two possibilities for this equation to be true:

1. sin(2x) = 0
2. cos(x) = 0

Let's solve each possibility separately:

1. sin(2x) = 0

For sin(2x) to be 0, 2x must be an integer multiple of π (pi) because the sine function is 0 at integer multiples of π. So:

2x = n * π

where n is an integer.

Now, solve for x:

x = (n * π) / 2

2. cos(x) = 0

For cos(x) to be 0, x must be an odd integer multiple of π/2 because the cosine function is 0 at odd integer multiples of π/2. So:

x = (2n + 1) * (π / 2)

where n is an integer.

So, the solutions to the equation are:

x = (n * π) / 2 and x = (2n + 1) * (π / 2)

where n is an integer. These are the values of x that satisfy the given trigonometric equation.