Question

Sin8x = корень из 3/2

Answer 1

sin8x = √3 / 2

система:

sin8x = √3 / 2

sin (π - 8x) = √3 / 2

система:

8x = arcsin (√3 / 2)

π - 8x = arcsin (√3 / 2)

система:

8x = π/3

π - 8x = π/3

система:

8x = π/3 + 2πn

π - 8x = π/3 + 2πn

система:

x = π/24 + πn/4

x = π/12 - πn/4

Answer 2

To solve the equation sin(8x) = √(3/2), we need to find the values of x that satisfy the equation. We can use trigonometric identities and algebraic manipulations to find the solutions. First, let's rewrite √(3/2) as sin(π/3).

Recall the trigonometric identity: sin(π/3) = √(3)/2

So, the equation becomes sin(8x) = sin(π/3)

Now, we need to find the values of 8x that satisfy sin(8x) = sin(π/3).

Trigonometric identity: sin(A) = sin(B) if and only if A = B + 2kπ or A = π - B + 2kπ, where k is an integer.

In our case, A = 8x and B = π/3.

1. 8x = π/3 + 2kπ (k is an integer)
2. 8x = π - π/3 + 2kπ (k is an integer)

Let's solve each case:

Case 1: 8x = π/3 + 2kπ

To solve for x, divide both sides by 8: x = (π/3 + 2kπ) / 8

Case 2: 8x = π - π/3 + 2kπ

To solve for x, divide both sides by 8: x = (π - π/3 + 2kπ) / 8

These are the general solutions for x that satisfy the equation sin(8x) = √(3/2). Note that the "k" in the solutions represents an integer, and it will generate multiple solutions for x. To find specific values of x, you can plug in different integer values for k.

For example, if we take k = 0, then:

For Case 1: x = (π/3 + 2 * 0 * π) / 8 x = π/24

For Case 2: x = (π - π/3 + 2 * 0 * π) / 8 x = 7π/24

So, the solutions for the equation sin(8x) = √(3/2) are x = π/24 and x = 7π/24, and there are infinitely many solutions due to the presence of the integer "k" in the solutions.