Is it possible to rearrange this equation for the variable angle c in terms of...

cooloicons62

cooloicons62

Answered

2022-07-06

Is it possible to rearrange this equation for the variable angle c in terms of the variable angles a and b?
sin ( π a b c ) sin ( a ) = sin ( c ) sin ( b )
(a and b variables in the range 0 < a , b < π and c is in the range 0 < c < π 6 )

Answer & Explanation

Monserrat Cole

Monserrat Cole

Expert

2022-07-07Added 12 answers

Notice that sin ( π x ) = sin ( x ), which allows us to simplify a bit. We then have the sum of angles formula, which states that sin ( x + y ) = sin ( x ) cos ( y ) + cos ( x ) sin ( y ). This simplifies things down a bit:
sin ( π a b c ) = sin ( a + b + c ) = sin ( a + b ) cos ( c ) + cos ( a + b ) sin ( c )
Now we divide both sides by sin ( a ) sin ( c ) to get
sin ( a + b ) cot ( c ) + cos ( a + b ) = sin ( a ) sin ( b )
Subtract cos ( a + b ) from both sides:
sin ( a + b ) cot ( c ) = sin ( a ) sin ( b ) cos ( a + b )
And divide by sin ( a + b ).
cot ( c ) = 1 sin ( a + b ) ( sin ( a ) sin ( b ) cos ( a + b ) )
And then take inverse cotangent to find c.

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