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The equation you've provided is:

\[
\cos(6x) = \sin(2x + 18^\circ)
\]

To solve this equation, we can use the identity that relates sine and cosine:

\[
\sin(A) = \cos(90^\circ - A)
\]

Using this identity, we can rewrite the sine function on the right-hand side of the equation:

\[
\sin(2x + 18^\circ) = \cos\left(90^\circ - (2x + 18^\circ)\right)
\]
\[
= \cos(72^\circ - 2x)
\]

Now the equation becomes:

\[
\cos(6x) = \cos(72^\circ - 2x)
\]

Since the cosine function is periodic, two general conditions arise for equality:

1. \( 6x = 72^\circ - 2x + 360^\circ n \), where \( n \in \mathbb{Z} \) (the general solution).
2. \( 6x = -(72^\circ - 2x) + 360^\circ n \), also accounting for cosine's periodicity.

### Case 1: \( 6x = 72^\circ - 2x + 360^\circ n \)

\[
6x + 2x = 72^\circ + 360^\circ n
\]
\[
8x = 72^\circ + 360^\circ n
\]
\[
x = \frac{72^\circ + 360^\circ n}{8}
\]
\[
x = 9^\circ + 45^\circ n
\]

### Case 2: \( 6x = -(72^\circ - 2x) + 360^\circ n \)

\[
6x = -72^\circ + 2x + 360^\circ n
\]
\[
6x - 2x = -72^\circ + 360^\circ n
\]
\[
4x = -72^\circ + 360^\circ n
\]
\[
x = \frac{-72^\circ + 360^\circ n}{4}
\]
\[
x = -18^\circ + 90^\circ n
\]

### Final Solutions:
Thus, the general solutions for \( x \) are:

1. \( x = 9^\circ + 45^\circ n \)
2. \( x = -18^\circ + 90^\circ n \)

where \( n \) is an integer.

Would you like to go into more detail, or have any questions?

Here are some related questions:
1. What are the periodic properties of trigonometric functions?
2. How would the solution change if a different trigonometric identity was used?
3. How can these solutions be represented on a unit circle?
4. What happens if you graph both sides of the equation?
5. How would the solution approach change if radians were used instead of degrees?

**Tip:** Always check the domain and periodicity of trigonometric functions to ensure all solutions are accounted for.

Learn how to solve the trigonometric equation cos 6x = sin(2x + 18°) by applying trigonometric identities and periodicity principles. Detailed steps include converting sine to cosine using the identity sin(A) = cos(90° - A). Suitable for students in Grades 10-12.

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