Write (√2)sinA-(√2)cosA in the form rsin(A-x)

We are asked to write \( \sqrt{2} \sin A - \sqrt{2} \cos A \) in the form \( r \sin(A - x) \). This is a standard form conversion where we combine the sine and cosine terms into a single sine function.

### Step 1: Recall the formula for \( r \sin(A - x) \)

The general form \( r \sin(A - x) \) can be expanded using the angle subtraction identity:
\[
r \sin(A - x) = r [\sin A \cos x - \cos A \sin x]
\]
We want to match this with the expression \( \sqrt{2} \sin A - \sqrt{2} \cos A \).

### Step 2: Identify coefficients

From the expression \( \sqrt{2} \sin A - \sqrt{2} \cos A \), we can see:
\[
a = \sqrt{2} \quad \text{(coefficient of } \sin A\text{)}
\]
\[
b = -\sqrt{2} \quad \text{(coefficient of } \cos A\text{)}
\]

### Step 3: Compute \( r \)

We use the formula \( r = \sqrt{a^2 + b^2} \) to find the amplitude \( r \):
\[
r = \sqrt{(\sqrt{2})^2 + (-\sqrt{2})^2} = \sqrt{2 + 2} = \sqrt{4} = 2
\]

### Step 4: Compute \( x \)

Next, we find the phase shift \( x \) using the relations:
\[
\cos x = \frac{a}{r} = \frac{\sqrt{2}}{2}, \quad \sin x = \frac{-b}{r} = \frac{\sqrt{2}}{2}
\]
Since both \( \cos x \) and \( \sin x \) are positive, \( x \) lies in the first quadrant. Specifically, \( \cos x = \sin x = \frac{\sqrt{2}}{2} \), which corresponds to:
\[
x = 45^\circ \quad \text{or} \quad x = \frac{\pi}{4} \text{ radians}
\]

### Step 5: Write the final expression

Thus, the expression \( \sqrt{2} \sin A - \sqrt{2} \cos A \) can be written in the form \( r \sin(A - x) \) as:
\[
2 \sin\left(A - \frac{\pi}{4}\right)
\]

### Final Answer:
\[
\sqrt{2} \sin A - \sqrt{2} \cos A = 2 \sin\left(A - \frac{\pi}{4}\right)
\]

Would you like further details or have any questions?

### Related Questions:
1. How can we convert \( r \cos(A - x) \) into a different trigonometric form?
2. What is the geometric meaning of converting to the form \( r \sin(A - x) \)?
3. How does the phase shift \( x \) affect the graph of \( \sin(A - x) \)?
4. Can you express \( 3 \sin A + 4 \cos A \) in the form \( r \sin(A - x) \)?
5. How would you convert \( \sin A + \sqrt{3} \cos A \) into the form \( r \cos(A + x) \)?

### Tip:
The conversion to the form \( r \sin(A - x) \) is useful in simplifying trigonometric expressions, especially in solving equations and analyzing waveforms.

Learn how to express √2sinA - √2cosA in the form rsin(A - x) using trigonometric identities. This problem explores key trigonometric concepts such as angle subtraction formulas and amplitude calculation. Suitable for high school students in Grades 10-12.

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