Solving for a variable inside a sin()

  1. sin( degtorad( (180 - (180 - 360/x))/2 ) ) = y/z

    degtorad(degrees) means the the degrees inside the parenthesis are converted to radians.

    How do you solve for x?

    Thank you.
     
  2. jcsd
  3. you can eliminate the 180's by using properties of sine
    sin (180+ x) = -sin x = sin(-x)

    also,
    sin(90+x) = cos(x)
     
  4. sin( degtorad( (180 - (180 - 360/x))/2 ) ) = y/z

    Is this right?

    sin( (180 - (180 - 360/x))/2 )
    = sin( ( (180 - 360/x))/2 )
    = sin( (180 - 360/x) /2 )

    sin( (180 - 360/x) /2 )
    = sin( ( 360/x) /2 )
    = sin( ( 180/x) )
    = sin( 180/x )

    sin( 180/x ) = y/z

    um, then what?
     
  5. sin( (180 - 360/x) /2 ) = sin (90 - 180/x) = cos(-180/x) = cos (180/x)

    Are you familiar with arccos (or cos^{-1})?

    By the way, what exactly are y and z?
     
  6. HallsofIvy

    HallsofIvy 40,391
    Staff Emeritus
    Science Advisor

    Shorn of all the other things, arcsin( ) (also written sin-1( )) is defined as the inverse of sin( ) and arccos() (also written cos-1( )) is defined as the inverse of cos( ) (well, principal value). That is, arcsin(sin(x))= x and arccos(cos(x))= x.

    You have to be a bit careful about that: since sin(x) and cos(x) are not "one-to-one" they don't have inverses, strictly speaking. Given an x between -1 and 1, there exist an infinite number of y such that sin(y)= x or cos(y)= x. Arcsin(x) always gives the value, y, between -pi/2 and pi/2 such that sin(y)= x and arccos(x) always gives the value, y, between 0 and pi such that cos(y)= x.
     
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