General Solution of Trigonometric Equations: sin2x -1

Trigonometric equations are a fundamental part of mathematics, with a wide range of applications in various fields. The equation (sin(2x) -1) is a specific type of equation that requires careful manipulation and understanding of trigonometric identities and periodic properties. This article will explore how to solve this equation, provide a detailed explanation, and discuss the implications of the solution.

Understanding the Equation

The equation (sin(2x) -1) requires us to find the values of (x) that satisfy this condition. We start by understanding that the sine function equals -1 at specific points. These points are determined by the periodic nature of the sine function.

Sine Function Analysis

The sine function, (sin(theta)), equals -1 at the following points:

(frac{3pi}{2} 2kpi) where (k) is any integer. This periodicity of the sine function is crucial in solving the equation.

Deriving the General Solution

Given that (sin(2x) -1), we can set:

(2x frac{3pi}{2} 2kpi)

Dividing both sides by 2, we obtain:

(x frac{3pi}{4} kpi)

Thus, the general solution for the equation is:

(x frac{3pi}{4} kpi) for any integer (k).

Alternative Solution Methods

Varying methods can be used to solve (sin(2x) -1). One alternative is to use the trigonometric identity:

(sin(2x) -sinleft(frac{pi}{2}right))

This leads to:

(sin(2x) sinleft(-frac{pi}{2}right))

Thus, the general solution can also be expressed as:

(2x 2npi - frac{pi}{2}) or (2x 2npi frac{3pi}{2})

Dividing both equations by 2, we get:

(x npi - frac{pi}{4}) or (x npi frac{3pi}{4})

Where (n) is an integer.

Graphical Interpretation

Understanding the graphical representation of (sin(2x)) and (sin(x)) helps in visualizing the solution. The sine curve with the factor 2 is compressed and oscillates twice as fast as the standard sine curve:

The period of (sin(2x)) is half that of (sin(x)). Thus, at (x frac{3pi}{4}), the value of (sin(2x)) is -1.

Conclusion

Understanding the solution of (sin(2x) -1) through multiple methods reinforces the importance of trigonometric identities and periodic properties. Mastering these concepts is crucial for solving a wide range of trigonometric equations. For further exploration and practical applications, consider studying related equations and their graphs.