Textbook Question
Solve each equation for x, where x is restricted to the given interval.
y = ― 2 cos 5x , for x in [0, π/5]
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5. Inverse Trigonometric Functions and Basic Trigonometric Equations
Inverse Sine, Cosine, & Tangent
Problem 21
Textbook Question
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 guidance1
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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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
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.
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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.
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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.
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