Certainly! Let's find the exact value of [tex]\( \tan \left( \frac{\pi}{8} \right) \)[/tex] using the double-angle formula for tangent and the fact that [tex]\( \tan \left( \frac{\pi}{4} \right) = 1 \)[/tex].
### Step-by-step Solution:
1. Double-Angle Formula:
The double-angle formula for tangent is given by:
[tex]\[
\tan(2x) = \frac{2 \tan(x)}{1 - \tan^2(x)}
\][/tex]
2. Given Information:
We know that [tex]\( \tan \left( \frac{\pi}{4} \right) = 1 \)[/tex].
3. Set Up the Problem:
Let [tex]\( x = \frac{\pi}{8} \)[/tex]. Then [tex]\( 2x = \frac{\pi}{4} \)[/tex].
4. Substitute into the Double-Angle Formula:
We need to find [tex]\( \tan \left( \frac{\pi}{8} \right) \)[/tex], which we'll denote as [tex]\( y \)[/tex]. Using the given fact [tex]\( \tan \left( \frac{\pi}{4} \right) = 1 \)[/tex], we get:
[tex]\[
1 = \frac{2 \tan \left( \frac{\pi}{8} \right)}{1 - \tan^2 \left( \frac{\pi}{8} \right)}
\][/tex]
Substituting [tex]\( y \)[/tex] for [tex]\( \tan \left( \frac{\pi}{8} \right) \)[/tex], we have:
[tex]\[
1 = \frac{2y}{1 - y^2}
\][/tex]
5. Solve for [tex]\( y \)[/tex]:
Rearrange and solve the equation:
[tex]\[
1 - y^2 = 2y
\][/tex]
[tex]\[
y^2 + 2y - 1 = 0
\][/tex]
We now have a quadratic equation. To solve for [tex]\( y \)[/tex], we use the quadratic formula:
[tex]\[
y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\][/tex]
where [tex]\( a = 1 \)[/tex], [tex]\( b = 2 \)[/tex], and [tex]\( c = -1 \)[/tex].
6. Apply the Quadratic Formula:
[tex]\[
y = \frac{-2 \pm \sqrt{2^2 - 4 \cdot 1 \cdot (-1)}}{2 \cdot 1}
\][/tex]
[tex]\[
y = \frac{-2 \pm \sqrt{4 + 4}}{2}
\][/tex]
[tex]\[
y = \frac{-2 \pm \sqrt{8}}{2}
\][/tex]
[tex]\[
y = \frac{-2 \pm 2\sqrt{2}}{2}
\][/tex]
[tex]\[
y = -1 \pm \sqrt{2}
\][/tex]
7. Select the Positive Value:
Since [tex]\( y = \tan \left( \frac{\pi}{8} \right) \)[/tex] must be positive, we choose the positive solution:
[tex]\[
y = -1 + \sqrt{2}
\][/tex]
### Conclusion:
Thus, the exact value of [tex]\( \tan \left( \frac{\pi}{8} \right) \)[/tex] is:
[tex]\[
\tan \left( \frac{\pi}{8} \right) = -1 + \sqrt{2}
\][/tex]
Numerically, this evaluates to approximately [tex]\( 0.41421356237309515 \)[/tex].
