Problem 35 - Precalculus

Question

For the following exercises, let \(f(x)=\sin x\) On \([0,2 \pi),\) the maximum value(s) of the function occur(s) at what \(x\) -value(s)?

Step-by-Step Solution

Answer

The maximum occurs at \(x = \frac{\pi}{2}\).

Step 1: Understand the Function

We have the function \(f(x) = \sin x\). Our task is to find where this function reaches a maximum within the interval \([0, 2\pi)\).

Step 2: Recall the Properties of Sine Function

The sine function, \(\sin x\), is periodic and oscillates between -1 and 1. Within one full period \([0, 2\pi)\), its maximum value is 1.

Step 3: Determine where \(\sin x = 1\)

The sine function reaches its maximum value of 1 at \(x = \frac{\pi}{2}\) within the interval \([0, 2\pi)\).

Step 4: Conclude Where the Maximum Occurs

Thus, the maximum value of \(f(x) = \sin x\) on the interval \([0, 2\pi)\) occurs at \(x = \frac{\pi}{2}\).

Key Concepts

sine function

The sine function, denoted as \( \sin x \), is a foundational concept in trigonometry that describes a unique wave-like pattern. This function is based on a unit circle where the output, or value of \( \sin x \), corresponds to the y-coordinate of a point on the circle. The sine function is immensely helpful in modeling cyclical and oscillatory phenomena such as sound waves, light waves, and tidal patterns.

In mathematics, the sine function is typically defined using radians. A single complete cycle of the function ranges from \(0\) to \(2\pi\), or about 6.28 radians. This cycle represents a crucial concept in understanding many real-world applications. The key features of the sine function include:

• Oscillation: The sine function smoothly oscillates between -1 and 1, creating a regular waveform.
• Symmetry: This function is symmetric about the origin, classifying it as an odd function, described by \(-\sin x = \sin (-x)\).

Understanding the sine function's periodic and oscillatory nature is vital for analyzing any periodic motion.

periodic function

A periodic function is a function that repeats its values at regular intervals or periods. The sine function \( \sin x \) is a classic example of a periodic function. Its period is \(2\pi\), meaning every \(2\pi\) units along the x-axis, the function repeats its pattern.

Recognizing a function as periodic is crucial for predicting future behavior based on observed cycles. This characteristic is valuable in various scientific and engineering fields, where systems often follow cyclical patterns.

• Cycle Length: For \( \sin x \), the cycle length is \(0\) to \(2\pi\).
• Repetition: The function value is identical once every period \(2\pi\).

In summary, periodic functions like \( \sin x \) help reveal underlying patterns in complex data sets by identifying the frequency at which these patterns repeat.

maximum value

In the context of trigonometric functions, the term "maximum value" refers to the peak value a function reaches within a given interval. For the sine function \( \sin x \), its maximum value within one full cycle \([0, 2\pi)\) is 1.

Finding the maximum value of \( \sin x \) within any interval involves identifying where it hits 1. This occurs at an angle \(x = \frac{\pi}{2}\), due to the properties of the sine wave derived from the unit circle.

• Location of Maximum Value: In a standard interval \([0, 2\pi)\), the sine function's maximum is specifically at \(x = \frac{\pi}{2}\).
• Interpreting Maximum: This max value is related to the highest point on the unit circle's sine wave.

Understanding how to locate these maxima is essential in solving optimization problems in physics and engineering and in predicting natural cyclical behaviors.