To find [tex]\(\sin 62^{\circ}\)[/tex], we need to compare the value of [tex]\(\sin 62^{\circ}\)[/tex] to the given choices and determine which one is closest.
First, consider the sine values of the choices provided. We convert the fractions to their decimal form for easy comparison:
1. [tex]\(\frac{8}{17} \approx 0.4706\)[/tex]
2. [tex]\(\frac{15}{17} \approx 0.8824\)[/tex]
3. [tex]\(\frac{15}{8} \approx 1.8750\)[/tex]
4. [tex]\(\frac{8}{15} \approx 0.5333\)[/tex]
Now, we know from trigonometric tables or a calculator that the value of [tex]\(\sin 62^{\circ}\)[/tex] is approximately [tex]\(0.8824\)[/tex].
Comparing the value [tex]\(0.8824\)[/tex] to our given choices:
- [tex]\(\frac{8}{17} \approx 0.4706\)[/tex]
- [tex]\(\frac{15}{17} \approx 0.8824\)[/tex]
- [tex]\(\frac{15}{8} \approx 1.8750\)[/tex]
- [tex]\(\frac{8}{15} \approx 0.5333\)[/tex]
It's clear that the fraction [tex]\(\frac{15}{17}\)[/tex] is the closest to the actual value of [tex]\(\sin 62^{\circ}\)[/tex].
Therefore, the correct answer is [tex]\( \boxed{\frac{15}{17}} \)[/tex], which corresponds to option B.
