Problem 27 - Precalculus Essentials

Question

Find a cofunction with the same value as the given expression. $$\cos \frac{2 \pi}{5}$$

Step-by-Step Solution

Answer

The cofunction with the same value as the given expression \(\cos \frac{2 \pi}{5}\) is \(\sin \frac{\pi}{10}\).

Step 1: Identity Recall

Recall the definition of a cofunction: for a given angle, the trigonometric functions sine and cosine are cofunctions. This means they yield the same value for complementary angles. In other words, \(\sin \theta = \cos \left(\frac{\pi}{2} - \theta \right)\). Here, \(\theta = \frac{2 \pi}{5}\).

Step 2: Calculate Complementary Angle

Subtract the given angle from \(\frac{\pi}{2}\) to get the complementary angle.\(\frac{\pi}{2}\) - \(\frac{2 \pi}{5} = \frac{5 \pi - 4 \pi}{10} = \frac{\pi}{10}\)

Step 3: Express as Cofunction

Replace \(\theta\) by the complementary angle in the sine function to get the equivalent cofunction. So, \(\sin \frac{\pi}{10} = \cos \frac{2 \pi}{5}\).