Solving for a variable inside a sin()

1. Apr 4, 2006

Crusty

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. Apr 4, 2006

Euclid

you can eliminate the 180's by using properties of sine
sin (180+ x) = -sin x = sin(-x)

also,
sin(90+x) = cos(x)

3. Apr 4, 2006

Crusty

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?

4. Apr 4, 2006

Euclid

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?

5. Apr 5, 2006

HallsofIvy

Staff Emeritus
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.