# What is angle 0<A<2pi if sin7A=sin5A?

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`sina=sinb==>a=(-1)^nb+npi` :

`n=0` `7A=5A==>2A=0==>A=0` which is not in the domain.

`n=1` `7A=-5a+pi==>12A=pi==>A=pi/12`

`n=2` `7A=5A+2pi==>2A=2pi==>A=pi`

`n=3` `7A=-5A+3pi==>12A=3pi==>A=(3pi)/12=pi/4`

`n=4` `7A=5A+4pi==>2A=4pi==>A=2pi` not in domain.

You will notice for further even n, A will no longer be in the domain. For even n the list is `0,pi,2pi,3pi,` etc...

`n=5` `7A=-5A+5pi==>2A=5pi==>A=(5pi)/12`

`n=7` `7A=-5a+7pi==>12A=7pi==>A=(7pi)/12`

You will notice the pattern for odd n: `A=(npi)/12` . To remain in the domain n<25.

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**The complete solution:**

`A=(npi)/12` for odd `1<=n<=23` and `A=pi` for a total of 13 solutions.

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**Sources:**

You need to remember that if `sin a = sin b =gt a = (-1)^n*b + npi` , hence, comparing the given equation to the general form sin a = sin b yields:

`7A = (-1)^n*(5A) + npi`

If n is an even number => `7A = 5A + npi`

`7A- 5A = npi =gt 2A = npi =gt A = npi/2`

Since `A in (0,2pi) =gt A = pi/2`

If n is an odd number => `7A = -5A + npi + pi`

`12A = npi + pi`

If n`= 0 =gt A = pi/12`

If n `= 1 =gt 12A = 2pi =gt A = pi/6`

If n`= 2=gt 12A = 3pi =gt A = pi/4`

If n`= 3=gt 12A = 4pi =gt A = pi/3`

If n`= 4=gt 12A = 5pi =gt A = 5pi/12`

If n`= 5=gt 12A = 6pi =gt A = pi/2`

If n`= 6=gt 12A = 7pi =gt A = 7pi/12`

If n`= 7=gt 12A = 8pi=gt A = 2pi/3`

If `n= 8=gt 12A = 9pi =gt A = 3pi/4`

If n`= 9=gt 12A = 10pi =gt A = 5pi/6`

If n`= 10=gt 12A = 11pi =gt A = 11pi/12`

If n`= 11=gt 12A = 12pi =gt A = pi`

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If n`= 23=gt 12A = 24pi =gt A = 2pi`

**Hence, evaluating the possible solutions to the given equation yields `pi/6 ; pi/4 ; pi/3 ; pi/2 ;7pi/12 ; 2pi/3 ; 3pi/4 ; 5pi/6 ; 11pi/12 ; pi.... ; 2pi.` **