Determine the general solution for sin(×-30°)=cos 2×?

To find the general solution for sin(x-30°) = cos(2x), we can use the trigonometric identity sin(x-30°) = sin(90° - (x-30°)) = sin(60° + x).

Therefore, the equation sin(60° + x) = cos(2x) can be rewritten as sin(60°)cos(x) + cos(60°)sin(x) = cos(2x).

Using the trigonometric identities sin(60°) = √3/2, cos(60°) = 1/2, and cos(2x) = 1 - 2sin^2(x), we get:

√3/2 * cos(x) + 1/2 * sin(x) = 1 - 2sin^2(x).

Multiplying through by 2 to clear the fractions, we get:

√3 * cos(x) + sin(x) = 2 - 4sin^2(x).

Rearranging terms, we get:

4sin^2(x) + sin(x) - √3cos(x) - 2 = 0.

This is a quadratic equation in sin(x). Solving for sin(x) using the quadratic formula, we get:

sin(x) = [-1 ± √(1 + 4(4)(√3)(2))]/(8).

sin(x) = [-1 ± √(1 + 32√3)]/8.

Therefore, the general solution for sin(x-30°) = cos(2x) is:

x = arcsin([-1 ± √(1 + 32√3)]/8) + 2πn, where n is an integer.

Step 1: Rewrite cos 2x in terms of cos x

Using the double-angle formula for cosine, we have:

```
cos 2x = 2 cos^2 x - 1
```

Step 2: Substitute cos 2x into the original equation

```
sin(x - 30°) = 2 cos^2 x - 1
```

Step 3: Solve for cos x

```
2 cos^2 x - sin(x - 30°) = 1
2 cos^2 x = 1 + sin(x - 30°)
cos^2 x = 1/2 + sin(x - 30°)/2
```

Step 4: Use the half-angle formula for cosine

The half-angle formula for cosine is:

```
cos (x/2) = sqrt((1 + cos x) / 2)
```

Substituting the expression for cos^2 x, we get:

```
cos (x/2) = sqrt((1 + 1 + sin(x - 30°))/4)
cos (x/2) = sqrt((2 + sin(x - 30°))/4)
```

Step 5: Solve for x

Taking the inverse cosine of both sides, we get:

```
x/2 = cos^-1 (sqrt((2 + sin(x - 30°))/4))
x = 2 cos^-1 (sqrt((2 + sin(x - 30°))/4))
```

Step 6: Add 2nπ to the solution

To account for all possible solutions, we add 2nπ to the solution, where n is an integer:

```
x = 2 cos^-1 (sqrt((2 + sin(x - 30°))/4)) + 2nπ
```

This is the general solution to the equation sin(×-30°)=cos 2×.