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 &lt; a , b &lt; π and c is in the range 0 &lt; c &lt; π 6 )

Question

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  &amp;lt;  a  ,  b  &amp;lt;  π and c is in the range   0  &amp;lt;  c  &amp;lt;      π    6  )

Monserrat Cole · Accepted Answer

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.