Solving the Equation Arccos(x) Arctan(x): A Step-by-Step Guide

Trigonometry can often present complex equations that may seem daunting at first, but with a systematic approach, they can be solved neatly. One such equation is Arccos(x) Arctan(x). Below is a detailed guide on how to solve this equation, along with the reasoning and steps involved.

Understanding the Problem

This equation involves inverse trigonometric functions, specifically arccosine (cos-1) and arctangent (tan-1). The functions arccos(x) and arctan(x) return angles, and the equation suggests that these angles are equal for some value of x.

Method to Solve the Equation

The first step is to set y arccos(x). This substitution simplifies the equation:

Let y arccos(x). Thus, x cos(y). Substitute x in the original equation to get y arctan(cos(y)). Next, we need to analyze the ranges of the functions. The range of arccos(x) is [0, π] and the range of arctan(x) is (-π/2, π/2). Therefore, for the equation to hold, y must lie in the interval [0, π/2].

Testing Specific Values

Let's test a few specific values of x:

When x 0, arccos(0) π/2 and arctan(0) 0. This does not satisfy the equation. When x 1, arccos(1) 0 and arctan(1) π/4. This also does not satisfy the equation. When x 1/√2, arccos(1/√2) π/4 and arctan(1/√2) π/4. This satisfies the equation.

From these tests, we conclude that one solution to the equation arccos(x) arctan(x) is:

(x frac{1}{√2} approx 0.7071)

Alternative Methods

Several other methods have been proposed to solve the same equation:

Using the identity (tan(arccos(x)) frac{sqrt{1 - x^2}}{x}), we get (x arctan(1/sqrt{1 - x^2})). Squaring both sides, we obtain: (x^2 frac{1}{1 - x^2}) (x^4 - x^2 - 1 0) Solving the characteristic equation, we get: (x^2 frac{-1 sqrt{5}}{2}) Therefore, (x sqrt{frac{-1 sqrt{5}}{2}} approx 0.78615)

Another Approach

We can think of a right triangle with opposite side x and adjacent side 1. The hypotenuse is (sqrt{1 x^2}). Using the identity (arctan(x) arccos(frac{1}{sqrt{1 x^2}})), we get:

(x arccos(frac{1}{sqrt{1 x^2}})) Squaring both sides: (x^2(1 x^2) 1) (x^4 x^2 - 1 0) Solving for (x^2): (x^2 frac{-1 sqrt{5}}{2}) Therefore, (x sqrt{frac{-1 sqrt{5}}{2}} approx 0.78615)

Conclusion

After testing and solving with different methods, we find that the solution to the equation Arccos(x) Arctan(x) is:

(x sqrt{frac{-1 sqrt{5}}{2}})

This is an exact solution but it is also approximately 0.78615.