Solve each equation for x, where x is restricted to the given interval.
y = 5 cos x , for x in [0, π]

Verified step by step guidance

1. Step 1: Start with the equation y = 5 \cos(x).
2. Step 2: Divide both sides of the equation by 5 to isolate \cos(x), resulting in \cos(x) = \frac{y}{5}.
3. Step 3: Determine the values of x for which \cos(x) = \frac{y}{5} within the interval [0, \pi].
4. Step 4: Use the inverse cosine function to find x, i.e., x = \cos^{-1}\left(\frac{y}{5}\right).
5. Step 5: Verify that the solution x is within the interval [0, \pi] and adjust if necessary.

Key Concepts

Cosine Function

The cosine function, denoted as cos(x), is a fundamental trigonometric function that relates the angle x to the ratio of the adjacent side to the hypotenuse in a right triangle. It oscillates between -1 and 1, with a period of 2π. Understanding its behavior is crucial for solving equations involving cosine, especially within specified intervals.

Solving Trigonometric Equations

Solving trigonometric equations involves finding the values of the variable (in this case, x) that satisfy the equation. This often requires using inverse trigonometric functions, identities, and understanding the periodic nature of trigonometric functions. The solution must also respect any given restrictions on the variable, such as the interval [0, π].

Interval Notation

Interval notation is a mathematical notation used to represent a range of values. In this context, [0, π] indicates that x can take any value from 0 to π, inclusive. Understanding interval notation is essential for determining valid solutions to the equation, as it restricts the possible values of x to a specific range.