Problem 25 - Precalculus Essentials

Question

Find a cofunction with the same value as the given expression. $$\tan \frac{\pi}{9}$$

Step-by-Step Solution

Answer

The cofunction with the same value as the given expression \(\tan \frac{\pi}{9}\) is \(\cot \frac{4\pi}{9}\).

Step 1 - Identify the cofunction of tangent

Firstly, recall the cofunction identity for tangent and cotangent. The cofunction identity states that for any acute angle \( \alpha \), \(\tan(\frac{\pi}{2} - \alpha) = \cot(\alpha)\). This means that the cofunction of the tangent of an angle is the cotangent of the complement of the angle.

Step 2 - Apply the cofunction identity

Plug the given angle \(\frac{\pi}{9}\) into the formula. The cofunction of the tangent is the cotangent, so the cofunction of the expression \(\tan \frac{\pi}{9}\) would be found by replacing the angle in the tangent function with its complement in the cotangent function. The complement of the angle \(\frac{\pi}{9}\) is \(\frac{\pi}{2} - \frac{\pi}{9}\), which simplifies to \(\frac{4\pi}{9}\). The cotangent of this angle is \(\cot \frac{4\pi}{9}\).

Step 3 - Conclude the solution

So, the cofunction of \(\tan \frac{\pi}{9}\) is \(\cot \frac{4\pi}{9}\). As we can see, once we understand what a cofunction is, finding a cofunction for a given angle is a straightforward process.