Write each function value in terms of the cofunction of a complementary angle.
sin (2π/5)

Verified step by step guidance

1. Recall the cofunction identity for sine and cosine: \(\sin(\theta) = \cos\left(\frac{\pi}{2} - \theta\right)\), where \(\theta\) is an angle in radians.
2. Identify the given angle \(\theta = \frac{2\pi}{5}\) and recognize that the complementary angle to \(\theta\) is \(\frac{\pi}{2} - \theta\).
3. Substitute \(\theta = \frac{2\pi}{5}\) into the complementary angle expression to get \(\frac{\pi}{2} - \frac{2\pi}{5}\).
4. Simplify the expression for the complementary angle by finding a common denominator: \(\frac{\pi}{2} = \frac{5\pi}{10}\) and \(\frac{2\pi}{5} = \frac{4\pi}{10}\), so the complementary angle is \(\frac{5\pi}{10} - \frac{4\pi}{10} = \frac{\pi}{10}\).
5. Express \(\sin\left(\frac{2\pi}{5}\right)\) as the cosine of the complementary angle: \(\sin\left(\frac{2\pi}{5}\right) = \cos\left(\frac{\pi}{10}\right)\).

Key Concepts

Cofunction Identity

Cofunction identities relate trigonometric functions of complementary angles, where the sum of the angles is π/2 radians (90°). For example, sin(θ) = cos(π/2 - θ). This allows expressing one function in terms of the cofunction of its complementary angle.

Complementary Angles

Two angles are complementary if their sum equals π/2 radians (90°). Understanding this relationship is essential for applying cofunction identities, as it defines the angle substitution needed to rewrite trigonometric functions.

Radian Measure

Radian measure is a way to express angles based on the radius of a circle. Since the question uses radians (2π/5), familiarity with radian values and conversions is important to correctly identify complementary angles and apply cofunction identities.