How Do I Solve Trigonometric Equations Using a Graphic Calculator?

[danne89]

Hi! I know, sort of, the algorithm of solving ei. sin x = sin 30 degrees. But I don't quite understand it. I would be very happy if someone could explain it for me, or redirect me to a site that does.

Danne

 

[HallsofIvy]

I have no idea what you mean. If you mean solving the equation sin(x)= sin(30 degrees) then the best I can come up with is not an "algorithm" but understanding:
certain x= 30 degrees is a solution (x= a is always a solution to f(x)= f(a) but not necessarily the only one.). Now think about the position on the unit circle. sin(θ) can be defined as the y coordinate of the point a distance θ around the unit circle from (1,0). But the horizontal line corresponding to that coordinate will, in general, cross the circle twice. It's pretty easy to see that if one is at 30 degrees, by symmetry, the other will be at 180-30= 150 degrees. Now use the fact that sine is periodic with period 360 degrees: sin(x)= sin(30) for x= 30+ 360n and x= 150+ 360n where n can be any integer.

I'm not at all comfortable solving equations like that in degrees rather than radians but you were the one that said "sin 30 degrees". I also have a feeling that you might be asking about FINDING sin 30 degrees. There is no one "algorithm" for that. You might go back to whichever of many equivalent definitions you are using for sin(x) or use Taylor's polynomial or the "Cordic" method which is what most calculators and computers use.

 

[danne89]

Sorry if I expressed me unclear. But I get pretty much what I wanted. Thank you.

 

[danne89]

Hmm. I've still problem with this. Look over this please.
sin(4x) = sin(6x)
4x=6x+360n
0=2x+360n
-2x=360n
2x=-360n
x=-180n
And
4x=180-6x+360n
10x=180+360n
x=18+36n

 

[Hurkyl]

Looks good to me. To be extra sure, you might try plotting the graph of sin 4x - sin 6x, and see if you have all of the first few roots.

 

[danne89]

Sure I would if I had a graphic calculator. *dreaming*