Solving Trigonometric Equations: The Case of tan θ × tan 3θ 1

In this article, we will delve into solving trigonometric equations with a focus on the specific problem of determining the value of tan 2θ given the condition tan θ × tan 3θ 1. The solution employs various trigonometric identities and properties, making it a comprehensive exploration of this mathematical challenge.

Introduction to Trigonometric Identities

Trigonometric identities are fundamental relationships involving trigonometric functions that hold true for all values of the variables involved. They are invaluable in solving a variety of problems in mathematics, physics, and engineering. In this context, we will use several identities to break down and simplify the given equation, leading us to the solution.

Solving tan θ × tan 3θ 1

The problem at hand is to determine the value of tan 2θ when tan θ × tan 3θ 1. Let's start by using a key trigonometric identity to simplify the problem.

First, we know that:

tan x cot 3x

Given the equation tan θ × tan 3θ 1, we can express tan 3θ as cot θ. Therefore:

tan 3θ 1/tan θ cot θ

Using the identity cot θ tan(π/2 - θ), we can rewrite the equation as:

tan 3θ tan(π/2 - θ)

Equating the arguments of the tangent function, we get:

3θ π/2 - θ

Solving for θ, we find:

4θ π/2

2θ π/4

Therefore, tan 2θ tan(π/4) 1.

Another Method of Solving

As an alternative approach, let's consider another method to solve the same problem. Starting from the equation:

tan θ × tan 3θ 1

Using the identity sin A/cos A × sin 3A/cos 3A 1, we can rewrite the equation as:

sin A/sin 3A cos 3A/cos A

Multiplying both sides by cos A × cos 3A, we get:

cos 3A sin 3A

Thus, we have:

cos 3A - sin 3A 0

This implies:

cos 3A sin 3A

Using the identity cos A sin(π/2 - A), we get:

3A π/2 - A

Solving for A, we find:

4A π/2

2A π/4

Therefore, tan 2θ tan(π/4) 1.

Another Technique: Expanding Trigonometric Functions

As another technique, let's expand the trigonometric functions directly. Starting from the equation:

tan 2θ - θ (tan 3θ - tan θ)/(1 tan 3θ × tan θ)

Given the condition tan 3θ 1/tan θ, the denominator becomes:

1 1 2

Thus, the equation simplifies to:

tan 2θ - θ (1 - tan^2 θ) / (2 tan θ)

Simplifying further, we find:

tan 2θ (1 - tan^2 θ) / (2 tan θ)

Finally, we have:

tan 2θ 1 / tan 2θ

Squaring both sides, we get:

tan^2 2θ 1

Therefore, tan 2θ 1 or tan 2θ -1.

Conclusion

In conclusion, the value of tan 2θ given the condition tan θ × tan 3θ 1 can be determined by using trigonometric identities and properties. Through different methods, we have shown that tan 2θ is indeed 1. This comprehensive exploration not only provides the solution but also reinforces the importance of understanding and applying trigonometric identities in solving complex problems.