In Exercises 85–96, use a calculator to solve each equation, correct to four decimal places, on the interval [0, 2𝝅). 2 cos x = ﹣ ------ 5

Verified step by step guidance

1. Step 1: Recognize that the equation is \( \cos x = -\frac{2}{5} \).
2. Step 2: Use the inverse cosine function to find the reference angle: \( x = \cos^{-1}\left(-\frac{2}{5}\right) \).
3. Step 3: Since cosine is negative in the second and third quadrants, determine the angles in these quadrants that correspond to the reference angle.
4. Step 4: For the second quadrant, calculate \( x = \pi - \cos^{-1}\left(-\frac{2}{5}\right) \).
5. Step 5: For the third quadrant, calculate \( x = \pi + \cos^{-1}\left(-\frac{2}{5}\right) \).

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 its values repeat every 2π radians. Understanding the behavior of the cosine function is essential for solving equations involving cos(x).

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) = a, then x can be found using x = arccos(a). These functions are crucial for solving equations where the trigonometric function is set equal to a specific value, as in the given equation.

Calculator Use for Trigonometric Equations

Using a calculator to solve trigonometric equations involves inputting the equation in a way that the calculator can interpret. It is important to ensure the calculator is set to the correct mode (degrees or radians) based on the context of the problem. Additionally, understanding how to round answers to a specified number of decimal places is necessary for providing precise solutions.