To solve the expression $2\sin x - \sqrt{3} \cos x$, you can rewrite it in the form $R \sin(x + \phi)$, where $R$ and $\phi$ are constants. Here’s how to do it:

1. Calculate $R$: $R = \sqrt{(2)^2 + (-\sqrt{3})^2} = \sqrt{4 + 3} = \sqrt{7}$

2. Find $\phi$ using the tangent function: $\tan \phi = \frac{-\sqrt{3}}{2}$ This gives $\phi = \tan^{-1}\left(-\frac{\sqrt{3}}{2}\right)$.

3. Rewrite the expression: $2\sin x - \sqrt{3} \cos x = \sqrt{7} \left(\sin x \cos \phi + \cos x \sin \phi\right)$

4. Combine into a single sine function: $= \sqrt{7} \sin\left(x + \phi\right)$

This transformation helps in analyzing the maximum and minimum values of the expression, as well as solving equations involving it.

Questions to Consider:

1. What are the maximum and minimum values of the expression $2\sin x - \sqrt{3} \cos x$?
2. How can we find specific values of $x$ that satisfy $2\sin x - \sqrt{3} \cos x = 0$?
3. Can we graph the function $y = 2\sin x - \sqrt{3} \cos x$ to visualize its behavior?
4. What does the angle $\phi$ represent in this context?
5. How can we apply this transformation technique to other similar expressions?

Tip:

When dealing with trigonometric expressions, converting to the sine or cosine of a sum can simplify the analysis significantly.