Solving Trigonometric Equations Using Identities: An Example with Cos3x - Sin3x

In this article, we will walk through the process of solving the trigonometric equation cos^3x - sin^3x 1. We will utilize various trigonometric identities to simplify and solve the equation step-by-step. By the end of this article, you will understand the methods used and be able to apply similar techniques to solve other trigonometric equations.

Introduction to Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions (sine, cosine, and tangent) that are always true for all values of the variables involved. These identities are essential for solving complex trigonometric equations. Some of the fundamental identities we will use in this process include:

cos^2x sin^2x 1 a^3 - b^3 (a - b)(a^2 ab b^2)

Solving the Equation: cos^3x - sin^3x 1

To solve the equation cos^3x - sin^3x 1 using trigonometric identities, we can start by applying the difference of cubes formula. The difference of cubes can be factored as follows:

a^3 - b^3 a - ba^2 ab b^2

In our case, we let:

a cosx b sinx

Thus, we can rewrite the equation as:

cosx - sinxcos^2x cosxsinx sin^2x 1

We know that:

cos^2x sin^2x 1

Therefore, we can simplify the second factor:

cos^2x sin^2x 1 implies cos^2x sin^2x - cosxsinx 1 - cosxsinx

Substituting this back into our equation gives:

cosx - sinx(1 - cosxsinx) 1

Let's denote u cosx - sinx. Then, our equation becomes:

u(1 - cosxsinx) 1

Now we need to consider the possible values for u and 1 - cosxsinx.

Case 1: If u 1

cosx - sinx 1 implies cosx 1 sinx

This case does not yield valid solutions since cosx cannot exceed 1.

Case 2: If u -1

cosx - sinx -1 implies cosx 1 sinx

This leads to:

sin^2x cosx 1^2

Expanding this gives:

sin^2x cos^2x 2cosx 1

Substituting sin^2x 1 - cos^2x gives:

1 - cos^2x cos^2x 2cosx 1

Simplifying, we get:

0 2cos^2x 2cosx

Factoring out 2cosx gives:

2cosx(cosx 1) 0

This gives us two solutions:

cosx 0, which gives x π/2 nπ cosx 1 0, which gives cosx -1, or x π 2nπ

Case 3: If 1 - cosxsinx 0

cosxsinx -1

This case is impossible since cosxsinx has a maximum value of 1/2 and cannot equal -1.

Thus, the solutions to the equation cos^3x - sin^3x 1 are:

x π/2 nπ and x π 2nπ for n ∈ Z.

Conclusion and Further Applications

In this article, we have demonstrated how to solve the equation cos^3x - sin^3x 1 using trigonometric identities. This method can be applied to similar problems involving trigonometric functions. Understanding and practicing these techniques will enhance your ability to solve complex trigonometric equations, making you a more skilled problem-solver in mathematics.