Problem 20 - Calculus Early Transcendentals: Pearson New International Edition

Question

\(g(\theta)=|\sin \theta|, 0<\theta<2 \pi\)

Step-by-Step Solution

Answer

\(g(\theta)\) is the absolute value of \(\sin\theta\), remaining non-negative and oscillates between 0 and 1.

Step 1: Understand the Function

The function given is \(g(\theta) = |\sin \theta|\). This is the absolute value of the sine function, meaning it will always produce non-negative outcomes.

Step 2: Determine the Non-negative Output

For any angle \(\theta\), \(\sin \theta\) ranges from -1 to 1. The absolute value \(|\sin \theta|\) will thus have the same magnitude as \(\sin \theta\), but it will always be positive or zero. For example, when \(\sin \theta = -0.5\), \(| \sin \theta | = 0.5\).

Step 3: Identify Key Points Within the Interval

Given that the interval is \(0 < \theta < 2\pi\), find key points where \(\sin\theta = 0\), \(\sin\theta = 1\), and \(\sin\theta = -1\):- For \(\sin\theta = 0\): \(\theta = 0, \pi, 2\pi\), but exclude 0 and 2\pi as they are not within the interval.- For \(\sin\theta = 1\): \(\theta = \frac{\pi}{2}\).- For \(\sin\theta = -1\): \(\theta = \frac{3\pi}{2}\).

Step 4: Analyze Behavior in Each Interval Segment

Since \(g(\theta) = |\sin \theta|\), we observe the behavior in each segment:- From \(0\) to \(\frac{\pi}{2}\), \(\sin \theta\) is positive and equals \(|\sin \theta|\).- From \(\frac{\pi}{2}\) to \(\pi\), \(\sin \theta\) is still positive.- From \(\pi\) to \(\frac{3\pi}{2}\), \(\sin \theta\) is negative but becomes positive when taking the absolute value.- From \(\frac{3\pi}{2}\) to \(2\pi\), \(\sin \theta\) is negative, and \(|\sin \theta|\) makes it positive as well.

Step 5: Sketch the Graph for Better Understanding

Sketching the graph can show that within the interval from \(0\) to \(2\pi\), the \(|\sin\theta|\) resembles the typical sine wave but reflected above the x-axis. The graph consists of arcs that peak at \(1\) and touch the x-axis at \(\theta = \pi\).

Step 6: Finalize the Behavior Description

The absolute value function \(g(\theta) = |\sin \theta|\) oscillates between \(0\) and \(1\) within the interval \(0 < \theta < 2\pi\). The function reaches \(1\) at \(\theta = \frac{\pi}{2}\) and \(\theta = \frac{3\pi}{2}\), and crosses the x-axis at \(\theta = \pi\).

Key Concepts

absolute value

The concept of absolute value is quite straightforward yet impactful, especially when applied to functions like the sine function. In mathematics, the absolute value of a number refers to its distance from zero on the number line, regardless of its direction. It is represented using vertical bars \( |x| \). Hence, for any real number or expression \(x\), \( |x| \) is always non-negative.

• If \(x\) is positive or zero, then \( |x| = x\).
• If \(x\) is negative, \( |x| = -x\), which converts \(x\) to a positive value.

When applied to a function like \(g(\theta) = |\sin \theta|\), it indicates that we take the sine of \(\theta\) and ensure the result is always non-negative. This is crucial when analyzing behaviors and patterns because, irrespective of the sine of \(\theta\), which can naturally range from \(-1\) to \(1\), the output will always be from \(0\) to \(1\). This transformation reflects any negative parts of the sine function to positive, altering its graph without changing its repeated pattern within the interval.

sine function

The sine function is one of the fundamental trigonometric functions, primarily used to relate the angle of a right-angled triangle to the ratios of its sides. In a circle described in radians, it helps define the vertical position of a point on the circumference. The sine of an angle \(\theta\), written as \(\sin \theta\), results in values between \(-1\) and \(1\).

• For angles where \(\theta\) is \(0\) or integral multiples of \(\pi\), \(\sin \theta\) equals \(0\).
• When \(\theta\) is \(\frac{\pi}{2}\), \(\sin \theta\) achieves a maximum value of \(1\), representing the peak of its wave.
• Conversely, \(\sin\theta\) reaches a minimum value of \(-1\) at \(\theta = \frac{3\pi}{2}\).

The wave-like pattern of sine is continuous and periodic with a period of \(2\pi\), meaning it repeats every \(2\pi\) units. This property makes it predictable and valuable across multiple areas of mathematics and applied sciences. It naturally oscillates, and such oscillations, when paired with the concept of absolute value, transform into a more uniform wave above the x-axis, as seen in \(g(\theta) = |\sin \theta|\).

interval analysis

Interval analysis is an essential tool in understanding the behavior and characteristics of functions over specific ranges. In this exercise, we focus on the interval \(0 < \theta < 2\pi\) for the function \(g(\theta) = |\sin \theta|\). By analyzing how \(\theta\) behaves within this range, we can better understand the patterns in the function's output.

• Within \(0 < \theta < \frac{\pi}{2}\), \(\sin \theta\) steadily increases to its maximum of \(1\).
• From \(\frac{\pi}{2}\) to \(\pi\), it then decreases back to \(0\).
• In the interval \(\pi\) to \(\frac{3\pi}{2}\), \(\sin \theta\) becomes negative, but \(|\sin \theta|\) flips this negativity positive.
• Finally, as \(\theta\) progresses from \(\frac{3\pi}{2}\) to \(2\pi\), \(\sin \theta\) returns to zero.

By examining these segments, one can deduce important points along the interval, such as when the function reaches zero, peaks, and its symmetry. This kind of analysis is invaluable for sketching graphs and anticipating function behavior, particularly for trigonometric functions that inherently possess repeating periodicity.