Final answer:
There are no real solutions to the equation √3 sin(20) = 2 /√3 because the sine of an angle cannot exceed the range of [-1, 1].
Step-by-step explanation:
The equation to solve is √3 sin(20) = 2 /√3. To find the solutions, we must first isolate sin(20).
- Multiply both sides by √3 to get: sin(20) = √3 * (2/√3) = 2.
- Since the sine function has a range of [-1, 1], sin(20) cannot equal 2, which means there is no solution to this equation in the real number system.
If we were working within a system that allows for complex numbers, we could explore solutions where the sine value exceeds the standard range of the sine function, but this requires knowledge beyond standard high school trigonometry.
To list six solutions or provide a general formula for all the solutions, we would typically use the fact that sine is periodic with a period of 2π, but since there is no real solution, this is not applicable here.
It is essential to remember to check your answer and see if it is reasonable, especially when working with trigonometric functions where the range of possible values can limit the solutions available. In this case, the requested equality is outside the range of the sine function, confirming why there are no real solutions.