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.