To find the exact value of [tex]\(\cos 2x\)[/tex] given that [tex]\(\sin x = -\frac{12}{13}\)[/tex] and [tex]\(\cos x > 0\)[/tex], follow these steps:
1. Understand the given information:
- [tex]\(\sin x = -\frac{12}{13}\)[/tex]
- [tex]\(\cos x > 0\)[/tex]
2. Use the Pythagorean identity to find [tex]\(\cos x\)[/tex]:
[tex]\[
\sin^2 x + \cos^2 x = 1
\][/tex]
Substitute [tex]\(\sin x = -\frac{12}{13}\)[/tex]:
[tex]\[
\left(-\frac{12}{13}\right)^2 + \cos^2 x = 1
\][/tex]
[tex]\[
\frac{144}{169} + \cos^2 x = 1
\][/tex]
[tex]\[
\cos^2 x = 1 - \frac{144}{169}
\][/tex]
[tex]\[
\cos^2 x = \frac{169}{169} - \frac{144}{169}
\][/tex]
[tex]\[
\cos^2 x = \frac{25}{169}
\][/tex]
3. Determine [tex]\(\cos x\)[/tex]:
Since [tex]\(\cos x > 0\)[/tex],
[tex]\[
\cos x = \sqrt{\frac{25}{169}} = \frac{5}{13}
\][/tex]
4. Use the double-angle formula for cosine:
The double-angle formula for cosine is:
[tex]\[
\cos 2x = 2\cos^2 x - 1
\][/tex]
Substitute [tex]\(\cos x = \frac{5}{13}\)[/tex]:
[tex]\[
\cos^2 x = \left(\frac{5}{13}\right)^2 = \frac{25}{169}
\][/tex]
[tex]\[
\cos 2x = 2 \cdot \frac{25}{169} - 1
\][/tex]
[tex]\[
\cos 2x = \frac{50}{169} - 1
\][/tex]
[tex]\[
\cos 2x = \frac{50}{169} - \frac{169}{169}
\][/tex]
[tex]\[
\cos 2x = \frac{50 - 169}{169}
\][/tex]
[tex]\[
\cos 2x = \frac{-119}{169}
\][/tex]
5. Check the provided options for the answer:
The options were:
- [tex]\(-\frac{119}{169}\)[/tex]
- [tex]\(-\frac{144}{169}\)[/tex]
- [tex]\(\frac{119}{169}\)[/tex]
- [tex]\(\frac{144}{169}\)[/tex]
Given the calculation, the exact solution for [tex]\(\cos 2x\)[/tex] is:
[tex]\[
\cos 2x = -\frac{119}{169}
\][/tex]
So, the correct answer is [tex]\(-\frac{119}{169}\)[/tex].
