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

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Recall the cofunction identity for sine and cosine: \(\sin(\theta) = \cos\left(\frac{\pi}{2} - \theta\right)\), where \(\theta\) is an angle in radians.

Identify the given angle \(\theta = \frac{2\pi}{5}\) and recognize that the complementary angle to \(\theta\) is \(\frac{\pi}{2} - \theta\).

Substitute \(\theta = \frac{2\pi}{5}\) into the complementary angle expression to get \(\frac{\pi}{2} - \frac{2\pi}{5}\).

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 \(\frac{5\pi}{10} - \frac{4\pi}{10} = \frac{\pi}{10}\).

Express \(\sin\left(\frac{2\pi}{5}\right)\) as the cosine of the complementary angle: \(\sin\left(\frac{2\pi}{5}\right) = \cos\left(\frac{\pi}{10}\right)\).

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

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.

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