Problem 61 - Calculus Single Variable

Question

A function \(f\) is said to have period \(p\) if there is a smallest positive number \(p\) such that \(f(x+p)=f(x)\) for all \(x\) in the domain of \(f\). Find the period of the function defined by the given expression. \(\tan (x)\)

Step-by-Step Solution

Answer

The period of \( \tan(x) \) is \( \pi \).

Step 1: Understand the Definition of Period

The period of a function is the smallest positive number \( p \) for which \( f(x+p) = f(x) \) for every \( x \) in the function's domain. We need to apply this definition to the tangent function \( \tan(x) \).

Step 2: Basic Property of the Tangent Function

Recognize that the tangent function is related to the sine and cosine functions. Recall the identity \( \tan(x) = \frac{\sin(x)}{\cos(x)} \). Since sine and cosine have period \( 2\pi \), we will examine how \( \tan(x) \) behaves with similar transformations.

Step 3: Determine One Period

For the tangent function, \( \tan(x + \pi) = \frac{\sin(x + \pi)}{\cos(x + \pi)} = \frac{-\sin(x)}{-\cos(x)} = \frac{\sin(x)}{\cos(x)} = \tan(x) \). This simplification shows that over a shift of \( \pi \), the function returns to its original form.

Step 4: Check the Conditions for the Period

Earlier calculations demonstrate that \( \tan(x + \pi) = \tan(x) \) for all \( x \), meaning the function is periodic with period \( \pi \). Check smaller positive subperiods, such as \( \frac{\pi}{2} \), to ensure no smaller period exists, but find that \( \tan(x + \frac{\pi}{2}) eq \tan(x) \).

Step 5: Conclusion

Given that \( \tan(x) \) repeats every \( \pi \) without smaller repetitions, we conclude that the period \( p \) of \( \tan(x) \) is \( \pi \).

Key Concepts

Periodicity

Periodicity in mathematics refers to the characteristic of a function to repeat its values at regular intervals. The smallest positive interval at which a function repeats is known as the period of the function. Knowing the period is crucial for understanding the behavior of trigonometric functions. It helps predict the values of these functions over infinite domains.

In trigonometric functions like sine, cosine, and tangent, each has a specific period indicative of its repetitive nature. For example:

• Sine and cosine functions repeat every \(2\pi\).
• The tangent function, however, has a shorter period of \(\pi\).

When working with periodic functions, identifying the period allows for simplifying calculations and graphing over extended ranges without examining every individual value.

Tangent Function

The tangent function, denoted as \(\tan(x)\), is a fundamental trigonometric function. It is defined in terms of sine and cosine functions as \(\tan(x) = \frac{\sin(x)}{\cos(x)}\). This definition offers insights into some distinctive properties of the tangent function:

• Unlike sine and cosine, which have period \(2\pi\), the tangent function has a period of \(\pi\).
• It is undefined at points where the cosine function equals zero, specifically at \(x = \frac{\pi}{2} + n\pi\), where \(n\) is an integer.
• This leads to vertical asymptotes at the undefined points, resulting in a distinctive wave-like graph with repeating patterns every \(\pi\).

The tangent function's period is half that of the other two primary functions due to how \(\pi\)-shifts result in an identical function value, defined by the identity \(\tan(x + \pi) = \tan(x)\). Understanding the tangent's period and points of discontinuity is vital for graphing and calculating problems requiring trigonometric identities.

Mathematical Proof

In mathematics, proving a statement or conjecture requires a series of logical steps to demonstrate its truth. For the tangent function's periodic nature, the proof involves transformation and simplification through known identities.

The key steps in the proof for \(\tan(x)\)'s period involve:

• Expressing \(\tan(x + \pi)\) using the angle addition identities for sine and cosine.
• Simplifying to showcase how the signs in sine and cosine flip, leading to a negative division which simplifies back to the original tangent function.
• Ensuring no smaller period exists by testing subperiods, like \(\frac{\pi}{2}\), and finding they do not meet the equal function condition \(f(x+p)=f(x)\).

This logical structuring affirms that \(\pi\) is indeed the smallest period for \(\tan(x)\), confirmed by both transformational analysis and verification under the periodicity definition.

Sine and Cosine Relation

Sine and cosine functions are the backbone of trigonometric concepts, providing foundational relations to many other functions. The tangent function's relationship with sine and cosine can be crucial for understanding its properties and behaviors. The key relation is given by \(\tan(x) = \frac{\sin(x)}{\cos(x)}\). This indicates:

• Whenever \(\cos(x) = 0\), \(\tan(x)\) is undefined, which is essential for understanding the vertical asymptotes of the tangent function.
• Sine and cosine have identical periods of \(2\pi\), but their influence on the tangent function leads to its repetition every \(\pi\).
• This differentiation in period from \(2\pi\) to \(\pi\) is due to the effect of the sine and cosine ratio doubling back to repeat its pattern over a lesser interval.

Understanding this relationship doesn't just underline proofs and calculations but also aids in invaluable predictions about function behavior for broader applications in trigonometry and calculus.