in the interval of 0<θ<360 solve for θ , the equation (1) , I would just say tanθ=−1 (scernario 1) and solve that.
sinθ=−cosθ(1)
but what if i said sinθ+cosθ=0 then cosθ(tanθ+1)=0 (scenario 2) . I'd have solutions for cosθ=0 as well as tanθ=−1 , why is doing this method wrong since arithmetically going through the workings out it seems fine , but the solutions for cosθ=0 are wrong?
I'm trying to get a verdict on if the 2nd method is still right or wrong ? i have a feeling its wrong since the solutions dont work, but the process of me getting to cos(θ)⋅[tanθ+1]=0 seemed sound that I wouldnt realise it was wrong ?
How do I stop myself from carrying out this sort of working out ,is there a way for me to intuitively prove its wrong even before the equation ends up in the form of cos(θ)⋅[tanθ+1]=0 because if i carried out a similar working out for this question " Solve for theta ,sinθ=3sinθcosθ " and solving it similarly like so I'd end up with sinθ(1−3cosθ)=0 (Scenario 3) ,and it would give the right solutions.
So as you can see in all the cases whenever i divided by a trig function you're automatically assuming its not =0 , in some cases this will lose solutions as in the case of sinθ=3sinθcosθ where dividing through by sinθ would make you lose solutions to get 1=3cosθ , but in other case such as in scernario 1 where i divide by cosθ to get tanθ and scenario 2 where i divided by cosθ to get tanθ and +1, this wont make me lose solutions? in all the cases whenever i divided by a trig function you're automatically assuming its not =0
so how am i fundamentally supposed to know mid way through doing my working out if dividing through by a trig will make me lose solutions or not.......
the last two paragraphs were edits to the question .
