Solving Trigonometric Equations: A Comprehensive Guide

Trigonometric equations are a fundamental part of mathematics that involve finding the values of angles or sides in a triangle based on the relationships between their sine, cosine, and tangent. This article will explore the process of solving equations like tan-1x 4π - cos-1x, providing a step-by-step breakdown and explaining the nuances of these equations.

Overview

Trigonometric equations can be complex due to the nature of the trigonometric functions involved. This article will focus on two trigonometric identities, tan-1x and cos-1x, and how they interact to form a solvable problem. We’ll cover the methods for solving these equations and the steps to find the correct solutions.

Introduction to Trigonometric Equations

Trigonometric equations involve the use of trigonometric functions like sine, cosine, tangent, and their inverse functions. These equations can be utilized in various fields, including physics, engineering, and calculus. In this section, we will introduce the basic concepts of inverse trigonometric functions and how they apply to the given equation.

The Problem and Solution Method

The problem to solve is: What is the value of x in the equation tan-1x 4π - cos-1x? To solve this equation, we will follow a series of steps, including taking the cosine of both sides and using the properties of trigonometric identities.

1. Taking Cosines of Both Sides

Let's start by taking the cosine of both sides of the equation:

#34;±1/√(1   x2)  x#34;

After squaring both sides, we get:

1/(1 x2) x2

Multiplying both sides by (1 x2) gives:

1 x2(1 x2) - x2

Expanding and simplifying, we get the polynomial equation:

x4 - x2 - 1 0

Using the quadratic formula, this can be further simplified to:

x ±√((√5 - 1)/2)

2. Checking Quadrants and Solutiions

Given the range of tan-1x and cos-1x, we need to consider the sign of x. Depending on whether x is positive or negative, the equation can be simplified differently. For positive x, we

use arctan(x) in the first quadrant and cos-1(x) in the fourth quadrant. For negative x, we use arctan(x) in the second quadrant and cos-1(x) in the third quadrant.

Thus, the final solutions are:

x ±√((√5 - 1)/2)

Here, the negative value must be excluded for real solutions, leaving:

x √((√5 - 1)/2)

Conclusion

In conclusion, solving trigonometric equations can be challenging but is made simpler by using the appropriate identities and properties of trigonometric functions. The final answer to the given problem is x √((√5 - 1)/2). Understanding the nuances of these equations and the methods used to solve them is crucial for advanced mathematical applications.