Solving Trigonometric Equations: General Solution of cos4x cos2x

Introduction to Trigonometric Equations

Trigonometric equations involve trigonometric functions and their arguments. These equations are widely studied in mathematics and are often encountered in various scientific and engineering disciplines. A common type of trigonometric equation we'll analyze in this article is cos4x cos2x. Understanding how to solve these types of equations is crucial for many practical applications.

Step-by-Step Solution of cos4x cos2x

Given: cos4x cos2x

Step 1: Express cos4x using the double angle formula for cosine.

cos2A cosA2 - sinA2

cos4x cos22x - sin22x

Step 2: Set the equation:

cos22x - sin22x cos2x

Step 3: Use the identity cos2A - sin2A 2cos2A - 1 1 - 2sin2A.

cos22x - (1 - cos22x) cos2x

2cos22x - 1 cos2x

Step 4: Set the quadratic equation:

2cos22x - cos2x - 1 0

Step 5: Solve the quadratic equation. Let's set cos2x y:

2y2 - y - 1 0

Solving for y, we get:

y 1 or y -1/2

Step 6: Substitute the values back:

If cos2x 1, then:

2x 2nπ rarr; x nπ

If cos2x -1/2, then:

2x 2nπ ± π/3 rarr; x nπ ± 60°

So, the general solution to cos4x cos2x is:

x nπ for n ∈ Z x nπ ± π/3 for n ∈ Z

Graphical Interpretation of cos4x cos2x

When plotting the graph of cos4x, it will repeat its values every 2π/4 π/2. Similarly, cos2x repeats every 2π/2 π. This periodic nature means that at certain points, the values of these functions will be the same, leading to solutions of the equation.

Conclusion

The general solution of the equation cos4x cos2x involves both the periodic solutions of cosine functions and solving a resulting quadratic equation. By understanding these steps, we can solve similar trigonometric equations more effectively.

Related Keywords:

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Keywords: cos4x, trigonometric equations, general solution