Problem 40 - Precalculus

Question

For the following exercises, let \(f(x)=\cos x\) On \([0,2 \pi),\) find the \(x\) -intercepts of \(f(x)=\cos x\)

Step-by-Step Solution

Answer

The x-intercepts are at \( x = \frac{\pi}{2} \) and \( x = \frac{3\pi}{2} \).

Step 1: Understand the function and interval

The function given is \( f(x) = \cos x \). We are interested in finding the x-intercepts of this function on the interval \([0, 2\pi)\). The x-intercepts are the points where the graph of the function crosses the x-axis, which occurs when \( f(x) = 0 \).

Step 2: Set the cosine function to zero

To find the x-intercepts, set \( \cos x = 0 \). This equation needs to be solved for \( x \) within the interval \([0, 2\pi)\).

Step 3: Solve for x

The cosine function \( \cos x \) is zero at odd multiples of \( \frac{\pi}{2} \), specifically at \( x = \frac{\pi}{2} + k\pi \) where \( k \) is an integer. In the interval \([0, 2\pi)\), these values are: \( x = \frac{\pi}{2} \) and \( x = \frac{3\pi}{2} \).

Step 4: Verify solutions within the interval

Confirm that the solutions \( x = \frac{\pi}{2} \) and \( x = \frac{3\pi}{2} \) fall within the given interval \([0, 2\pi)\). Both values satisfy this condition because they are between 0 and \( 2\pi \).

Step 5: Conclude the x-intercepts

The x-intercepts of the function \( f(x) = \cos x \) on the interval \([0, 2\pi)\) occur at \( x = \frac{\pi}{2} \) and \( x = \frac{3\pi}{2} \).

Key Concepts

Cosine Function

The cosine function, denoted as \( \cos x \), is one of the fundamental trigonometric functions. It plays a key role in mathematics, especially in geometry and periodic phenomena.
In the unit circle, the cosine of an angle \( x \) is the x-coordinate of the corresponding point on the circle.
This is because the unit circle has a radius of 1, which gives us a perfect representation of the cosine function's values as points wrap around the circle.
When graphed, the

• Cosine function creates a wave-like pattern.
• It oscillates between -1 and 1.
• The period of the graph is \( 2\pi \), meaning it repeats every \(2\pi\) units.

The peaks of the cosine wave occur at multiples of \(2\pi\), where the value is 1, while the troughs (lowest points) occur at odd multiples of \(\pi\), where the value is -1. Understanding the behavior of the cosine function in both its graph and its unit circle representation is crucial for solving trigonometric problems.

x-intercepts

The x-intercepts of a function are the points where the function's graph crosses the x-axis. At these points, the output value of the function is zero.
For the cosine function \( f(x) = \cos x \), finding the x-intercepts involves setting the function equal to zero and solving for \( x \).
The cosine function equals zero at certain specific points:

• These points are exactly the odd multiples of \( \frac{\pi}{2} \).
• Mathematically, it's expressed as \( x = \frac{\pi}{2} + k\pi \), where \( k \) is an integer.

On the interval \([0, 2\pi)\), we look for these x-intercepts that fall within these bounds. The solutions are \( x = \frac{\pi}{2} \) and \( x = \frac{3\pi}{2} \).
These values satisfy both the condition of the cosine function being zero and the requirement that they lie within the specified range. It's essential to check these points carefully to ensure they fit both the function's behavior and the interval constraints.

Intervals in Trigonometry

Intervals in trigonometry are used to define specific ranges for angles or values over which we examine the behavior of trigonometric functions.
In our example, the interval \([0, 2\pi)\) is particularly important because it represents one full cycle of a trigonometric curve like the cosine wave.
Intervals help in focusing on:

• Specific sections of the trigonometric functions' graphs.
• Finding precise x-intercepts, maxima, and minima within these cycles.

Intervals often come with specific notations. A closed bracket, such as \([ \), implies inclusion, meaning the endpoint is part of the range. An open bracket, \(( \), implies exclusion. Here, \( [0, 2\pi) \) includes 0 but excludes \( 2\pi \), emphasizing the periodic nature of trig functions. Recognizing how intervals guide the solving of trigonometric equations helps in accurately controlling the scope and identifying the needed solutions for various problems.