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We are given that \( \sin x = \frac{5}{13} \) and that the angle \( x \) is in the second quadrant. The goal is to find \( \cos 2x \).

### Step 1: Determine \( \cos x \)
Since \( \sin x = \frac{5}{13} \), we can use the Pythagorean identity to find \( \cos x \):

\[
\sin^2 x + \cos^2 x = 1
\]

Substituting \( \sin x = \frac{5}{13} \):

\[
\left( \frac{5}{13} \right)^2 + \cos^2 x = 1
\]

\[
\frac{25}{169} + \cos^2 x = 1
\]

\[
\cos^2 x = 1 - \frac{25}{169} = \frac{144}{169}
\]

Taking the square root of both sides:

\[
\cos x = \pm \frac{12}{13}
\]

Since \( x \) is in the second quadrant, \( \cos x \) is negative, so:

\[
\cos x = -\frac{12}{13}
\]

### Step 2: Use the double angle formula for \( \cos 2x \)
The formula for \( \cos 2x \) is:

\[
\cos 2x = 2 \cos^2 x - 1
\]

Substitute \( \cos x = -\frac{12}{13} \):

\[
\cos^2 x = \left( -\frac{12}{13} \right)^2 = \frac{144}{169}
\]

Now apply it to the formula:

\[
\cos 2x = 2 \times \frac{144}{169} - 1
\]

\[
\cos 2x = \frac{288}{169} - \frac{169}{169} = \frac{119}{169}
\]

Thus, \( \cos 2x = \frac{119}{169} \).

Would you like further explanation on any step? Here are 5 related questions that can extend your understanding:

1. How would the result change if the angle \( x \) was in the third quadrant?
2. What is the formula for \( \sin 2x \), and how would you calculate it given \( \sin x \) and \( \cos x \)?
3. Can you derive the double angle formulas from the sum of angles identities?
4. Why is the sign of \( \cos x \) negative in the second quadrant?
5. How can you determine the quadrant of an angle using its sine and cosine values?

**Tip:** Always be mindful of the quadrant when calculating trigonometric functions. It determines the sign of sine, cosine, and tangent values.

If sin x = 5/13 and angle x is in the second quadrant, find cos 2x.

Learn how to find cos 2x when sin x = 5/13 and the angle x is in the second quadrant. This problem involves using the Pythagorean identity and the double angle formula for cosine. Suitable for high school students studying trigonometry.

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