To determine the form [tex]$\tan \frac{\pi}{12} = -\sqrt{\frac{a + \sqrt{b}}{c}}$[/tex], we need to break down the value we obtained step by step. Given the numerical result and knowing the [tex]$\tan\left(\frac{7 \pi}{12}\right)$[/tex] corresponds to [tex]\(-\sqrt{\frac{a + \sqrt{b}}{c}}\)[/tex], we can match the given form step-by-step:
1. Let’s start by addressing the numerical result: [tex]\(\tan\left(\frac{7 \pi}{12}\right) = -3.7320508075688794\)[/tex]. We see that this value can be expressed in the form of [tex]\(-\sqrt{\frac{a + \sqrt{b}}{c}}\)[/tex].
2. Our goal is to find the constants [tex]\( a \)[/tex], [tex]\( b \)[/tex], and [tex]\( c \)[/tex]. We are given:
[tex]\[
\sqrt{\frac{4 + \sqrt{3}}{2}}
\][/tex]
We simplify:
[tex]\[
\sqrt{\frac{4 + \sqrt{3}}{2}}
\][/tex]
Here, the expression we are given is:
[tex]\[
-\sqrt{\frac{4 + \sqrt{3}}{2}}
\][/tex]
3. To equate this to the result:
- We observe that the structure of [tex]\(\sqrt{\frac{a + \sqrt{b}}{c}}\)[/tex] matches with [tex]\( a = 4 \)[/tex], [tex]\( b = 3 \)[/tex], and [tex]\( c = 2 \)[/tex].
4. Verification:
- Calculating the inside: [tex]\(\frac{4 + \sqrt{3}}{2}\)[/tex] results in:
[tex]\[
\frac{4}{2} + \frac{\sqrt{3}}{2} = 2 + \frac{\sqrt{3}}{2}
\][/tex]
5. Then, taking the square root, we have:
-[tex]\(\sqrt{2 + \frac{\sqrt{3}}{2}}\)[/tex]
Thus, we find:
[tex]\[ a = 4 \][/tex]
[tex]\[ b = 3 \][/tex]
[tex]\[ c = 2 \][/tex]
When we reassemble everything according to the simplified form, [tex]\(\tan \frac{7\pi}{12} = -\sqrt{\frac{4 + \sqrt{3}}{2}}\)[/tex].
