Consider the trigonometric equation sin3x - 2 = 0 . Use transformations to explain why there are no solutions.

Question

Consider the trigonometric equation sin3x - 2 = 0 . Use transformations to explain why there are no solutions.

Answer 1

-1 ≤ sinx ≤ 1

 

Horizontally compress the graph of y = sinx  to obtain the graph of       y = sin(3x).  Horizontal compression does not change the range of the function, so -1 ≤ sin(3x) ≤ 1.  

 

Translate the graph of y = sin(3x) two units downward to obtain the graph of y = sin(3x) - 2.

 

So, -1 - 2 ≤ sin(3x) - 2 ≤ 1- 2

 

         -3 ≤ sin(3x) - 2 ≤ -1

 

Since y = sin(3x) - 2 has a maximum value of -1, it is not possible for sin(3x) - 2 to equal 0.  

Answer 2

This says that sin3x = 2, but the value of sin () can never be larger than 1. Therefore there is no solution.