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. Identify the given equation: \(y = \cos(x + 3)\), and the interval for \(x\) is \([-3, \pi - 3]\).
2. Since \(y = \cos(x + 3)\), let’s introduce a substitution to simplify the expression. Define a new variable \(\theta = x + 3\).
3. Rewrite the interval for \(x\) in terms of \(\theta\): since \(x\) ranges from \(-3\) to \(\pi - 3\), then \(\theta = x + 3\) ranges from \(-3 + 3 = 0\) to \((\pi - 3) + 3 = \pi\). So, \(\theta \in [0, \pi]\).
4. Solve the equation for \(\theta\) by setting \(y = \cos(\theta)\) equal to the desired value (if given) or analyze the behavior of \(\cos(\theta)\) on the interval \([0, \pi]\). If the problem requires solving \(\cos(\theta) = y_0\), use the inverse cosine function: \(\theta = \arccos(y_0)\) and consider all solutions within \([0, \pi]\).
5. After finding the solutions for \(\theta\), convert back to \(x\) by using \(x = \theta - 3\). Make sure the solutions for \(x\) lie within the original interval \([-3, \pi - 3]\).

Key Concepts

Cosine Function and Its Properties

The cosine function, cos(x), is a periodic trigonometric function with period 2π. It oscillates between -1 and 1 and is even, meaning cos(-x) = cos(x). Understanding its graph and behavior helps in solving equations involving shifts and intervals.

Solving Trigonometric Equations

Solving equations like cos(x + 3) = y involves finding all angles x that satisfy the equation within a given interval. This requires using inverse trigonometric functions and considering the periodic nature of cosine to find all valid solutions.

Interval Restrictions and Domain Considerations

When solving trigonometric equations, restricting the variable x to a specific interval, such as [−3, π−3], limits the possible solutions. It is essential to check which solutions fall within this domain to provide the correct answer set.