https://file.aiquickdraw.com/mathbot/file/1726458475129mvqc7elq.jpg

The equation shown in the image is:

\[
\sqrt{3} \sin(2x) - \cos(2x) = ?
\]

This expression can be simplified using trigonometric identities. Specifically, we recognize that it closely resembles the form of a sine addition formula:

\[
a \sin(\theta) - b \cos(\theta) = R \sin(\theta + \phi)
\]

Where \(R = \sqrt{a^2 + b^2}\) and \(\tan(\phi) = \frac{b}{a}\).

In this case:
- \(a = \sqrt{3}\)
- \(b = 1\)

1. Find \(R\):
\[
R = \sqrt{(\sqrt{3})^2 + (-1)^2} = \sqrt{3 + 1} = \sqrt{4} = 2
\]

2. Find \(\phi\) using \(\tan(\phi) = \frac{b}{a} = \frac{-1}{\sqrt{3}}\), so:
\[
\phi = -\frac{\pi}{6}
\]

Thus, we can rewrite the expression as:

\[
\sqrt{3} \sin(2x) - \cos(2x) = 2 \sin\left(2x - \frac{\pi}{6}\right)
\]

Therefore, the simplified form is:

\[
\boxed{2 \sin\left(2x - \frac{\pi}{6}\right)}
\]

Would you like further details or have any questions?

Here are 5 related questions to explore this topic further:
1. How do you derive the sine addition formula?
2. Can this result help solve equations involving sine and cosine functions?
3. What are other common identities used in trigonometric simplifications?
4. How would you apply this formula to solve real-world problems?
5. How does the phase shift \( \frac{\pi}{6} \) affect the graph of this function?

**Tip**: Whenever dealing with trigonometric expressions, identifying patterns that match addition or subtraction identities can simplify the work significantly.

Solve the equation √3 sin(2x) - cos(2x).

Learn how to simplify and solve the trigonometric expression √3 sin(2x) - cos(2x) using the sine addition formula and key trigonometric identities. Suitable for high school students in Grades 10-12.

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