To determine which equation can be used to solve for [tex]\( b \)[/tex], let's evaluate each of the given options. We know some important trigonometric values for a [tex]\(30^\circ\)[/tex] angle:
1. [tex]\(\sin(30^\circ) = \frac{1}{2}\)[/tex]
2. [tex]\(\tan(30^\circ) = \frac{1}{\sqrt{3}}\)[/tex]
Now let's plug these values into each equation to find the corresponding value of [tex]\( b \)[/tex]:
1. Option 1: [tex]\( b = 8 \tan(30^\circ) \)[/tex]
[tex]\[
\tan(30^\circ) = \frac{1}{\sqrt{3}}
\][/tex]
[tex]\[
b = 8 \times \frac{1}{\sqrt{3}} \approx 4.6188
\][/tex]
2. Option 2: [tex]\( b = \frac{8}{\tan(30^\circ)} \)[/tex]
[tex]\[
\tan(30^\circ) = \frac{1}{\sqrt{3}}
\][/tex]
[tex]\[
b = \frac{8}{\frac{1}{\sqrt{3}}} = 8 \sqrt{3} \approx 13.8564
\][/tex]
3. Option 3: [tex]\( b = 8 \sin(30^\circ) \)[/tex]
[tex]\[
\sin(30^\circ) = \frac{1}{2}
\][/tex]
[tex]\[
b = 8 \times \frac{1}{2} = 4
\][/tex]
4. Option 4: [tex]\( b = 6 - \frac{8}{\sin(30^\circ)} \)[/tex]
[tex]\[
\sin(30^\circ) = \frac{1}{2}
\][/tex]
[tex]\[
b = 6 - \frac{8}{\frac{1}{2}} = 6 - 16 = -10
\][/tex]
From our calculations, the results for each option are as follows:
1. [tex]\( b \approx 4.6188 \)[/tex]
2. [tex]\( b \approx 13.8564 \)[/tex]
3. [tex]\( b = 4 \)[/tex]
4. [tex]\( b = -10 \)[/tex]
Thus, these are the values corresponding to each equation:
- [tex]\( b = 8 \tan(30^\circ) \approx 4.6188 \)[/tex]
- [tex]\( b = \frac{8}{\tan(30^\circ)} \approx 13.8564 \)[/tex]
- [tex]\( b = 8 \sin(30^\circ) = 4 \)[/tex]
- [tex]\( b = 6 - \frac{8}{\sin(30^\circ)} = -10 \)[/tex]
So, depending on the specific value needed for [tex]\( b \)[/tex], any of these equations could theoretically be used. However, each equation solves for a different value of [tex]\( b \)[/tex].
