SUMMARY
The correct general solutions for the equation sin(x)sin(y) = 1 are given by the equations x = (4n + 1)(π/2) and y = (4m + 1)(π/2), as well as x = (4n - 1)(π/2) and y = (4m - 1)(π/2), where n and m are integers. The first proposed solution, x = n(π) + (π/2)(-1)^n + 1 and y = m(π) + (π/2)(-1)^m + 1, is invalid because it allows for the possibility of sin(x) = -sin(y), resulting in sin(x)sin(y) = -1. The valid solutions ensure that sin(x) and sin(y) are both equal to 1 or -1, confirming the correctness of the second and third equations.
PREREQUISITES
- Understanding of trigonometric functions, specifically sine.
- Familiarity with integer sequences and their applications in equations.
- Knowledge of periodic functions and their properties.
- Basic algebraic manipulation skills for solving equations.
NEXT STEPS
- Study the properties of sine functions and their periodicity.
- Explore integer sequences and their role in trigonometric equations.
- Learn about the implications of negative values in trigonometric identities.
- Investigate other trigonometric equations and their general solutions.
USEFUL FOR
Students studying trigonometry, mathematicians solving equations involving sine functions, and educators teaching advanced algebra concepts.