Solve each equation for x, where x is restricted to the given interval.
y = cos (x + 3) , for x in [―3, π―3]

Verified step by step guidance

1. Step 1: Start by understanding the equation y = \cos(x + 3). We need to solve for x within the interval [-3, \pi - 3].
2. Step 2: Recognize that the cosine function is periodic with a period of 2\pi. This means that \cos(\theta) = \cos(\theta + 2k\pi) for any integer k.
3. Step 3: To solve for x, set \cos(x + 3) = y. This implies x + 3 = \cos^{-1}(y) + 2k\pi or x + 3 = -\cos^{-1}(y) + 2k\pi, where k is an integer.
4. Step 4: Solve for x by isolating it: x = \cos^{-1}(y) - 3 + 2k\pi or x = -\cos^{-1}(y) - 3 + 2k\pi.
5. Step 5: Determine the values of k such that x is within the interval [-3, \pi - 3]. Evaluate both expressions for x and check which values satisfy the interval condition.

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 it repeats its values every 2π units. Understanding the properties of the cosine function, including its range of values from -1 to 1, is essential for solving equations involving cosine.

Solving Trigonometric Equations

Solving trigonometric equations involves finding the values of the variable that satisfy the equation. This often requires using inverse trigonometric functions, identities, and understanding the periodic nature of trigonometric functions. In this case, we need to isolate x and consider the specified interval to find valid solutions.

Interval Notation

Interval notation is a mathematical notation used to represent a range of values. In this context, the interval [―3, π―3] indicates that x can take any value from -3 to π - 3, inclusive. Understanding how to interpret and apply interval notation is crucial for determining which solutions to the trigonometric equation are valid within the specified range.