Given [tex]\(\cos \theta = \frac{12}{15}\)[/tex], we need to find [tex]\(\sin \theta\)[/tex] among the provided options.
Step-by-step solution:
1. Simplify [tex]\(\cos \theta\)[/tex]:
[tex]\[\cos \theta = \frac{12}{15} = \frac{4}{5}\][/tex]
So, [tex]\(\cos \theta = \frac{4}{5}\)[/tex].
2. Apply the Pythagorean identity:
The Pythagorean identity states that [tex]\(\sin^2 \theta + \cos^2 \theta = 1\)[/tex]. We replace [tex]\(\cos \theta\)[/tex] with [tex]\(\frac{4}{5}\)[/tex] and solve for [tex]\(\sin \theta\)[/tex]:
[tex]\[\sin^2 \theta + \left(\frac{4}{5}\right)^2 = 1\][/tex]
[tex]\[\sin^2 \theta + \frac{16}{25} = 1\][/tex]
[tex]\[\sin^2 \theta = 1 - \frac{16}{25}\][/tex]
[tex]\[\sin^2 \theta = \frac{25}{25} - \frac{16}{25}\][/tex]
[tex]\[\sin^2 \theta = \frac{9}{25}\][/tex]
3. Solve for [tex]\(\sin \theta\)[/tex]:
Take the square root of both sides:
[tex]\[\sin \theta = \sqrt{\frac{9}{25}}\][/tex]
[tex]\[\sin \theta = \frac{\sqrt{9}}{\sqrt{25}}\][/tex]
[tex]\[\sin \theta = \frac{3}{5}\][/tex]
4. Rationalize the result:
Simplify [tex]\(\frac{3}{5}\)[/tex]. To match it with one of the provided options, let's check which option corresponds to [tex]\(\frac{3}{5}\)[/tex]. We convert the provided options into decimal form for easier comparison:
- Option A: [tex]\(\frac{9}{12} = 0.75\)[/tex]
- Option B: [tex]\(\frac{15}{12} = 1.25\)[/tex]
- Option C: [tex]\(\frac{12}{9} = 1.33\)[/tex]
- Option D: [tex]\(\frac{9}{15} = 0.6\)[/tex]
5. Compare the result with the given options:
[tex]\[\frac{3}{5} = 0.6\][/tex]
The fractional form that matches 0.6 is Option D: [tex]\(\frac{9}{15}\)[/tex].
Therefore, the correct answer is:
[tex]\[ \boxed{D} \][/tex]
