To solve the equation $\sqrt{2} \cos \left(x - \frac{\pi}{4}\right) + \left(\sqrt{3} - 1\right) \sin x = 2$ for $x$ in the interval $[0, 2\pi]$, let's proceed as follows:

1. Expand the Cosine Term: We can expand $\cos \left(x - \frac{\pi}{4}\right)$ using the cosine difference identity: $\cos \left(x - \frac{\pi}{4}\right) = \cos x \cos \frac{\pi}{4} + \sin x \sin \frac{\pi}{4}$ Since $\cos \frac{\pi}{4} = \sin \frac{\pi}{4} = \frac{\sqrt{2}}{2}$, we have: $\cos \left(x - \frac{\pi}{4}\right) = \frac{\sqrt{2}}{2} \cos x + \frac{\sqrt{2}}{2} \sin x$

2. Substitute and Simplify: Substitute this into the equation: $\sqrt{2} \left(\frac{\sqrt{2}}{2} \cos x + \frac{\sqrt{2}}{2} \sin x\right) + \left(\sqrt{3} - 1\right) \sin x = 2$ Simplify the first term: $\sqrt{2} \cdot \frac{\sqrt{2}}{2} \cos x + \sqrt{2} \cdot \frac{\sqrt{2}}{2} \sin x = \cos x + \sin x$ The equation becomes: $\cos x + \sin x + \left(\sqrt{3} - 1\right) \sin x = 2$ Combine the sine terms: $\cos x + \left(1 + \sqrt{3} - 1\right) \sin x = 2$ Simplify further: $\cos x + \sqrt{3} \sin x = 2$

3. Express in Terms of a Single Sine or Cosine Function: Rewrite $\cos x + \sqrt{3} \sin x$ as $R \cos(x - \alpha)$ for some $R$ and $\alpha$, where: $R = \sqrt{1^2 + (\sqrt{3})^2} = \sqrt{1 + 3} = 2$ and $\cos \alpha = \frac{1}{2}, \quad \sin \alpha = \frac{\sqrt{3}}{2} \Rightarrow \alpha = \frac{\pi}{3}$ Thus, $\cos x + \sqrt{3} \sin x = 2 \cos \left( x - \frac{\pi}{3} \right)$ Our equation now simplifies to: $2 \cos \left( x - \frac{\pi}{3} \right) = 2$ Divide both sides by 2: $\cos \left( x - \frac{\pi}{3} \right) = 1$

4. Solve for $x$: The equation $\cos \left( x - \frac{\pi}{3} \right) = 1$ holds when $x - \frac{\pi}{3} = 2k\pi \quad \text{for integers } k$ Solving for $x$: $x = \frac{\pi}{3} + 2k\pi$ Within the interval $[0, 2\pi]$, this solution only holds for $k = 0$, so $x = \frac{\pi}{3}$

Solution

The solution to the equation within $[0, 2\pi]$ is: $x = \frac{\pi}{3}$

Would you like more details on any step?

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Here are some related questions you might find useful:

1. How do trigonometric identities help in solving equations like this?
2. What does it mean to express an equation in terms of a single cosine or sine function?
3. Why does $\cos(x - \frac{\pi}{3}) = 1$ only have certain solutions?
4. How do we determine the interval for solutions in trigonometric equations?
5. How do we handle more complex trigonometric equations involving multiple terms?

Tip: For equations like this, simplifying expressions with identities can make them far easier to solve directly.