SOLUTION: If cos α / cos β = a, sin α / sin β = b, then sinēβ is equal to

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Question 1040397: If cos α / cos β = a, sin α / sin β = b, then sinēβ is equal to
Answer by ikleyn(52810) About Me  (Show Source):
You can put this solution on YOUR website!
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If cos(alpha)/cos(beta) = a, sin(alpha)/sin(beta) = b, then sinē(beta) is equal to
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From   sin%28alpha%29%2Fsin%28beta%29 = b  you have  sin%28beta%29 = sin%28alpha%29%2Fb  and then  sin%5E2%28beta%29 = sin%5E2%28alpha%29%2Fb%5E2.     (1)

Having this, I would say the problem is just solved, because I just expressed  sin%5E2%28beta%29. 

But I understand that you want me to express  sin%5E2%28beta%29  via  "a"  and "b"  and exclude  alpha,  although 
it is not stated directly and implicitly in the problem formulation (which is the author's fault).

So I will continue.


From the condition, we have this line of identities 

b%5E2 = sin%5E2%28alpha%29%2Fsin%5E2%28beta%29  --->  sin%5E2%28alpha%29 = b%5E2%2Asin%5E2%28beta%29                                        (2)

and this line 

a%5E2 = cos%5E2%28alpha%29%2Fcos%5E2%28beta%29 = %281-sin%5E2%28alpha%29%29%2F%281-sin%5E2%28beta%29%29  --->  1-sin%5E2%28alpha%29 = a%5E2%2A%281-sin%5E2%28beta%29%29  --->  

1-sin%5E2%28alpha%29 = a%5E2+-a%5E2%2Asin%5E2%28beta%29  --->  sin%5E2%28beta%29 = %28a%5E2+-+1+%2B+sin%5E2%28alpha%29%29%2Fa%5E2.                  (3)

Next substitute the last identity of (3) into the last identity of (2). You will get

sin%5E2%28alpha%29 = %28b%5E2%2Fa%5E2%29%2A%28a%5E2-1%2Bsin%5E2%28alpha%29%29  --->  %281-%28b%5E2%2Fa%5E2%29%29%2Asin%5E2%28alpha%29 = %28b%5E2%2Fa%5E2%29%2A%28a%5E2-1%29  --->  sin%5E2%28alpha%29 = %28b%5E2%2A%28a%5E2-1%29%29%2F%28a%5E2-b%5E2%29.     (4)

Now we can complete (1) by substituting the found value of sin%5E2%28alpha%29 from (4) into (1). You will get

sin%5E2%28beta%29 = %28a%5E2-1%29%2F%28a%5E2-b%5E2%29.

Answer.  sin%5E2%28beta%29 = %28a%5E2-1%29%2F%28a%5E2-b%5E2%29.