What is [tex] \sin 67^{\circ} ? [/tex]

A. [tex] \frac{5}{12} [/tex]

B. [tex] \frac{12}{5} [/tex]

C. [tex] \frac{12}{13} [/tex]

D. [tex] \frac{5}{13} [/tex]



Answer :

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]