We are given the equation:

\[
\sin(A + 30^\circ) = \cos(14A)
\]

We can use a trigonometric identity to rewrite this equation in terms of sine and cosine of the same angle. Specifically, we use the identity:

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

This transforms the right-hand side of the equation into a sine function:

\[
\sin(A + 30^\circ) = \sin(90^\circ - 14A)
\]

Now, we have the equation:

\[
\sin(A + 30^\circ) = \sin(90^\circ - 14A)
\]

For two sine functions to be equal, the angles must satisfy the following conditions:

1. \( A + 30^\circ = 90^\circ - 14A + 360^\circ k \), where \( k \) is an integer representing the periodicity of the sine function, or
2. \( A + 30^\circ = 180^\circ - (90^\circ - 14A) + 360^\circ k = 90^\circ + 14A + 360^\circ k \)

### Solving case 1:
\[
A + 30^\circ = 90^\circ - 14A + 360^\circ k
\]
Rearrange the terms:
\[
A + 14A = 90^\circ - 30^\circ + 360^\circ k
\]
\[
15A = 60^\circ + 360^\circ k
\]
\[
A = \frac{60^\circ + 360^\circ k}{15}
\]
\[
A = 4^\circ + 24^\circ k
\]

### Solving case 2:
\[
A + 30^\circ = 90^\circ + 14A + 360^\circ k
\]
Rearranging terms:
\[
A - 14A = 90^\circ - 30^\circ + 360^\circ k
\]
\[
-13A = 60^\circ + 360^\circ k
\]
\[
A = \frac{-60^\circ - 360^\circ k}{13}
\]
\[
A = -\frac{60^\circ + 360^\circ k}{13}
\]

### Final solution:
The general solutions for \(A\) are:

1. \( A = 4^\circ + 24^\circ k \)
2. \( A = -\frac{60^\circ + 360^\circ k}{13} \)

Where \( k \) is any integer. Depending on the value of \(k\), different values of \(A\) can be found.

Do you want to explore specific values for \(A\) or any particular case?

Here are 5 related questions:
1. How do we use trigonometric identities to solve equations?
2. What is the general solution for sine and cosine functions?
3. How does the periodicity of trigonometric functions affect the solution?
4. Can we solve the equation graphically?
5. How do we handle negative angles in trigonometry?

**Tip**: Always check if the equation has multiple solutions due to the periodic nature of trigonometric functions.

Learn how to solve the trigonometric equation sin(A + 30°) = cos(14A) step-by-step. This problem explores key trigonometric identities and periodic functions. Suitable for students in Grades 10-12.

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