Question

Let $$ f(x)=\sin x \text { and } g(x)=\cos x $$ Show that \(f\) and \(g\) are differentiable on \(E^{1},\) with $$ f^{\prime}(p)=\cos p \text { and } g^{\prime}(p)=-\sin p \text { for each } p \in E^{1} . $$ Hence prove for \(n=0,1,2, \ldots\) that $$ f^{(n)}(p)=\sin \left(p+\frac{n \pi}{2}\right) \text { and } g^{(n)}(p)=\cos \left(p+\frac{n \pi}{2}\right) . $$

Short answer

Both \( f \) and \( g \) are differentiable, with \( f'(p) = \cos p \) and \( g'(p) = -\sin p \).\nThe derivatives cycle every four steps, aligning with \( f^{(n)}(p) = \sin(p + \frac{n\pi}{2}) \) and \( g^{(n)}(p) = \cos(p + \frac{n\pi}{2}) \).

Step-by-step solution

Differentiate f(x) = sin(x)

Start by differentiating the function \( f(x) = \sin x \).\ The derivative of \( \sin x \) with respect to \( x \) is \( \cos x \).\ Therefore, \( f'(x) = \cos x \). This shows that \( f \) is differentiable, and specifically, \( f'(p) = \cos p \) for any \( p \in E^1 \).

Differentiate g(x) = cos(x)

Now differentiate the function \( g(x) = \cos x \).\ The derivative of \( \cos x \) with respect to \( x \) is \( -\sin x \).\ Thus, \( g'(x) = -\sin x \). This ensures that \( g \) is differentiable, and specifically, \( g'(p) = -\sin p \) for any \( p \in E^1 \).

Establish Base Case for n = 0

For \( n=0 \), the assertions are \( f^{(0)}(p) = \sin p \) and \( g^{(0)}(p) = \cos p \).\ This is trivially true since \( f^{(0)}(p) = f(p) = \sin p \) and \( g^{(0)}(p) = g(p) = \cos p \).

Recognize Pattern of Derivatives

Recognize that the pattern of the derivatives repeats every four differentiations.\ For \( f(x) = \sin x \), the sequence of derivatives is:\\( f^{(1)}(p) = \cos p, \ f^{(2)}(p) = -\sin p, \ f^{(3)}(p) = -\cos p, \ f^{(4)}(p) = \sin p \), and then repeats.\ Similarly, \( g(x) = \cos x \) has derivatives:\\( g^{(1)}(p) = -\sin p, \ g^{(2)}(p) = -\cos p, \ g^{(3)}(p) = \sin p, \ g^{(4)}(p) = \cos p \), and then repeats.\ These patterns align with the form \( f^{(n)}(p) = \sin\left(p + \frac{n\pi}{2}\right) \) and \( g^{(n)}(p) = \cos\left(p + \frac{n\pi}{2}\right) \).

Generalize for Any n

Since the derivatives of \( \sin x \) and \( \cos x \) cycle every four steps, use this periodic pattern to generalize: \For \( f(x): \ f^{(n)}(p) = \sin\left(p + \frac{n\pi}{2}\right) \) and for \( g(x): \ g^{(n)}(p) = \cos\left(p + \frac{n\pi}{2}\right) \).\ Use the properties of addition within trigonometric functions, particularly the periodicity of sine and cosine, to establish these identities.

Key concepts

Trigonometric Functions

Trigonometric functions are mathematical functions that relate the angles of a triangle to the ratios of its sides. Two of the most common trigonometric functions are sine \( f(x) = \sin x \) and cosine \( g(x) = \cos x \). These functions are fundamental in many areas of mathematics and science, particularly in studying periodic phenomena such as sound and light waves.
Trigonometric functions have several key properties:

• Periodicity: Both sine and cosine functions repeat their values at regular intervals, specifically \( 2\pi \). This property is what makes them incredibly useful for modeling periodic events.
• Amplitude: The maximum value of both sine and cosine functions is 1, and the minimum is -1, making the amplitude 1.
• Phase Shift: When the input to a sine or cosine function is modified by addition or subtraction of a constant, the graph of the function shifts horizontally. This is essential for understanding changes in wave patterns.

Understanding these properties is crucial to working with trigonometric functions, particularly when analyzing their derivatives and applications.

Derivatives

Derivatives are central in calculus and represent the rate of change of a function with respect to a variable. To understand differentiability, it's essential to grasp how derivatives are computed for trigonometric functions.
For the function \( f(x) = \sin x \), the derivative is \( f'(x) = \cos x \), meaning that at any point \( x \), the rate of change of sine is cosine.
Similarly, the derivative of \( g(x) = \cos x \) is \( g'(x) = -\sin x \). This indicates the rate of change of cosine is the negative sine.
Derivatives of trigonometric functions exhibit particular patterns:

• They cycle through a fixed sequence with each subsequent differentiation.
• For instance, every fourth derivative of \( \sin x \) returns to itself, creating a repeating cycle: \( \sin x, \cos x, -\sin x, -\cos x \).

These fundamental properties give derivatives of trigonometric functions a predictable structure, simplifying the process of higher-order differentiation.

Mathematical Analysis

Mathematical analysis involves rigorous exploration of real and complex number calculus, among other mathematical topics, such as differentiation and integration.
When analyzing functions like sine and cosine for their differentiability, their smoothness and continuous nature become evident. This ensures that these functions have derivatives everywhere on their domain.
In this exercise, mathematical analysis allows us to generalize and predict patterns across multiple derivatives. This involves looking at how derivatives develop as you differentiate the trigonometric functions repeatedly. Using rigorous methods typical in analysis, you can prove continuously differentiable functions, ensuring all derivatives are well-defined.
This analytical approach, through boomions and broader configurations, reveal inherent characteristics in these functions.

• Examine Patterns: Recognizing derivative cycles and periodicity aids in the deep understanding of functional behavior.
• Confirm Theorems: Validating that \( f^{(n)}(p) = \sin(p + \frac{n\pi}{2}) \) and \( g^{(n)}(p) = \cos(p + \frac{n\pi}{2}) \) can be achieved through such analysis.

Mathematical analysis thus forms the backbone of understanding how trigonometric functions behave under differentiation.

Periodicity

Periodicity is a key characteristic of trigonometric functions, referring to how these functions repeat their values in a consistent, cyclical fashion. For both \( \sin x \) and \( \cos x \), the period is \( 2\pi \), meaning the functions complete one full cycle of values every \( 2\pi \) units.
This periodic nature is particularly useful in predicting future values once a cycle is known, a concept that extends to their derivatives.

• Cyclic Behavior: As trigonometric functions are differentiated, certain patterns appear. For instance, the derivatives of sine cycle every four steps, as do the derivatives of cosine.
• Generalization: This repetition allows us to write general formulas such as \( f^{(n)}(p) = \sin(p + \frac{n\pi}{2}) \) and \( g^{(n)}(p) = \cos(p + \frac{n\pi}{2}) \), which encapsulate this periodic behavior for any integer \( n \).
• Applications: Understanding periodicity is crucial for applications beyond mathematics, including engineering, physics, and even music, where cyclic patterns are prevalent.

Periodicity provides a powerful tool for both theoretical exploration and practical application of trigonometric functions in various fields, revealing patterns that are significant in both mathematical and natural phenomena.