Textbook Question
Solve each equation for x.
y = 1/2 tan (3x + 2), for x in [-2/3 - π/6, -2/3 + π/6]
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5. Inverse Trigonometric Functions and Basic Trigonometric Equations
Inverse Sine, Cosine, & Tangent
Problem 6.7
Textbook Question
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 guidance1
Identify the equation given: \(y = 5 \cos x\), and the interval for \(x\) is \([0, \pi]\).
Since \(y\) is expressed in terms of \(\cos x\), isolate \(\cos x\) by dividing both sides by 5: \(\cos x = \frac{y}{5}\).
Determine the possible values of \(y\) such that \(\cos x = \frac{y}{5}\) is valid, remembering that \(\cos x\) ranges between \(-1\) and \(1\).
Use the inverse cosine function to solve for \(x\): \(x = \arccos\left(\frac{y}{5}\right)\), ensuring that the solutions lie within the interval \([0, \pi]\).
Check if there are any additional solutions within the interval \([0, \pi]\) by considering the properties of the cosine function on this interval, and write down all valid \(x\) values.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Basic Trigonometric Functions
Trigonometric functions like cosine relate an angle to the ratio of sides in a right triangle or points on the unit circle. Understanding the cosine function's behavior, especially its values between 0 and π, is essential for solving equations involving cos x.
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Solving Trigonometric Equations
Solving equations like y = 5 cos x involves isolating the trigonometric function and finding all angle solutions within the specified interval. This requires knowledge of inverse trigonometric functions and how to interpret multiple solutions in a given domain.
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Domain and Interval Restrictions
The interval restriction (x in [0, π]) limits the possible solutions to those angles within this range. Understanding how to apply domain constraints ensures that only valid solutions are considered when solving trigonometric equations.
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