Find the exact values of s in the given interval that satisfy the given condition.

[0 , 2π) ; cos² s = 1/2

검증된 단계별 안내

Start with the given equation: \(\cos^{2} s = \frac{1}{2}\).

Take the square root of both sides to solve for \(\cos s\): \(\cos s = \pm \sqrt{\frac{1}{2}}\).

Simplify the square root: \(\cos s = \pm \frac{\sqrt{2}}{2}\).

Recall the unit circle values where \(\cos s = \pm \frac{\sqrt{2}}{2}\), which correspond to angles \(s = \frac{\pi}{4}, \frac{3\pi}{4}, \frac{5\pi}{4}, \frac{7\pi}{4}\) within the interval \([0, 2\pi)\).

List these values as the exact solutions for \(s\) in the given interval.

비슷한 문제에 대한 검증된 영상 답변:

이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.

도움이 되었나요?

주요 개념

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

Trigonometric Functions and Their Squares

The cosine function, cos(s), measures the horizontal coordinate of a point on the unit circle at angle s. Squaring the cosine, cos²(s), means taking the square of this value, which is always non-negative and ranges from 0 to 1. Understanding how cos²(s) relates to cos(s) is essential for solving equations involving squared trigonometric functions.

Unit Circle and Angle Intervals

The unit circle represents all angles s from 0 to 2π radians, corresponding to one full rotation. Knowing how cosine values correspond to points on the unit circle helps identify all angles s within the interval [0, 2π) that satisfy the equation. This includes recognizing symmetry and periodicity of cosine.

Solving Trigonometric Equations

To solve equations like cos²(s) = 1/2, one must first find cos(s) = ±√(1/2) = ±√2/2. Then, determine all angles s in the given interval where cosine equals these values. This involves using inverse cosine functions and considering all solutions within the specified domain.

Find the exact values of s in the given interval that satisfy the given condition.[0 , 2π) ; cos² s = 1/2