Problem 12 - Calculus with Concepts in Calculus

Question

Solve the inequality for \(x\) in \([0,2 \pi)\). $$ \cos x \leq 0 $$

Step-by-Step Solution

Answer

The solution is \( x \in [\frac{\pi}{2}, \frac{3\pi}{2}] \).

Step 1: Understand the Range

The problem asks us to solve the inequality \( \cos x \leq 0 \) for \( x \) in the interval \([0, 2\pi)\). This means we need to find the values of \( x \) where the cosine function is less than or equal to zero, within one full rotation around the unit circle.

Step 2: Identify the Key Points on the Unit Circle

The cosine function is zero at \( x = \frac{\pi}{2} \) and \( x = \frac{3\pi}{2} \). Cosine is negative in the second and third quadrants. Therefore, we need to consider the interval where the cosine function is non-positive.

Step 3: Determine the Interval for Cosine Values

The cosine function is negative in the interval \( (\frac{\pi}{2}, \frac{3\pi}{2}) \). It is equal to zero exactly at the points \( x = \frac{\pi}{2} \) and \( x = \frac{3\pi}{2} \). Therefore, the solution to the inequality \( \cos x \leq 0 \) includes these points and the interval between them.

Step 4: Combine the Intervals

Thus, the inequality \( \cos x \leq 0 \) is satisfied for \( x \) in the interval \( [\frac{\pi}{2}, \frac{3\pi}{2}] \). This is because the cosine is negative between these points and zero at exactly those points, satisfying the non-positivity condition.

Key Concepts

Cosine Function

The cosine function is a fundamental trigonometric function represented as \( \cos(x) \). It measures the horizontal distance of a point on the unit circle from the origin at a given angle \( x \), usually expressed in radians.
Angles in trigonometry often emerge from a rotation starting at the positive x-axis, moving counter-clockwise.
The cosine of an angle in standard position specifically indicates how far along the x-axis this point lies.The key characteristics of the cosine function include:

• Periodicity: The cosine function has a period of \( 2\pi \), meaning it repeats every \( 2\pi \) radians.
• Amplitude: For the basic cosine function, the amplitude is \( 1 \), which represents the maximum deviation from its mean value, zero.
• Symmetry: It is an even function, satisfying \( \cos(-x) = \cos(x) \).

Understanding where the cosine function is non-positive (negative or zero) helps us solve inequalities like \( \cos x \leq 0 \).

Unit Circle

The unit circle is a circle with a radius of 1, centered at the origin of a coordinate plane. It plays a crucial role in trigonometry for defining the trigonometric functions for all angles.
The circumference of the unit circle is \( 2\pi \) in radians, which indicates a full 360-degree rotation.
The unit circle allows us to visualize how trigonometric functions behave, specifically:

• Points on the unit circle have coordinates \((\cos(x), \sin(x))\), where \( \cos(x) \) is the x-coordinate and \( \sin(x) \) is the y-coordinate.
• The angles increase as you move counter-clockwise from the positive x-axis, starting at zero.
This visualization helps us easily identify specific angle measurements, such as where the cosine function equals zero or becomes negative.

Interval Notation

Interval notation is a concise way of expressing a range of values, often used in mathematics to detail solutions to inequalities.
It describes continuous subsets of real numbers, allowing mathematicians to communicate efficiently:

• A closed interval, \([a, b]\), includes the endpoints \(a\) and \(b\).
• An open interval, \((a, b)\), excludes the endpoints.
• Half-open intervals such as \((a, b]\) or \([a, b)\) include only one endpoint.

In our inequality problem \( \cos x \leq 0 \), we use interval notation \([\frac{\pi}{2}, \frac{3\pi}{2}]\) to indicate where the cosine function is non-positive in the specified range. Each bracket or parenthesis communicates precise information about inclusivity or exclusivity at the boundary points.

Quadrants of the Unit Circle

The unit circle is divided into four regions known as quadrants. Each quadrant represents a 90-degree slice of the circle, or \( \frac{\pi}{2} \) radians.
The quadrants help determine the sign of trigonometric functions based on the angle's location:

• First Quadrant: from \(0\) to \( \frac{\pi}{2} \) – positive cosine values.
• Second Quadrant: from \( \frac{\pi}{2} \) to \( \pi \) – negative cosine values.
• Third Quadrant: from \( \pi \) to \( \frac{3\pi}{2} \) – also negative cosine values.
• Fourth Quadrant: from \( \frac{3\pi}{2} \) to \( 2\pi \) – positive cosine values again.

For the inequality \( \cos x \leq 0 \), the solution \([\frac{\pi}{2}, \frac{3\pi}{2}]\) means the relevant x-values lie in the second and third quadrants, where the cosine function is either zero or negative.