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The equation provided is:

\[
\sin(x) = \cos(x + 30^\circ)
\]

To solve for \(x\), we can use a trigonometric identity that relates sine and cosine. We know that:

\[
\cos(\theta) = \sin\left(90^\circ - \theta\right)
\]

Using this, we can rewrite the right-hand side of the equation:

\[
\cos(x + 30^\circ) = \sin\left(90^\circ - (x + 30^\circ)\right)
\]

Simplifying:

\[
\cos(x + 30^\circ) = \sin(60^\circ - x)
\]

Now the equation becomes:

\[
\sin(x) = \sin(60^\circ - x)
\]

For sine functions to be equal, the angles must either be the same or differ by an integer multiple of \(180^\circ\), as \(\sin(\theta) = \sin(\pi - \theta)\). So, we have two cases:

1. \(x = 60^\circ - x + 360^\circ k\) where \(k\) is any integer.
2. \(x = 180^\circ - (60^\circ - x) + 360^\circ k\).

Let's solve each case:

### Case 1: \(x = 60^\circ - x + 360^\circ k\)
\[
x + x = 60^\circ + 360^\circ k
\]
\[
2x = 60^\circ + 360^\circ k
\]
\[
x = 30^\circ + 180^\circ k
\]

### Case 2: \(x = 180^\circ - (60^\circ - x) + 360^\circ k\)
\[
x = 180^\circ - 60^\circ + x + 360^\circ k
\]
\[
x = 120^\circ + x + 360^\circ k
\]
\[
0 = 120^\circ + 360^\circ k
\]

This equation has no solutions.

Thus, the general solution is:

\[
x = 30^\circ + 180^\circ k
\]

Where \(k\) is any integer.

Would you like any further clarifications or details on this?

Here are 5 related questions you might find interesting:
1. How would the solution change if the angle on the right-hand side were different, say \(x + 45^\circ\)?
2. What is the general approach for solving equations involving sine and cosine?
3. Can this equation have any solutions in radians instead of degrees?
4. What would the solution be if you included restrictions on the domain of \(x\)?
5. How does the sine function behave compared to the cosine function geometrically?

**Tip:** Always remember that trigonometric functions are periodic, so there are often multiple solutions due to the periodicity of sine and cosine.

Solve for x in the equation sin(x) = cos(x + 30°).

Explore the detailed step-by-step solution to solve the trigonometric equation sin(x) = cos(x + 30°). This problem involves using key trigonometric identities and solving equations related to sine and cosine functions. Suitable for high school students in Grades 10-12.

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