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Trigonometry confusion

If I have something like 5sinxcosx=3cosx, dividing by cosx would eliminate a pair of solutions. But in what situations is it fine for me to divide by a trig function without getting rid of a solution?
you will always lose part of your solution :redface:

the only dividing should be by numbers.
We can't divide by 0. We can divide by everything else. So you can only divide by something if it's nonzero. If it could be zero, then you can divide by it (mentioning 'if nonzero') but you must consider the zero case as well.

So in this case you note that cosx=0 is a solution, and solve that, and then you can divide by cosx in the original eqn to get the solutions for which cosx is nonzero.
Alternatively you can bring everything to one side and factorise, but it's good to know why/when you can divide by variables.
(edited 6 years ago)
Reply 3
Avatar for MR1999
MR1999
21
Original post by cotkhd
If I have something like 5sinxcosx=3cosx, dividing by cosx would eliminate a pair of solutions. But in what situations is it fine for me to divide by a trig function without getting rid of a solution?


It's never okay to remove any variable from an equation by division because that variable could be equal to zero.

In fact, if you have the same variable on both sides of your equation, then that variable will always be considered equal to zero, so we can never cancel variables using division, only multiplicative constants.

What you must always do is bring the equation to standard form and then factorise. Never think about cancelling any common factors.
(edited 6 years ago)
Reply 4
Avatar for S.G.
S.G.
22
Original post by cotkhd
If I have something like 5sinxcosx=3cosx, dividing by cosx would eliminate a pair of solutions. But in what situations is it fine for me to divide by a trig function without getting rid of a solution?

Instead of dividing, factorise.

cosx(5sinx-3) = 0

No solutions are lost.
Reply 5
Avatar for cotkhd
cotkhd
OP
10
Original post by Desmos
It's never okay to remove any variable from an equation by division because that variable could be equal to zero.

In fact, if you have the same variable on both sides of your equation, then that variable will always be considered equal to zero, so we can never cancel variables using division, only multiplicative constants.

What you must always do is bring the equation to standard form and then factorise. Never think about cancelling any common factors.


So there wouldn't be any scenarios in which I divide by a trig func?
Reply 6
Avatar for MR1999
MR1999
21
Original post by cotkhd
So there wouldn't be any scenarios in which I divide by a trig func?


There would never be a scenario where you should divide by any variable, except if the question explicitly or implicitly states that said variable ≠ 0.

Of course, you could do what @IrrationalRoot said, but I would not recommend doing that ever, especially seeing as you're new to this concept.
(edited 6 years ago)
Reply 7
Avatar for S.G.
S.G.
22
Original post by SGHD26716
Instead of dividing, factorise.

cosx(5sinx-3) = 0

No solutions are lost.


Original post by cotkhd
So there wouldn't be any scenarios in which I divide by a trig func?


Just factorise mate. See above
Original post by cotkhd
If I have something like 5sinxcosx=3cosx, dividing by cosx would eliminate a pair of solutions. But in what situations is it fine for me to divide by a trig function without getting rid of a solution?


You can divide by a trig function only when you know that it cannot be 0 alongside the solutions to the equation at hand.

For example, in sin⁡(x)=cos⁡(x)\sin(x)=\cos(x) you can divide both sides by cosine to get tan because cos⁡(x)=0\cos(x)=0 only for x=π2+kπx=\frac{\pi}{2}+k\pi for integers kk, however none of these values of xx will give sin⁡(x)\sin(x) as 0. They will all rather give the LHS to be 1 for even k and -1 for odd k. So we clearly do not have 0=00=0 therefore cos⁡(x)=0\cos(x)=0 is NOT a solution to this equation, therefore you CAN divide by it.

In your example however, cos⁡(x)=0\cos(x)=0 is indeed a solution so you cannot divide by cosine here.
(edited 6 years ago)

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