In Exercises 25–30, use an identity to find the value of each expression. Do not use a calculator. sin² 𝜋 + cos² 𝜋 10 10

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Recall the Pythagorean identity in trigonometry: \(\sin^2 x + \cos^2 x = 1\) for any angle \(x\).

Identify the angle given in the problem, which is \(\frac{\pi}{10}\).

Apply the identity directly by substituting \(x = \frac{\pi}{10}\) into the formula: \(\sin^2 \left(\frac{\pi}{10}\right) + \cos^2 \left(\frac{\pi}{10}\right)\).

Since the identity holds for all angles, the expression simplifies to 1 without needing to calculate the sine or cosine values individually.

Therefore, the value of \(\sin^2 \frac{\pi}{10} + \cos^2 \frac{\pi}{10}\) is 1.

비슷한 문제에 대한 검증된 영상 답변:

이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.

영상 길이:

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주요 개념

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

Pythagorean Identity

The Pythagorean identity states that for any angle θ, sin²θ + cos²θ = 1. This fundamental trigonometric identity is derived from the Pythagorean theorem and is essential for simplifying expressions involving sine and cosine squared terms.

Evaluating Trigonometric Functions at Specific Angles

Understanding the values of sine and cosine at key angles, such as π (180 degrees), helps in directly substituting and simplifying expressions. For example, sin(π) = 0 and cos(π) = -1, which are critical for evaluating the given expression without a calculator.

Using Identities to Simplify Expressions

Applying trigonometric identities allows simplification of complex expressions into simpler forms. Instead of calculating sine and cosine values separately, identities like sin²θ + cos²θ = 1 provide a straightforward way to find the value of the expression efficiently.