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

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1. Recognize that the equation \( \cos x = -\frac{1}{2} \) is asking for the angles \( x \) where the cosine value is \(-\frac{1}{2}\).
2. Recall that the cosine function is negative in the second and third quadrants of the unit circle.
3. Identify the reference angle where \( \cos \theta = \frac{1}{2} \), which is \( \theta = \frac{\pi}{3} \) or \( 60^\circ \).
4. Determine the angles in the second and third quadrants: \( x = \pi - \frac{\pi}{3} \) and \( x = \pi + \frac{\pi}{3} \).
5. Express the general solutions for \( x \) using the periodicity of the cosine function: \( x = \frac{2\pi}{3} + 2k\pi \) and \( x = \frac{4\pi}{3} + 2k\pi \), where \( k \) is any integer.

Key Concepts

Cosine Function

The cosine function, denoted as cos(x), is a fundamental trigonometric function that relates the angle x in a right triangle to the ratio of the length of the adjacent side to the hypotenuse. It is periodic with a period of 2π, meaning that cos(x) repeats its values every 2π radians. Understanding the behavior of the cosine function is essential for solving equations involving it.

Inverse Trigonometric Functions

Inverse trigonometric functions, such as arccos(x), are used to find angles when the value of a trigonometric function is known. For example, if cos(x) = -1/2, we can use arccos(-1/2) to find the principal angle. However, since trigonometric functions are periodic, multiple angles can satisfy the equation, necessitating a comprehensive approach to find all solutions.

General Solutions of Trigonometric Equations

When solving trigonometric equations, it is important to find all possible solutions within a specified interval or in general terms. For cosine equations, the general solution can be expressed as x = arccos(value) + 2nπ or x = -arccos(value) + 2nπ, where n is any integer. This accounts for the periodic nature of the cosine function and ensures that all angles that satisfy the equation are included.