Problem 35 - Calculus with Concepts in Calculus

Question

For a function \(f\), if there is a smallest positive number \(a\) such that \(f(x+a)=f(x)\) for all \(x\) in the domain of \(f\), then \(a\) is called the period of \(f .\) Plot cach of the following functions, and from the graph guess the period. Then prove that your guess is correct. a. \(\tan x\) b. \(\sin 3 x\) c. \(|\sin x|\) d. \(\cos \left(-2 x+\frac{\pi}{2}\right)\)

Step-by-Step Solution

Answer

a: \(\pi\), b: \(\frac{2\pi}{3}\), c: \(\pi\), d: \(\pi\).

Step 1: Identify the Period of \(\tan x\)

For the tangent function \(\tan x\), observe that it has vertical asymptotes at odd multiples of \(\frac{\pi}{2}\). The basic period of \(\tan x\) is \(\pi\), as it repeats its values every \(\pi\) units. So, from the periodic property \(\tan(x + \pi) = \tan x\), we identify that \(\tan x\) has a period of \(\pi\).

Step 2: Prove the Periodicity of \(\tan x\)

To prove that the period of \(\tan x\) is \(\pi\), consider the addition formula for tangent: \(\tan(x + \pi) = \frac{\tan x + \tan \pi}{1 - \tan x \tan \pi}\). Since \(\tan \pi = 0\), this simplifies to \(\tan(x + \pi) = \tan x\). Hence, the period of \(\tan x\) is indeed \(\pi\).

Step 3: Identify the Period of \(\sin 3x\)

The sine function \(\sin x\) has a period of \(2\pi\). In the function \(\sin 3x\), the period is affected by the coefficient of \(x\). Generally, for \(\sin bx\), the period is \(\frac{2\pi}{b}\). Thus, the period of \(\sin 3x\) is \(\frac{2\pi}{3}\).

Step 4: Prove the Periodicity of \(\sin 3x\)

For \(\sin 3x\), substitute \(x\) with \(x + \frac{2\pi}{3}\): \(\sin 3(x + \frac{2\pi}{3}) = \sin(3x + 2\pi)\). Since \(\sin(\theta + 2\pi) = \sin \theta\), we find \(\sin(3x + 2\pi) = \sin 3x\), confirming that the period is \(\frac{2\pi}{3}\).

Step 5: Identify the Period of \(|\sin x|\)

The absolute value function \(|\sin x|\) modifies the sine wave by reflecting negative values to positive values, retaining the same period. The period of \(\sin x\) is \(2\pi\), and so \(|\sin x|\) repeats every \(\pi\) units due to symmetry.

Step 6: Prove the Periodicity of \(|\sin x|\)

To establish \(|\sin x|\) has a period of \(\pi\), note \(\sin(x + \pi) = -\sin x\) thus \(|\sin(x + \pi)| = |-\sin x| = |\sin x|\). Therefore, \(|\sin x|\) repeats every \(\pi\) units.

Step 7: Identify the Period of \(\cos(-2x + \frac{\pi}{2})\)

The cosine function has a period of \(2\pi\). The transformation involves \(\cos(bx + c)\), specifically here \(-2\pi\), which suggests the period is \(\frac{2\pi}{|b|}\). Thus, the period is \(\frac{2\pi}{2} = \pi\).

Step 8: Prove the Periodicity of \(\cos(-2x + \frac{\pi}{2})\)

For \(\cos(-2x + \frac{\pi}{2})\), substitute \(x + \pi\): \(\cos(-2(x + \pi) + \frac{\pi}{2}) = \cos(-2x - 2\pi + \frac{\pi}{2})\). Simplifying, \(\cos(-2x + \frac{\pi}{2})\) confirms the period of \(\pi\) as it returns to the original formula.

Key Concepts

Trigonometric Identities

Understanding trigonometric identities is key to mastering periodic functions. These identities are equations that hold true for all values of the variable where both sides of the equation are defined. For example, the periodicity identity for sine states that

• \( \sin(x + 2\pi) = \sin x \)
• \( \cos(x + 2\pi) = \cos x \)

These identities show that sine and cosine functions repeat every \( 2\pi \) units. Recognizing these patterns is essential for proving periodicity of trigonometric functions.
Besides, identities like the Pythagorean Identity \( \sin^2 x + \cos^2 x = 1 \) help in verifying properties and working through transformations of these functions.

Graphing Trigonometric Functions

When graphing trigonometric functions like \( \sin x \) and \( \cos x \), you need to understand their baseline shapes and modifications. Both sine and cosine waves are smooth, continuous oscillations displaying distinctive peaks and troughs. Their graphs have the same shape but are shifted relative to each other horizontally.
For instance,

• The \( \sin x \) graph starts at the origin, rising to 1 at \( \frac{\pi}{2} \), descending back to 0 at \( \pi \), continuing towards -1 at \( \frac{3\pi}{2} \), and finally completing the cycle at \( 2\pi \).
• The \( \cos x \) graph begins at 1, but follows a similar path, reaching extreme points at multiples of \( \frac{\pi}{2} \) but mirrored in terms of the starting point.

Graph transformations often involve changes in amplitude, period, or phase, and help in examining less straightforward functions like \( \sin 3x \) or \( \cos(-2x + \frac{\pi}{2}) \).

Proving Trigonometric Properties

Proving trigonometric properties often involves the strategic use of identities and transformations. To prove the periodicity of functions like \( \tan x \) or \( \sin 3x \), you substitute the function argument with an increment that represents a full period, then simplify using identities.
For example, the identity \( \tan(x + \pi) = \tan x \) demonstrates that \( \tan x \) has a period of \( \pi \), as adding \( \pi \) does not change its value.
Similarly, transforming \( \sin 3(x + \frac{2\pi}{3}) \) into \( \sin 3x \) using the identity \( \sin(\theta + 2\pi) = \sin \theta \) confirms the period of \( \frac{2\pi}{3} \). Such proofs are key in understanding the behavior of trigonometric functions beyond basic observations from their graphs.

Sine and Cosine Functions

The sine and cosine functions are fundamental to trigonometry, providing the basis for understanding periodic phenomena. These functions describe smooth, periodic oscillations that repeat every \( 2\pi \) units.
The sine function \( \sin x \) measures the vertical coordinate of a point on the unit circle, while \( \cos x \) measures the horizontal coordinate. This rotation around the circle explains their periodic nature.

• Different transformations, such as multiplying the angle (like in \( \sin 3x \)), change the speed of oscillation, effectively altering the period of the function.
• Cosine with transformations (like \( \cos(-2x + \frac{\pi}{2}) \)) can shift or stretch the graph, making understanding transformations crucial in graphing more complex or modified trigonometric functions.

Mastering these core characteristics equips students with the tools to handle questions on graphing and analyzing trigonometric functions, ensuring a robust mathematical foundation.