In Exercises 50–53, find all solutions of each equation. cos x = ﹣1/2

검증된 단계별 안내

Recognize that the equation is \(\cos x = -\frac{1}{2}\). We need to find all angles \(x\) where the cosine value is \(-\frac{1}{2}\).

Recall the unit circle values where \(\cos x = \pm \frac{1}{2}\). Specifically, \(\cos x = \frac{1}{2}\) at \(x = \frac{\pi}{3}\) and \(x = \frac{5\pi}{3}\), so for \(\cos x = -\frac{1}{2}\), the solutions will be in the second and third quadrants.

Identify the reference angle \(\theta\) such that \(\cos \theta = \frac{1}{2}\). This reference angle is \(\theta = \frac{\pi}{3}\).

Write the general solutions for \(\cos x = -\frac{1}{2}\) using the reference angle and the fact that cosine is negative in the second and third quadrants: \(x = \pi - \frac{\pi}{3} + 2k\pi\) and \(x = \pi + \frac{\pi}{3} + 2k\pi\), where \(k\) is any integer.

Simplify the expressions to get \(x = \frac{2\pi}{3} + 2k\pi\) and \(x = \frac{4\pi}{3} + 2k\pi\). These represent all solutions to the equation \(\cos x = -\frac{1}{2}\).

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주요 개념

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

Basic Trigonometric Equations

Solving trigonometric equations involves finding all angles x that satisfy the given equation within a specified domain. For example, solving cos x = -1/2 means identifying all angles where the cosine value equals -0.5, considering the periodic nature of cosine.

Unit Circle and Reference Angles

The unit circle helps visualize cosine values as the x-coordinate of points on the circle. To solve cos x = -1/2, one finds the reference angle where cosine is 1/2 and then determines the angles in the appropriate quadrants where cosine is negative (second and third quadrants).

General Solution for Cosine Equations

Because cosine is periodic with period 2π, the general solutions for cos x = a are x = ±θ + 2kπ, where θ is the reference angle and k is any integer. This accounts for all possible solutions over the real numbers.