AcosX + BcosX?

Question

whay cant i just solve such and equation ex-

cosx + 2sinx = 0 take 2sinx to the left and divide by cos and make it 2tanx and solve it??

why do we have to reduce such a term to Rcos(x-y)

i know how do do the sum but my question is why do it like this, i knw the solutions we get are different if i use the tan method which is wrong!!, but why is it wrong??? and why do i have to write it in the other from

thanks, just wanna clear the doubt in my head

Answer 1

there are 2 aspects

A cos x + B sin x = C

if c = 0 then you can devide by cos x or sin x and solve the equation.

but if c not equal to zero then u cannot solve in this way and you have to reduce to form R(cos (x-y)) or R sin (x-y).

if you use the tan method the value cannot be wrong

Answer 2

in general you might be wanting to find out more than just where the zeros are

if you were trying to find cosx+2sinx = 1 then simply dividing by cos doesn't get you far : 1 + 2tanx = secx and also means that you have to deal separately with special cases where cosx = 0.

Answer 3

I'm not sure I understand you. Who tells you that you have to reduce to cos(x-y) ? I am not even sure this helps here? cos(x-y) is equal to (cosXcosY + sinXsinY) and in your equation you have no product?
You might have some doubt dividing by cosX but in this case cosX = 0 is not a solution, so go for it! :)

Answer 4

If
cosx + 2sinx = 0
2sinx = - cosx
tanx = -1/2
x = 153.43°,333.43°