https://file.aiquickdraw.com/mathbot/file/1728952890404cqlgn92v.jpg

Let's solve the given trigonometric equation:

The equation is:

\[
2 \sin(x) - \sqrt{2} = 0
\]

### Step-by-step solution:

1. **Isolate** \( \sin(x) \):
   \[
   2 \sin(x) = \sqrt{2}
   \]
   \[
   \sin(x) = \frac{\sqrt{2}}{2}
   \]

2. **Find the general solutions** for \( \sin(x) = \frac{\sqrt{2}}{2} \).

   The value \( \sin(x) = \frac{\sqrt{2}}{2} \) occurs at:
   \[
   x = 45^\circ \text{ and } x = 135^\circ \text{ (in degrees) }
   \]
   or in radians:
   \[
   x = \frac{\pi}{4} \text{ and } x = \frac{3\pi}{4}
   \]

### Now, let's address both ranges separately:

#### 1. **For the range [0°, 360°]:**

The solutions are:
\[
x = 45^\circ, 135^\circ
\]

#### 2. **For the range [0, 2π]:**

The solutions are:
\[
x = \frac{\pi}{4}, \frac{3\pi}{4}
\]

### Final answers:

- The solutions in degrees \([0^\circ, 360^\circ]\) are: 
  \[
  45^\circ, 135^\circ
  \]

- The solutions in radians \([0, 2\pi]\) are: 
  \[
  \frac{\pi}{4}, \frac{3\pi}{4}
  \]

---

Would you like more details or have any questions?

Here are 5 related questions to expand on this topic:
1. How can we find the general solution for \( \sin(x) = k \)?
2. What are the key differences between solving in degrees versus radians?
3. How does the unit circle help in determining sine and cosine values?
4. Can we have negative solutions for sine? If so, how do they fit into the solution set?
5. How do transformations like \( 2 \sin(x) \) affect the sine graph?

**Tip:** Always check your calculator's mode (degrees vs radians) when solving trigonometric equations.

Solve, finding all solutions in [0,2π) or [0°,360°): 2sin(x) - √2 = 0.

Learn how to solve the trigonometric equation 2sin(x) - √2 = 0 by isolating sin(x) and finding exact solutions within [0°, 360°) and [0, 2π). This example demonstrates how to work with the sine function and unit circle concepts. Suitable for students in Grades 10-12.

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