In Exercises 1–10, use substitution to determine whether the given x-value is a solution of the equation. 1 2𝝅cos x = ﹣ ------- , x = --------- 2 3

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1. Identify the given equation: \( \cos x = -\frac{1}{2} \).
2. Identify the given value of \( x \): \( x = \frac{2\pi}{3} \).
3. Substitute \( x = \frac{2\pi}{3} \) into the equation: \( \cos\left(\frac{2\pi}{3}\right) \).
4. Recall the cosine value for \( \frac{2\pi}{3} \): \( \cos\left(\frac{2\pi}{3}\right) = -\frac{1}{2} \).
5. Compare the result of the substitution with the right side of the equation to determine if \( x = \frac{2\pi}{3} \) is a solution.

Key Concepts

Cosine Function

The cosine function is a fundamental trigonometric function defined as the ratio of the adjacent side to the hypotenuse in a right triangle. It is periodic with a period of 2π, meaning its values repeat every 2π radians. Understanding the cosine function is essential for solving equations involving angles, as it helps determine the relationship between angles and their corresponding side lengths.

Substitution Method

The substitution method involves replacing a variable in an equation with a specific value to check if the equation holds true. In this context, substituting the given x-value into the cosine equation allows us to verify if the left-hand side equals the right-hand side. This method is crucial for solving equations and verifying solutions in algebra and trigonometry.

Radian Measure

Radian measure is a way of measuring angles based on the radius of a circle. One radian is the angle formed when the arc length is equal to the radius of the circle. In this problem, the x-value is given in radians, which is important for accurately evaluating the cosine function and ensuring that the substitution aligns with the unit of measurement used in trigonometric calculations.