Solving Trigonometric Equations: Step-by-Step Guide with Example

Trigonometry equations are a common challenge in mathematics, especially for students and professionals who need to apply trigonometry in real-world problems. This article provides a comprehensive guide to solving the trigonometric equation 3 cos2 θ - 2√3 sinθ cosθ - 3sin2θ 0. We will walk through the steps, methods, and use various trigonometric identities to reach the final solution.

Introduction to Trigonometric Equations

Trigonometric equations involve trigonometric functions such as sine, cosine, and tangent. These equations can be complex and require a solid understanding of trigonometric identities and properties. Solving trigonometric equations often involves simplifying expressions, factoring, and applying inverse trigonometric functions.

Step-by-Step Solution to 3 cos2 θ - 2√3 sinθ cosθ - 3sin2θ 0

Step 1: Simplify the Equation

Start with the given equation: 3 cos2 θ - 2√3 sinθ cosθ - 3sin2θ 0 Factor out the common terms and simplify the equation: Implies: 3 cos2 θ - sin2 θ 2√3 sinθ cosθ

Step 2: Apply Trigonometric Identities

Use the double-angle identities: cos 2θ cos2 θ - sin2 θ and sin 2θ 2 sinθ cosθ Substitute these identities into the equation: Implies: 3 cos2 θ - sin2 θ 2√3 sin θ cosθ becomes 3 cos 2θ √3 sin 2θ

Step 3: Solve for the Tangent Function

Express the equation in terms of a single trigonometric function: Divide both sides by cos 2θ to isolate tan 2θ: 3 tan 2θ √3 Therefore, tan 2θ √3

Step 4: Find the Angles for Which Tan 2θ √3

Recall that tan θ √3 at θ π/3 nπ, where n is an integer. Therefore, 2θ π/3 nπ Solve for θ: θ (π/3 nπ)/2 or θ (π/6 nπ/2)

Final Solution

The solutions to the trigonometric equation 3 cos2 θ - 2√3 sin θ cosθ - 3sin2θ 0 are given by:

θ π/6 nπ/2 where n is an integer.

Conclusion

Solving trigonometric equations can be challenging, but with a solid understanding of trigonometric identities and methods, such as factoring, applying double-angle identities, and solving for tangent, you can tackle any trigonometric problem. This guide provides a clear step-by-step process for solving the given equation, demonstrating how to use these techniques effectively.

Key Concepts Covered:

Trigonometric Identities Factoring Double-Angle Formulas Tangent Function