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

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Identify the given equation: \(y = \cos(x + 3)\), and the interval for \(x\) is \([-3, \pi - 3]\).

Since \(y = \cos(x + 3)\), let’s introduce a substitution to simplify the expression. Define a new variable \(\theta = x + 3\).

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, \(\theta \in [0, \pi]\).

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 \([0, \pi]\).

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 \([-3, \pi - 3]\).

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