Cos theta + root 2 sin theta equal to 2 find the value of sin theta minus root 3 cos theta

Given the equation \( \cos \theta + \sqrt{2} \sin \theta = 2 \), we are asked to find the value of \( \sin \theta - \sqrt{3} \cos \theta \).

### Step 1: Use the given equation
We start by squaring both sides of the given equation:

\[
(\cos \theta + \sqrt{2} \sin \theta)^2 = 2^2
\]

Expanding the left side:

\[
\cos^2 \theta + 2\cos \theta \cdot \sqrt{2} \sin \theta + (\sqrt{2} \sin \theta)^2 = 4
\]

Simplifying:

\[
\cos^2 \theta + 2\sqrt{2} \cos \theta \sin \theta + 2\sin^2 \theta = 4
\]

Since \( \cos^2 \theta + \sin^2 \theta = 1 \), we can substitute this into the equation:

\[
1 + \sin^2 \theta + \cos^2 \theta + 2\sqrt{2} \cos \theta \sin \theta = 4
\]

Simplifying:

\[
3 + 2\sqrt{2} \cos \theta \sin \theta = 4
\]

Thus:

\[
2\sqrt{2} \cos \theta \sin \theta = 1
\]

### Step 2: Use trigonometric identities
We know from the identity for double angles that \( 2\cos \theta \sin \theta = \sin 2\theta \). Therefore:

\[
\sqrt{2} \sin 2\theta = 1
\]

\[
\sin 2\theta = \frac{1}{\sqrt{2}} = \frac{\sqrt{2}}{2}
\]

This implies \( 2\theta = \frac{\pi}{4} \) or \( 2\theta = \frac{3\pi}{4} \), and so on.

### Step 3: Find \( \sin \theta - \sqrt{3} \cos \theta \)
Given \( 2\theta = \frac{\pi}{4} \), then \( \theta = \frac{\pi}{8} \). Let's substitute into \( \sin \theta - \sqrt{3} \cos \theta \).

Using known trigonometric values:

\[
\sin \frac{\pi}{8} - \sqrt{3} \cos \frac{\pi}{8}
\]

While calculating exact values for these would involve deeper trigonometric computations or lookup, typically this expression might be better simplified by substituting values and approximating them.

### Step 4: Approximation or exact computation
Substituting approximate values:

\[
\sin \frac{\pi}{8} \approx 0.3827, \quad \cos \frac{\pi}{8} \approx 0.9239
\]

\[
0.3827 - \sqrt{3} \times 0.9239 \approx 0.3827 - 1.598 = -1.215
\]

Thus, \( \sin \theta - \sqrt{3} \cos \theta \) evaluates approximately to \( -1.215 \).

Do you want further details, clarifications, or any step revisited?

Here are 5 questions related to the problem:

1. How can we determine the exact value of \( \sin \frac{\pi}{8} \) and \( \cos \frac{\pi}{8} \)?
2. What trigonometric identity could simplify the expression \( \sin \theta - \sqrt{3} \cos \theta \)?
3. What if \( 2\theta = \frac{3\pi}{4} \)? How does this change the solution?
4. Can we use a different method to solve the original equation?
5. How does the sign of trigonometric functions change in different quadrants?

**Tip**: When working with trigonometric functions, always consider whether identities or double-angle formulas might simplify your calculations.

Learn how to find the value of sin theta - sqrt(3) cos theta from the given trigonometric equation cos theta + sqrt(2) sin theta = 2. Explore step-by-step solution, trigonometric identities, and double-angle formulas. Suitable for high school students studying trigonometry.

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