Certainly! Let's solve the equation [tex]\(\sin(\theta) = -0.6\)[/tex] to find six specific solutions. We are given the angles [tex]\(\theta = 0.64 + 2k\pi\)[/tex] and [tex]\(\theta = 2.5 + 2k\pi\)[/tex] for integer values of [tex]\(k\)[/tex].
To list six specific solutions, we need to generate values for [tex]\(k\)[/tex] ranging from 0 to 5 for both given angle forms.
### Step-by-Step Solution:
1. First Angle Form: [tex]\(\theta = 0.64 + 2k\pi\)[/tex]
Let's compute the values for [tex]\(k = 0, 1, 2, 3, 4, 5\)[/tex]:
- For [tex]\(k = 0\)[/tex]:
[tex]\[
\theta = 0.64 + 2 \cdot 0 \cdot \pi = 0.64
\][/tex]
- For [tex]\(k = 1\)[/tex]:
[tex]\[
\theta = 0.64 + 2 \cdot 1 \cdot \pi \approx 0.64 + 6.283185307179586 = 6.923185307179586
\][/tex]
- For [tex]\(k = 2\)[/tex]:
[tex]\[
\theta = 0.64 + 2 \cdot 2 \cdot \pi \approx 0.64 + 12.566370614359172 = 13.206370614359173
\][/tex]
- For [tex]\(k = 3\)[/tex]:
[tex]\[
\theta = 0.64 + 2 \cdot 3 \cdot \pi \approx 0.64 + 18.84955592153876 = 19.48955592153876
\][/tex]
- For [tex]\(k = 4\)[/tex]:
[tex]\[
\theta = 0.64 + 2 \cdot 4 \cdot \pi \approx 0.64 + 25.132741228718345 = 25.772741228718345
\][/tex]
- For [tex]\(k = 5\)[/tex]:
[tex]\[
\theta = 0.64 + 2 \cdot 5 \cdot \pi \approx 0.64 + 31.41592653589793 = 32.05592653589793
\][/tex]
2. Second Angle Form: [tex]\(\theta = 2.5 + 2k\pi\)[/tex]
Similarly, we compute the values for [tex]\(k = 0, 1, 2, 3, 4, 5\)[/tex]:
- For [tex]\(k = 0\)[/tex]:
[tex]\[
\theta = 2.5 + 2 \cdot 0 \cdot \pi = 2.5
\][/tex]
- For [tex]\(k = 1\)[/tex]:
[tex]\[
\theta = 2.5 + 2 \cdot 1 \cdot \pi \approx 2.5 + 6.283185307179586 = 8.783185307179586
\][/tex]
- For [tex]\(k = 2\)[/tex]:
[tex]\[
\theta = 2.5 + 2 \cdot 2 \cdot \pi \approx 2.5 + 12.566370614359172 = 15.066370614359172
\][/tex]
- For [tex]\(k = 3\)[/tex]:
[tex]\[
\theta = 2.5 + 2 \cdot 3 \cdot \pi \approx 2.5 + 18.84955592153876 = 21.34955592153876
\][/tex]
- For [tex]\(k = 4\)[/tex]:
[tex]\[
\theta = 2.5 + 2 \cdot 4 \cdot \pi \approx 2.5 + 25.132741228718345 = 27.632741228718345
\][/tex]
- For [tex]\(k = 5\)[/tex]:
[tex]\[
\theta = 2.5 + 2 \cdot 5 \cdot \pi \approx 2.5 + 31.41592653589793 = 33.91592653589793
\][/tex]
Therefore, the six specific solutions we found for [tex]\(\theta\)[/tex] are:
[tex]\[0.64, 6.923185307179586, 13.206370614359173, 19.48955592153876, 25.772741228718345, 32.05592653589793, 2.5, 8.783185307179586, 15.066370614359172, 21.34955592153876, 27.632741228718345, 33.91592653589793\][/tex]
These are the angles satisfying the given trigonometric condition [tex]\(\sin(\theta) = -0.6\)[/tex].
