Solving the Trigonometric Equation sec^2 θ 2 tan θ 4

In this article, we will solve the trigonometric equation sec^2 θ 2 tan θ 4 step-by-step, using trigonometric identities and standard algebraic techniques. This guide is perfect for high school students and anyone interested in deepening their understanding of trigonometric equations.

Step-by-Step Solution

The first step is to utilize the fundamental identity sec^2 θ 1 tan^2 θ.

Substitution

We start by substituting the identity into the given equation:

First, rewrite the equation:

sec^2 θ 2 tan θ 4

Using sec^2 θ 1 tan^2 θ, we get:

1 tan^2 θ 2 tan θ 4

Subtract 4 from both sides to set the equation to zero:

tan^2 θ 2 tan θ - 3 0

Factoring the Quadratic Equation

The next step is to solve the quadratic equation tan^2 θ 2 tan θ - 3 0. This can be done by factoring the quadratic expression:

tan^2 θ 3 tan θ - tan θ - 3 0

Group the terms:

(tan θ 3)(tan θ - 1) 0

This gives us two solutions:

tan θ 3 0 or tan θ - 1 0

Solving for θ

Let's solve each equation separately.

1. tan θ -3

Using the inverse tangent function:

θ arctan(-3)

Since the tangent function is negative in the 2nd and 4th quadrants, we have:

θ π - π/4 3π/4 (2nd quadrant)

θ 2π - π/4 7π/4 (4th quadrant)

2. tan θ 1

Using the inverse tangent function:

θ arctan(1) π/4

Since the tangent function is positive in the 1st and 3rd quadrants, we only take one solution in this case. Thus:

θ π/4

Conclusion

In summary, the solutions to the equation sec^2 θ 2 tan θ 4 are:

θ π/4 θ 3π/4 θ 7π/4

These solutions can be verified by substituting them back into the original equation. This problem demonstrates the power of trigonometric identities and quadratic equations in solving complex trigonometric problems.

Related Trigonometric Equations

Solving similar trigonometric equations can help develop a deeper understanding of trigonometric functions and techniques. Some related equations include:

sec^2 θ - tan^2 θ 1 1 cot^2 θ csc^2 θ sin^2 θ cos^2 θ 1

Understanding these identities and their applications can significantly improve your skills in solving trigonometric equations.