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# Best method of solving sin(ax)=sin(bx) ?

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1. Especially where a and b are non integers so you can't use identities.

You can't just inverse sine them since you either get the trivial x=0 or a contradiction.

Also the same for cos(ax)=cos(bx) and tan(ax)=tan(bx).
2. cos(ax)=cos(bx)

for this identity a and b would be equall
3. Subtract one of the terms from both sides.
You get sin(ax)-sin(bx)=0
Use identity sinA - sinB =2sin[(A-B)/2]cos[(a+b)/2]
In this case A=ax and B=bx
Then either sin[(a-b)x/2]=0 or cos[(a+b)x/2]=0
4. (Original post by Katiee224)
cos(ax)=cos(bx)

for this identity a and b would be equall
Are you sure?

Plotting cos(x/2) and cos(2x/3) shows a solution at 12pi/7 .
5. (Original post by Katiee224)
cos(ax)=cos(bx)

for this identity a and b would be equall
a =b, what about a=-b, a=b+2π.....
6. (Original post by Ano123)
Subtract one of the terms from both sides.
You get sin(ax)=-sinbx=0
Use identity sinA - sinB =2sin[(A-B)/2]cos[(a+b)/2]
In this case A=ax and B=bx
Then either sin[(a-b)x/2]=0 or cos[(a+b)x/2]=0
Oh thanks that's a great one, I almost never use these identities so it's easy to forget. Does one exist for tan?
7. (Original post by 16characterlimit)
Oh thanks that's a great one, I almost never use these identities so it's easy to forget. Does one exist for tan?

Convert tan to expression with sin and cos and then combine fractions.
8. (Original post by Ano123)
Convert tan to expression with sin and cos and then combine fractions.
Thanks.
9. It might be interesting if you try to find the general solution to the equation in terms of and b.
10. Not sure what the above answers are trying to achieve. The general solution to is or .

So substituting in and gets you what you need.
11. I am not great at basic mathematics, but I ave some understanding of the more advanced fields of maths.

If I understand the question correctly,

sin(ax)=sin(bx) tan(ax)=tan(bx) cos(ax)=cos(bx) the answer I think is

sin(ab)=sin(2x) tan(ab)=tan(2x) cos(ab)=cos(2x)

can someone tell me if I am right on this, as I would be well chuffed if I am.
12. (Original post by Zacken)
Not sure what the above answers are trying to achieve. The general solution to is or .

So substituting in and gets you what you need.
I agree that this is the neatest approach. However, you can also use a factor formula method as suggested earlier.

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Updated: June 11, 2016
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