sin(x − 30◦
) − cos(x − 30◦
) = 0 

We are given the equation:

\[
\sin(x - 30^\circ) - \cos(x - 30^\circ) = 0
\]

### Step 1: Move \(\cos(x - 30^\circ)\) to the other side:
\[
\sin(x - 30^\circ) = \cos(x - 30^\circ)
\]

### Step 2: Use the identity \(\sin(\theta) = \cos(90^\circ - \theta)\):
This tells us that:

\[
\sin(x - 30^\circ) = \cos(90^\circ - (x - 30^\circ))
\]
Simplifying the right side:

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

### Step 3: Set up the new equation:
Now we have:

\[
x - 30^\circ = 60^\circ - x
\]

### Step 4: Solve for \(x\):
Add \(x\) to both sides:

\[
2x - 30^\circ = 60^\circ
\]

Add \(30^\circ\) to both sides:

\[
2x = 90^\circ
\]

Finally, divide by 2:

\[
x = 45^\circ
\]

### Step 5: Check for additional solutions:
Since sine and cosine are periodic functions with a period of \(360^\circ\), we need to consider the general solution. The general solution for \(\sin(x - 30^\circ) = \cos(x - 30^\circ)\) occurs when the argument of the sine and cosine differ by multiples of \(360^\circ\). So, we can express the solution as:

\[
x = 45^\circ + 360^\circ n
\]

where \(n\) is any integer.

Thus, the general solution is:

\[
x = 45^\circ + 360^\circ n \quad \text{or} \quad x = 45^\circ + 360^\circ n
\]

Do you need more details or have any questions?

Here are five related questions:
1. What are the general solutions for \(\sin(x) = \cos(x)\)?
2. How do sine and cosine functions shift with angles?
3. What is the relationship between sine and cosine at complementary angles?
4. What are the periodic properties of sine and cosine functions?
5. Can we solve the same equation in radians instead of degrees?

**Tip:** Always remember that trigonometric functions repeat their values in cycles, so solutions often involve general forms with periodic terms.

Learn how to solve the trigonometric equation sin(x − 30°) − cos(x − 30°) = 0 with step-by-step solutions. This problem explores key concepts such as complementary angle identities, periodic functions, and the general solution of trigonometric equations. Suitable for students in Grades 10-12.

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