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Magazine Chapter 6: Problem 91
Approximate the solution to each inequality on the interval \([0,2 \pi]\). $$\cos x \geq 0.3$$
Short Answer
Expert verified
The solution is \([0, 1.2661] \cup [5.0171, 2\pi]\).
Step by step solution
01
Understand the Inequality
We are given the inequality \( \cos x \geq 0.3 \). This means we need to find the values of \( x \) such that the cosine of \( x \) is greater than or equal to 0.3 within the interval \([0, 2\pi]\).
02
Identify the Range for \(\cos x\)
The cosine function ranges from -1 to 1. Visualize the cosine curve to understand where the function is above or equal to 0.3. Cosine is a periodic function with a period of \(2\pi\).
03
Find Critical Points
Find the angles where \( \cos x = 0.3 \) using \( \arccos(0.3) \). Calculate: \( x = \arccos(0.3) \approx 1.2661 \) radians. Similarly, for the negative domain, because \( \cos x \) is an even function, \( x = 2\pi - \arccos(0.3) \approx 5.0171 \) radians.
04
Determine Interval Information
Since \( \cos x \) decreases from 1 to -1 over \([0,\pi]\) and increases from -1 to 1 over \([\pi, 2\pi]\), the intervals where \( \cos x \geq 0.3 \) are: from 0 to \(1.2661\) and from \(5.0171\) to \(2\pi = 6.2832\).
05
Write the Solution
Within the interval \([0, 2\pi]\), the solution to \( \cos x \geq 0.3 \) is the union of intervals: \([0, 1.2661] \cup [5.0171, 2\pi]\). Thus, \( x \) takes these values.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Cosine Function
The cosine function is one of the fundamental trigonometric functions, commonly denoted as \( \cos x \). It is generally defined on a unit circle, representing the x-coordinate of a point rotating around the circle. This function has a range between -1 and 1.
In a unit circle framework:
In a unit circle framework:
- When \( x = 0 \) or \( x = 2\pi \), the value of \( \cos x \) is 1.
- The cosine decreases to 0 at \( x = \frac{\pi}{2} \) and reaches -1 at \( x = \pi \).
- The function is symmetric, meaning that \( \cos(-x) = \cos x \).
Interval Notation
Interval notation is a method of representing a range of numbers. It is particularly useful in mathematics for expressing solutions to inequalities and can describe both open and closed intervals.
In interval notation:
In interval notation:
- \([a, b]\) indicates that the interval includes both endpoints \(a\) and \(b\).
- \((a, b)\) indicates an interval that does not include the endpoints.
- A combination, such as \([a, b)\), includes \(a\) but not \(b\).
Arccos Function
The arccosine function, denoted as \( \arccos(x) \), is the inverse function of the cosine. Its primary goal is to determine the angle \( x \) whose cosine value is \( y \).
Key points about the arccos function:
Key points about the arccos function:
- Its range is \([0, \pi]\), reflecting only the angle outputs from the positive part of the cosine curve.
- For example, \( \arccos(0.3) \approx 1.2661 \), means the angle with cosine 0.3 is about 1.2661 radians.
- Since the range only covers a half-cycle of the cosine function, permutations like \( 2\pi - \arccos(x) \) are used to find corresponding angles in other intervals.
Periodic Functions
Periodic functions are functions that repeat their values in regular intervals, known as periods. The cosine function is a prime example of such functions. It repeats every \(2\pi\), meaning:
- For any angle \( x \), \( \cos(x + 2\pi) = \cos x \).
- This repetitive nature makes it easy to predict the function's behavior over larger intervals.
- Understanding periodicity helps find solutions to equations or inequalities in specific ranges by relating them to other ranges where the solutions are known.
