Sure, let's find the exact value step-by-step.
1. Understanding the given expression:
We are asked to find the value of [tex]\(\tan \left(\cos^{-1}\left(\frac{1}{8}\right)\right)\)[/tex].
2. Let [tex]\(\theta\)[/tex] be such that [tex]\(\theta = \cos^{-1}\left(\frac{1}{8}\right)\)[/tex]:
This means [tex]\(\cos(\theta) = \frac{1}{8}\)[/tex].
3. Use the Pythagorean identity:
We know from trigonometric identities that [tex]\(\sin^2(\theta) + \cos^2(\theta) = 1\)[/tex].
Substituting [tex]\(\cos(\theta)=\frac{1}{8}\)[/tex] into this identity, we get:
[tex]\[
\sin^2(\theta) + \left(\frac{1}{8}\right)^2 = 1
\][/tex]
[tex]\[
\sin^2(\theta) + \frac{1}{64} = 1
\][/tex]
Solving for [tex]\(\sin^2(\theta)\)[/tex]:
[tex]\[
\sin^2(\theta) = 1 - \frac{1}{64}
\][/tex]
[tex]\[
\sin^2(\theta) = \frac{64}{64} - \frac{1}{64}
\][/tex]
[tex]\[
\sin^2(\theta) = \frac{63}{64}
\][/tex]
Therefore:
[tex]\[
\sin(\theta) = \sqrt{\frac{63}{64}}
\][/tex]
Since [tex]\(\sin(\theta)\)[/tex] must be positive (because [tex]\(\theta = \cos^{-1}(x)\)[/tex] for [tex]\(0 \leq x \leq \pi\)[/tex], [tex]\(\theta\)[/tex] is in the first or second quadrant where sine is positive):
[tex]\[
\sin(\theta) = \sqrt{\frac{63}{64}} = \frac{\sqrt{63}}{8}
\][/tex]
4. Find [tex]\(\tan(\theta)\)[/tex]:
Now, [tex]\(\tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)}\)[/tex].
From above, we found [tex]\(\sin(\theta) = \frac{\sqrt{63}}{8}\)[/tex] and [tex]\(\cos(\theta) = \frac{1}{8}\)[/tex].
So:
[tex]\[
\tan(\theta) = \frac{\frac{\sqrt{63}}{8}}{\frac{1}{8}} = \sqrt{63}
\][/tex]
Thus, the exact value of [tex]\(\tan \left(\cos^{-1}\left(\frac{1}{8}\right)\right)\)[/tex] is [tex]\(\sqrt{63}\)[/tex]. The numerical value, given the trigonometric function calculations, is approximately [tex]\(7.937253933193772\)[/tex].
To summarize, we evaluated the expression step-by-step, starting from identifying the angle, using trigonometric identities to find sine, and then calculating the tangent. These steps lead us to the given numerical result.
