Problem 25 - A Graphical Approach to Precalculus with Limits

Question

Solve \((\mathbf{a}) f(x)=0,(\mathbf{b}) f(x)>0,\) and \((\mathbf{c}) f(x)<0\) over the interval \([0,2 \pi)\) $$f(x)=-2 \cos x+1$$

Step-by-Step Solution

Answer

(a) \(x = \frac{\pi}{3}, \frac{5\pi}{3}\); (b) \(x \in \left(\frac{\pi}{3}, \frac{5\pi}{3}\right)\); (c) \(x \in \left[0, \frac{\pi}{3}\right) \cup \left(\frac{5\pi}{3}, 2\pi\right)\)."

Step 1: Set the Function Equal to Zero

To solve \( f(x) = 0 \), set \(-2 \cos x + 1 = 0\). This gives us the equation to solve for the zeros of the function.

Step 2: Solve for \( x \) When the Function is Zero

Rearrange the equation to find \( \cos x \):\[ -2 \cos x + 1 = 0 \Rightarrow 2 \cos x = 1 \Rightarrow \cos x = \frac{1}{2} \]. The solutions for \( \cos x = \frac{1}{2} \) are \( x = \frac{\pi}{3} \) and \( x = \frac{5\pi}{3} \) within the interval \([0, 2\pi)\).

Step 3: Find Where the Function is Greater Than Zero

Set \(-2 \cos x + 1 > 0\). Simplifying gives \( 2 \cos x < 1 \Rightarrow \cos x < \frac{1}{2} \). The solution to this inequality within \([0, 2\pi)\) is \( x \in \left(\frac{\pi}{3}, \frac{5\pi}{3}\right)\).

Step 4: Find Where the Function is Less Than Zero

Set \(-2 \cos x + 1 < 0\). Simplifying gives \( 2 \cos x > 1 \Rightarrow \cos x > \frac{1}{2} \). The solution to this inequality within \([0, 2\pi)\) is \( x \in \left[0, \frac{\pi}{3}\right) \cup \left(\frac{5\pi}{3}, 2\pi\right)\).

Key Concepts

Cosine Function

The cosine function is a fundamental part of trigonometry. It measures the x-coordinate of the point on the unit circle corresponding to an angle known as the angle of rotation.
In mathematical terms, the cosine of an angle \(x\), typically in radians, is defined using the adjacent side over hypotenuse in a right triangle:- **Formula**: \( \cos x = \text{adjacent} / \text{hypotenuse} \)More importantly, the cosine function has a few key characteristics:

• It is periodic with a period of \(2\pi\).
• The function oscillates between \(-1\) and \(1\).
• The function is symmetric about the y-axis, hence it is an even function with property \(\cos(-x) = \cos(x)\).

Knowing these properties is crucial in solving trigonometric equations, as they help identify potential solutions for \(x\) within a specified interval.

Inequalities

Inequalities are mathematical expressions used to compare the sizes of two values.
When we say one value is greater than or less than another, we are using inequalities:- **Greater than** is represented as \(>\)- **Less than** is represented as \(<\)In trigonometric equations, inequalities can be used to determine ranges of angles that satisfy a given condition. For example, solving the inequality \( \cos x < \frac{1}{2} \) means finding all the angles \(x\) where the cosine value is less than \(0.5\).To solve trigonometric inequalities, understanding of the periodic nature and specific values of trigonometric functions is essential. The unit circle is a valuable tool here, as it visually represents solution intervals and assists in determining which sections of the graph meet the inequality conditions.

Interval Notation

Interval notation is a shorthand way of writing a range of numbers that satisfy a particular condition. This system uses parentheses \(()\) and square brackets \([]\) to describe intervals:

• Parentheses denote that an endpoint is not included: \((a, b)\) means all numbers between \(a\) and \(b\), excluding \(a\) and \(b\).
• Square brackets indicate that an endpoint is included: \([a, b]\) includes both \(a\) and \(b\).

In trigonometry, interval notation is very helpful when expressing the domain of solutions, like those found in trigonometric inequalities.
For instance, in the scenario where the function \(-2 \cos x + 1 > 0\), the solution \(x \in (\frac{\pi}{3}, \frac{5\pi}{3})\) indicates that any \(x\) between \(\frac{\pi}{3}\) and \(\frac{5\pi}{3}\) satisfies the condition, but the endpoints are not included.
Understanding how to interpret and use interval notation is important for reporting the solutions correctly and concisely.