Solving Trigonometric Equations: sin x cos x and cos2x sin2x

In this article, we will explore how to solve trigonometric equations involving sine and cosine. Specifically, we will look at how to solve for ( x ) in the equations sin x cos x and cos2x sin2x.

1. Solving sin x cos x

First, let's start with the equation sin x cos x. We will use the fundamental trigonometric identity:

sin x tan x cos x

cos x cos x

Dividing both sides of the equation by cos x (noting that cos x is not zero), we get:

tan x 1

Now, let's consider the unit circle and an isosceles right triangle to determine where tan x 1.

Key Points to Remember:

tan x 1 when x π/4 nπ, where n is an integer.

This solution arises from the periodicity of the tangent function, which has a period of π.

Therefore, the general solution for sin x cos x is x π/4 nπ, where n is an integer.

Considering the interval [0, 2π], the specific solutions are x π/4 and x 5π/4.

2. Solving cos2x sin2x

Next, we will solve the equation cos2x sin2x. We can start by using the Pythagorean identity:

sin2x cos2x 1

cos2x 1 - sin2x

Substituting this into the equation gives:

1 - sin2x sin2x

Combining like terms, we get:

1 2sin2x

Solving for sin2x yields:

sin2x 1/2

Therefore, sin x ±sqrt(1/2) ±1/sqrt(2).

Using the inverse sine function, we find:

x ±π/4 nπ, where n is an integer.

Key Points to Remember:

sin x 1 or -1/sqrt(2) when x ±π/4 nπ, where n is an integer.

This solution arises from the periodicity and symmetry of the sine function.

The general solution for cos2x sin2x includes x ±π/4 nπ, where n is an integer.

Special solutions in the interval [0, 2π] are x ±π/4 and x ±5π/4.

3. Geometric Interpretation

From a geometric perspective, we can use the unit circle to visualize the solutions:

When x π/4, sin x cos x sqrt(2)/2.

When x 5π/4, sin x cos x -sqrt(2)/2.

Adding 360° (or 2π) to any of these angles does not change the values of sin x or cos x.

4. Summary

To summarize, we have solved the equations sin x cos x and cos2x sin2x using trigonometric identities and geometric interpretations. The general solutions are as follows:

sin x cos x: x π/4 nπ, where n is an integer.

cos2x sin2x: x ±π/4 nπ, where n is an integer.

5. Conclusion

Mastering these trigonometric equations is crucial for further studies in mathematics and physics. Understanding the periodicity and geometric interpretations provides a solid foundation for solving more complex trigonometric problems.

Further Reading and Resources

Unit Circle: Google Search - Unit Circle

Trigonometric Identities: Math is Fun - Trigonometric Identities

Solving Trigonometric Equations: Khan Academy - Trigonometric Equations