Problem 26 - Precalculus Essentials

Question

Find a cofunction with the same value as the given expression. $$\tan \frac{\pi}{7}$$

Step-by-Step Solution

Answer

The cofunction with the same value as the given expression \(\tan \frac{\pi}{7}\) is \(\cot \frac{5\pi}{14}\).

Step 1: Identify the given function

First, recognize that the given function is \(\tan \frac{\pi}{7}\). The cofunctions of tangent(tan) is cotangent(cot).

Step 2: Find Complement of the Function argument

To find the cofunction for the tangent function, compute the complement of the function's argument. A complementary angle is the angle subtracted from \(\pi/2\) or 90 degrees, therefore, the complement of \(\frac{\pi}{7}\) is \((\frac{\pi}{2} - \frac{\pi}{7}) = \frac{5\pi}{14}\).

Step 3: Apply the cofunction identity of tangent

The cofunction identity of the tangent is the cotangent. Since the cofunction of a function is given by the function at the complement of the angle, the cotangent of the complement of the function's original argument is equivalent to the original function's value. Therefore, \(\tan \frac{\pi}{7}\) = \(\cot \frac{5\pi}{14}\). So, \(\cot \frac{5\pi}{14}\) is the cofunction with the same value as \(\tan \frac{\pi}{7}\).