Properties of the Arccosine Function

Oct 13, 2025 by windailys.com 104 views

Introduction

In this article, we will delve into the world of definite integrals and explore the solution to a specific problem involving the arccosine function. The problem at hand is to show that the definite integral of $\arccos(\sin x/\cos 2x)$ from $0$ to $\pi/6$ is equal to $\pi^2/16$ . This problem has been suggested by Wolfram, but the indefinite integral has not been evaluated. We will use various mathematical techniques and properties of trigonometric functions to derive the solution.

Understanding the Problem

The given problem involves the definite integral of the arccosine function, which is defined as the inverse of the cosine function. The arccosine function returns the angle whose cosine is a given value. In this case, we are dealing with the expression $\arccos(\sin x/\cos 2x)$ , which involves the sine and cosine functions. Our goal is to evaluate the definite integral of this expression from $0$ to $\pi/6$ .

Graph of the Function

To better understand the problem, let's consider the graph of the function $y=\arccos(\sin x/\cos 2x)$ . The graph of this function is shown below:

[Insert graph here]

As we can see from the graph, the function has a maximum value at $x=0$ and a minimum value at $x=\pi/6$ . The graph also shows that the function is decreasing as $x$ increases from $0$ to $\pi/6$ .

Properties of the Arccosine Function

Before we proceed with the solution, let's recall some properties of the arccosine function. The arccosine function is defined as the inverse of the cosine function, and it returns the angle whose cosine is a given value. The range of the arccosine function is $[0,\pi]$ , and it is a decreasing function.

Solution

To solve the problem, we will use the following approach:

Trigonometric Identity: We will start by using the trigonometric identity $\sin x = 2\sin \frac{x}{2} \cos \frac{x}{2}$ to rewrite the expression $\sin x/\cos 2x$ .
Arccosine Function: We will then use the arccosine function to rewrite the expression $\arccos(\sin x/\cos 2x)$ in terms of the sine and cosine functions.
Definite Integral: We will then evaluate the definite integral of the resulting expression from $0$ to $\pi/6$ .

Step 1: Trigonometric Identity

We start by using the trigonometric identity $\sin x = 2\sin \frac{x}{2} \cos \frac{x}{2}$ to rewrite the expression $\sin x/\cos 2x$ .

\sin x = 2\sin \frac{x}{2} \cos \frac{x}{2}

Using this identity, we can rewrite the expression $\sin x/\cos 2x$ as follows:

\frac{\sin x}{\cos 2x} = \frac{2\sin \frac{x}{2} \cos \frac{x}{2}}{\cos 2x}

Step 2: Arccosine Function

We can then use the arccosine function to rewrite the expression $\arccos(\sin x/\cos 2x)$ in terms of the sine and cosine functions.

\arccos\left(\frac{\sin x}{\cos 2x}\right) = \arccos\left(\frac{2\sin \frac{x}{2} \cos \frac{x}{2}}{\cos 2x}\right)

Using the properties of the arccosine function, we can rewrite this expression as follows:

\arccos\left(\frac{\sin x}{\cos 2x}\right) = 2\arccos\left(\sin \frac{x}{2}\right) - \arccos\left(\cos 2x\right)

Step 3: Definite Integral

We can then evaluate the definite integral of the resulting expression from $0$ to $\pi/6$ .

I = \int_0^{\pi/6} \arccos\left(\frac{\sin x}{\cos 2x}\right) dx

Using the properties of the definite integral, we can rewrite this expression as follows:

I = \int_0^{\pi/6} \left(2\arccos\left(\sin \frac{x}{2}\right) - \arccos\left(\cos 2x\right)\right) dx

Evaluating this integral, we get:

I = \left[2\left(x - \frac{\pi}{2}\arccos\left(\sin \frac{x}{2}\right)\right) - \left(x - \frac{\pi}{2}\arccos\left(\cos 2x\right)\right)\right]_0^{\pi/6}