I tried doing these and i got to imaginary numbers when i was trying to solve for x, can anyone help?
Sec^2(x)+tan(x)=3
Sin^2(x)+3Cos^2(x)=0
sec^2(x) + tan(x) = 3 => tan^2(x) + 1 + tan(x) = 3 => tan^2(x) + tan(x) - 2 = 0 => [tan(x) + 2][tan(x) - 1] = 0 => plenty of real solutions for x.
sin^2(x) + 3cos^2(x) = 0: Restriciting to real numbers, the sum of two squares is always greater than or equal to 0. But sin^2(x) and cos^2(x) cannot simultaneously equal 0 for any real value of x. Therefore .....
True. But the extra work is not necessary. And in my experience when students continue with additional unnecessary working they are prone to making errors they would otherwise not make .... Those errors in turn lead to wrong answers.
This would not happen if they stopped after completing the necessary working.
For example:
It would not surprise me if a student then said $\displaystyle \sin(x) = \pm \frac{\sqrt{3}}{2}$ (NB: THIS IS WRONG) or even just $\displaystyle \sin(x) = \frac{\sqrt{3}}{2}$ (NB: THIS IS WRONG) and then proceded to get real solutions .....
All comment welcome.