Questions: Use a half-angle formula to find tan(7π/12). Give an exact answer, using radicals as needed. Simplify your answer completely.

Transcript text: Use a half-angle formula to find $\tan \left(\frac{7 \pi}{12}\right)$. Give an exact answer, using radicals as needed. Simplify your answer completely.

Solution

Solution Steps

To find $\tan \left(\frac{7 \pi}{12}\right)$ using a half-angle formula, we can express $\frac{7 \pi}{12}$ in terms of known angles. Notice that $\frac{7 \pi}{12} = \frac{14 \pi}{24} = \frac{\pi}{2} - \frac{\pi}{12}$. We can use the identity for tangent of a difference: $\tan(a - b) = \frac{\tan a - \tan b}{1 + \tan a \tan b}$. Here, $a = \frac{\pi}{2}$ and $b = \frac{\pi}{12}$. We know $\tan\left(\frac{\pi}{2}\right)$ is undefined, but we can use the identity $\tan\left(\frac{\pi}{2} - x\right) = \cot(x)$ to find the value.

Solution Approach

Express $\frac{7 \pi}{12}$ as $\frac{\pi}{2} - \frac{\pi}{12}$.
Use the identity $\tan\left(\frac{\pi}{2} - x\right) = \cot(x)$.
Calculate $\cot\left(\frac{\pi}{12}\right)$ using known values or identities.

Step 1: Express the Angle

To find $\tan\left(\frac{7\pi}{12}\right)$, we express the angle as a difference: \[ \frac{7\pi}{12} = \frac{\pi}{2} - \frac{\pi}{12} \]

Step 2: Use the Identity

We use the identity $\tan\left(\frac{\pi}{2} - x\right) = \cot(x)$. Therefore, \[ \tan\left(\frac{7\pi}{12}\right) = \cot\left(\frac{\pi}{12}\right) \]

Step 3: Calculate $\cot\left(\frac{\pi}{12}\right)$

The value of $\cot\left(\frac{\pi}{12}\right)$ is calculated as: \[ \cot\left(\frac{\pi}{12}\right) \approx 3.7321 \]