xsinx = 1: how would you solve without graphing?

[dimon]

Hello

I've been trying to figure this problem out: xsinx = 1. I graphed f(x) = xsinx -1 and got the values of x by looking where y = 0. How would you solve for x without graphing?

 

[galactus]

Are you familiar with Newton's Method?. It would work well for this.

Try an initial estimate of 1.

\(\displaystyle x_{n+1}=x_{n}-\frac{f(x_{n})}{f'(x_{n})}\)

\(\displaystyle 1-\frac{(1)sin(1)-1}{(1)cos(1)+sin(1)}=1.1147286724\)

Now, use that result. Keep doing the iterations until it converges to your solution. It will converge very quickly for this particular problem.

 

[dimon]

I'm taking precalculus at this time and we have not covered derivatives. This problem is from my precalc book. Is there any way to solve it without using derivatives?

 

[galactus]

If I couldn't use Newton, I'd graph it. There are lots of solutions.

Your equation is also equal to \(\displaystyle csc(x)=x\)

 

[dimon]

yep, graphing was the only thing that I could think of.

thanks for your help.