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You are asked to express \( 3\sin x - 2\cos x \) in the form \( k\sin(x + \alpha)^\circ \). Here's how to proceed:

### Step 1: General Formula
We can express a linear combination of sine and cosine functions as a single sine function using the following identity:

\[
R\sin(x + \alpha) = R\sin x \cos \alpha + R\cos x \sin \alpha
\]

We want to match this with the given expression \( 3\sin x - 2\cos x \). 

### Step 2: Match Coefficients
Comparing \( R\sin x \cos \alpha + R\cos x \sin \alpha \) with \( 3\sin x - 2\cos x \), we get:

\[
R\cos \alpha = 3 \quad \text{(coefficient of } \sin x \text{)}
\]
\[
R\sin \alpha = -2 \quad \text{(coefficient of } \cos x \text{)}
\]

### Step 3: Solve for \( R \) and \( \alpha \)
To find \( R \), we use the Pythagorean identity:

\[
R = \sqrt{(R\cos \alpha)^2 + (R\sin \alpha)^2} = \sqrt{3^2 + (-2)^2} = \sqrt{9 + 4} = \sqrt{13}
\]

So, \( R = \sqrt{13} \).

To find \( \alpha \), we take the tangent of \( \alpha \):

\[
\tan \alpha = \frac{R\sin \alpha}{R\cos \alpha} = \frac{-2}{3}
\]

Thus,

\[
\alpha = \tan^{-1} \left( \frac{-2}{3} \right) \approx -33.69^\circ
\]

### Final Expression
The expression \( 3\sin x - 2\cos x \) can be written as:

\[
\sqrt{13}\sin(x - 33.69^\circ)
\]

Would you like more details or have any questions on this?

Here are 5 related questions:
1. How do you apply similar transformations for other trigonometric expressions?
2. Can this approach work with functions involving \( \cos(x) \) instead of \( \sin(x) \)?
3. How do you find \( \alpha \) when the ratio is more complex than \( \frac{-2}{3} \)?
4. Can this method be used for other combinations of sine and cosine terms?
5. What would the transformation look like if the coefficients were both negative?

**Tip:** The identity \( R\sin(x + \alpha) \) helps simplify trigonometric expressions into a single sinusoidal form, often used in solving equations and analyzing oscillations.

Express 3sin(x) - 2cos(x) in the form ksin(x + α)°.

Learn how to express the trigonometric expression 3sin(x) - 2cos(x) in the form ksin(x + α). This involves using key trigonometric identities and understanding how to combine sine and cosine into a single sinusoidal function. Suitable for high school students.

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