Solving Trigonometric Equations: sin3x cos2x

Trigonometric equations are fundamental in mathematics and have numerous applications in physics, engineering, and more. The equation sin 3x cos 2x is a specific example that requires the use of trigonometric identities to solve. In this article, we will explore the solution to this equation using trigonometric identities and periodic properties of trigonometric functions.

Trigonometric Identities

One of the key techniques to solve trigonometric equations is to utilize trigonometric identities. The identity cos A sin (π/2 - A) is particularly useful in such scenarios. Here, we can rewrite the given equation using this identity.

Solving the Equation

Given the equation:

sin 3x cos 2x

First, we use the identity cos 2x sin (π/2 - 2x) to rewrite the equation as:

sin 3x sin (π/2 - 2x)

Now, we can set the arguments equal to each other, considering the periodic nature of the sine function:

First Case

3x π/2 - 2x 2kπ, k ∈ ?

Rearranging the equation, we get:

5x π/2 2kπ

Solving for x:

x π/10 (2kπ/5), k ∈ ?

Second Case

3x π - (π/2 - 2x) 2kπ, k ∈ ?

Rewriting the equation:

3x π - π/2 2x 2kπ, k ∈ ?

Simplifying:

3x π/2 2x 2kπ, k ∈ ?

Further simplification:

x π/2 2kπ, k ∈ ?

Combining both cases, the general solutions to the equation are:

x π/10 (2kπ/5), k ∈ ? and x π/2 2kπ, k ∈ ?

General Approach to Solving Similar Equations

For equations of the form sin Ax cos Bx, we can use the identity cos (π/2 - x) sin x to transform the equation. For instance, given the equation:

sin 2x cos 3x

We can use the identity:

sin 2x cos (π/2 - 2x)

This gives:

cos (π/2 - 2x) cos 3x

Solving for x:

π/2 - 2x 3x 2kπ, k ∈ ?

From this, we can solve for x:

5x π/2 2kπ

Therefore:

x π/10 (2kπ/5), k ∈ ?

Similarly, we can solve for x when:

3x π/2 - 2x 2kπ, k ∈ ?

Solving this, we get:

x π/10 2kπ, k ∈ ?

Conclusion

Solving trigonometric equations involves recognizing and applying key trigonometric identities. By understanding the periodic properties and using these identities, we can transform and solve a wide variety of trigonometric equations. The examples provided demonstrate how to handle such problems effectively, providing a valuable skillset in mathematics.