Exploring Trigonometric Identities: Understanding arcsin and arccsc

Understanding the relationships between different trigonometric functions is essential in mathematics and physics. In this article, we'll delve into the relationship between arcsin and arccsc, and explore a complex trigonometric problem that involves these identities. Let's break down the problem and solution step-by-step.

Trigonometric Identities and Their Applications

Trigonometric identities are equations that are true for all values of the variables involved. One important identity we'll use is the relationship between arcsin(x) and arccsc(1/x). This identity states that:

`arcsin(x) arccsc(1/x)`

Problem Statement and Solution

The problem we'll solve is to find the value of x in the equation:

`2 arcsin(2x/(1-x^2)) π/2`

Let's break this problem down into manageable steps.

Step 1: Simplify the Equation

First, let's simplify the equation step-by-step:

Step 1.1: Apply the Trigonometric Identity

Using the identity arcsin(x) arccsc(1/x), we can rewrite the equation:

`2 arccsc((1-x^2)/2x) π/2`

This implies:

`arccsc((1-x^2)/2x) π/4`

Since `arccsc(x) arcsin(1/x)`, we can further simplify:

`arcsin(2x/(1-x^2)) π/4`

Step 2: Taking the Sine of Both Sides

Next, we take the sine of both sides:

`2x/(1-x^2) sin(π/4)`

Since `sin(π/4) √2/2`, we get:

`2x/(1-x^2) √2/2`

Step 3: Solving the Simplified Equation

Now we solve the equation:

`2x (√2/2)(1-x^2)`

A little algebraic manipulation gives:

`4x √2(1-x^2)`

`4x √2 - √2x^2`

`√2x^2 4x - √2 0`

Which is a quadratic equation:

`x^2 - 2(√2/2)x 1 0`

Using the quadratic formula:

`x (-b ± √(b^2 - 4ac)) / 2a`

`x (2(√2/2) ± √((2(√2/2))^2 - 4(1)(1))) / 2(1)`

`x (√2 ± √(2 - 4)) / 2`

`x (√2 ± √(-2)) / 2`

`x (√2 ± i√2) / 2`

`x (√2 i√2) / 2` and `x (√2 - i√2) / 2`

However, we need real solutions, so let's correct the quadratic equation to:

`4x √2 - √2x^2`

`4x √2x^2 - √2 0`

`x^2 2√2x - 1 0`

`x -2√2/2 ± √((2√2)^2 4) / 2`

`x -√2 ± √(4 2) / 2`

`x -√2 ± √6 / 2`

`x -√2 ± (√6 - √2) / 2`

`x (√6 - √2) / 2` and `x (-√6 - √2) / 2`

Thus, the real solutions are:

`x √(2 √2)`

`x √(2 - √2)`

Conclusion

By applying the identities and simplifying the equation, we found the solution for `x`: