Solving the Equation sin(2ab) 1: A Comprehensive Guide

Have you ever encountered the equation sin(2ab) 1 and wondered how to solve it? This guide will walk you through the process step-by-step, ensuring you understand the underlying trigonometric principles and the method to find solutions.

Understanding the Equation

Let's start with the given equation:

sin(2ab) 1

This equation involves the sine function, which oscillates between -1 and 1. The value 1 represents the peak of the sine wave. Therefore, the equation sin(2ab) 1 is satisfied when the argument of the sine function, 2ab, is equal to π/2, 3π/2, 5π/2, ..., and so on, for all integers n.

General Solution

To express the solution more generally, we can write:

sin(2ab) sin(nπ/2)

where n is any integer. This means:

2ab nπ/2

from which we can solve for one of the variables, a or b.

Step-by-Step Solution

Start with the given equation:

sin(2ab) 1

Since sin(2ab) 1, 2ab must be an odd multiple of π/2. Therefore, we can write:

2ab (2n - 1)π/2

for all integers n.

From the above equation, we can express b in terms of a:

b [(2n - 1)π/2 - 2a] / 2a

Alternatively, for a given value of n, we can express a in terms of b:

a [(2n - 1)π/2 - 2b] / 2

It is important to note that the solutions are periodic, meaning that every time the argument of the sine function increases by π, the function repeats its values. Therefore, the solutions also repeat every π. Since the sine function is periodic with a period of π, we can generalize the above solution as:

2ab nπ/2 for all integers n.

Graphical Interpretation

To visualize the solutions, consider the equation 2ab π/2. This describes a line in the ab-plane given by:

b (π/2 - 2a) / 2a

or more simply:

b (π/2 - 2a) / 2a (π/4 - a) / a

This line represents the set of points (a, b) that satisfy the equation 2ab π/2.

Conclusion

In conclusion, to solve the equation sin(2ab) 1, we need to recognize that the argument of the sine function, 2ab, must be an odd multiple of π/2. This leads to a general solution involving periodicity and a line in the ab-plane. Understanding these principles allows you to find the set of (a, b) that make the equation true.