To determine the value of \(\sin 67^\circ\) and find the closest matching option from the choices given, follow these steps:
1. Understand the Problem: We need to find the value of \(\sin 67^\circ\) and identify the corresponding fraction from the provided options:
- A. \(\frac{5}{12}\)
- B. \(\frac{12}{5}\)
- C. \(\frac{12}{13}\)
- D. \(\frac{5}{13}\)
2. Recall that the sine of an angle in degrees can be calculated and approximated. Use trigonometric principles and tools to find \(\sin 67^\circ\).
3. Compare with Given Options:
- The value of \(\sin 67^\circ\) is approximately \(0.9205\).
4. Evaluate Each Option:
- Option A: \(\frac{5}{12} \approx 0.4167\)
- Option B: \(\frac{12}{5} = 2.4\) (This value is greater than 1 and thus incorrect as sine values range from -1 to 1)
- Option C: \(\frac{12}{13} \approx 0.9231\)
- Option D: \(\frac{5}{13} \approx 0.3846\)
5. Choosing the Closest Value:
- The value of \(\sin 67^\circ \approx 0.9205\) is closest to \(\frac{12}{13} \approx 0.9231\).
Therefore, the best match is option C: \(\frac{12}{13}\).
Thus, the correct answer is:
[tex]\[ \boxed{\frac{12}{13}} \][/tex]
