Simplify each expression. See Example 4.
cos² π/8 - 1/2

검증된 단계별 안내

Recognize that the expression is \(\cos^{2} \frac{\pi}{8} - \frac{1}{2}\), where \(\cos^{2} \theta\) means \((\cos \theta)^2\).

Recall the double-angle identity for cosine: \(\cos 2\theta = 2\cos^{2} \theta - 1\). This can be rearranged to express \(\cos^{2} \theta\) as \(\cos^{2} \theta = \frac{1 + \cos 2\theta}{2}\).

Apply this identity to \(\cos^{2} \frac{\pi}{8}\) by substituting \(\theta = \frac{\pi}{8}\), so \(\cos^{2} \frac{\pi}{8} = \frac{1 + \cos \frac{\pi}{4}}{2}\).

Substitute this back into the original expression: \(\cos^{2} \frac{\pi}{8} - \frac{1}{2} = \frac{1 + \cos \frac{\pi}{4}}{2} - \frac{1}{2}\).

Simplify the expression by combining the fractions: \(\frac{1 + \cos \frac{\pi}{4} - 1}{2} = \frac{\cos \frac{\pi}{4}}{2}\). This is the simplified form in terms of cosine.

비슷한 문제에 대한 검증된 영상 답변:

이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.

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 relationship allows us to express sine in terms of cosine and vice versa, which is useful for simplifying trigonometric expressions.

Half-Angle Formulas

Half-angle formulas express trigonometric functions of half an angle in terms of the full angle. For cosine, cos²(θ) can be rewritten using cos(2θ) as cos²θ = (1 + cos 2θ)/2, which helps simplify expressions involving squared trigonometric functions.

Angle Measurement in Radians

Angles in trigonometry are often measured in radians, where π radians equal 180 degrees. Understanding radian measure is essential for correctly applying formulas and evaluating trigonometric functions at specific angles like π/8.

Simplify each expression. See Example 4.cos² π/8 - 1/2