In Exercises 31–38, find a cofunction with the same value as the given expression. cos 2𝜋 5

Verified step by step guidance

1. Recall the cofunction identity for cosine and sine: \(\cos(\theta) = \sin\left(\frac{\pi}{2} - \theta\right)\).
2. Identify the given angle \(\theta\) in the expression: here, \(\theta = \frac{2\pi}{5}\).
3. Apply the cofunction identity by substituting \(\theta\) into the formula: \(\cos\left(\frac{2\pi}{5}\right) = \sin\left(\frac{\pi}{2} - \frac{2\pi}{5}\right)\).
4. Simplify the expression inside the sine function by finding a common denominator: \(\frac{\pi}{2} = \frac{5\pi}{10}\) and \(\frac{2\pi}{5} = \frac{4\pi}{10}\), so \(\frac{\pi}{2} - \frac{2\pi}{5} = \frac{5\pi}{10} - \frac{4\pi}{10} = \frac{\pi}{10}\).
5. Write the final cofunction expression: \(\cos\left(\frac{2\pi}{5}\right) = \sin\left(\frac{\pi}{10}\right)\).

Key Concepts

Cofunction Identity

Cofunction identities relate the trigonometric functions of complementary angles, such as sin(θ) = cos(90° - θ) or sin(θ) = cos(π/2 - θ) in radians. These identities allow us to express one trigonometric function in terms of another by using the complementary angle.

Radian Measure

Radian measure is a way to express angles based on the radius of a circle, where 2π radians equal 360 degrees. Understanding how to convert between radians and degrees or interpret angles in radians is essential for applying trigonometric identities correctly.

Cosine Function

The cosine function gives the x-coordinate of a point on the unit circle corresponding to a given angle. Knowing its properties, such as periodicity and symmetry, helps in identifying equivalent expressions and applying cofunction identities effectively.