1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

B Smallest number whose sin(x) in radian and degrees is equal?

  1. Jul 2, 2016 #1
    Question: what is the smallest positive real number x with the property that the sine of x degrees is equal to the sine of x radian?

    My try: 0.

    But zero isn't a positive number. How do I even begin to solve it? I tried taking inverse on both sides of sin(theta)=sin(x), but that didn't help.
     
  2. jcsd
  3. Jul 2, 2016 #2

    mfb

    User Avatar
    2016 Award

    Staff: Mentor

    It works, but first you have to figure out where approximately your solution is (did you make a sketch?), and express theta in terms of x or vice versa.
     
  4. Jul 2, 2016 #3

    fresh_42

    Staff: Mentor

    Do you know how to transform degrees in radiant and vice versa?
     
  5. Jul 2, 2016 #4
    Yeah.
     
  6. Jul 2, 2016 #5

    fresh_42

    Staff: Mentor

    Then write it down. Put a sin before both, recognize that the sin could be canceled by a arcsin and your answer should pop up.
     
  7. Jul 2, 2016 #6

    Charles Link

    User Avatar
    Homework Helper

    What you are asking for is a non-zero solution ## \sin(x)=\sin(x \pi/180) ## where the ## \sin ## function in both cases now acts on radians. Suggest you graph y vs. x for both and see where they intersect.
     
  8. Jul 2, 2016 #7
    Got it, thanks! :)
     
  9. Jul 2, 2016 #8

    Charles Link

    User Avatar
    Homework Helper

    Because the first function is cycling at a rate of almost 60 times the second function, the first nonzero intersection is going to occur near (just before) ## x=\pi ##. The reason is the second function is going up very slowly and the first one has made it back down to zero already. If you let ## x=\pi-\Delta ##, you can use trig identities and Taylor series. (## \theta=(\pi-\Delta)\pi/180 ## is small so that e.g. ## \sin(\theta)=\theta ## approximately.) This gives ## \sin(\pi-\Delta)= \sin(\Delta)=\Delta=(\pi- \Delta )\pi /180 ## which gives ## \Delta=\pi^2/180 ## (approximately) so that ## x=\pi- \pi^2/180 ## is a good approximate solution for ## x ##. Further refinement could be done with higher order terms or by plugging into a calculator near this point.
     
    Last edited: Jul 2, 2016
  10. Jul 3, 2016 #9

    Svein

    User Avatar
    Science Advisor

    To spell it out: You seek the values of x, where [itex] \sin(x)=\sin(\frac{x\cdot \pi}{180}), x>0[/itex]. One set of solutions is, of course [itex]x=\frac{x\cdot\pi}{180}+2n\pi [/itex], the least solution given by n=1: [itex] x=\frac{360\cdot\pi}{180-\pi}[/itex].

    Another set of solutions is given by [itex] x=\pi-\frac{x\cdot\pi}{180}+2n\pi[/itex]. Here the smallest solution is given by n=0: [itex]x=\pi-\frac{x\cdot\pi}{180} [/itex], or [itex]x=\frac{180\cdot\pi}{180+\pi} [/itex]. This solution is the smallest value for x.
     
  11. Jul 3, 2016 #10

    mfb

    User Avatar
    2016 Award

    Staff: Mentor

    Why approximating better than "a bit below pi", if the analytic solution can be found by solving a linear equation?
     
  12. Jul 3, 2016 #11

    Charles Link

    User Avatar
    Homework Helper

    Svein's solution(s) are rather clever=I totally overlooked them. In reading them, I wasn't surprised that his first one doesn't pick up the smallest value for x, but rather picks up the intersection near ## x=2 \pi ##. His second one gets the exact answer by solving a linear equation, but without knowing that the first one did not get the answer near ## x=\pi ##, that set of solutions could easily be overlooked. I did not expect an exact answer to this one. Very good solution by @Svein. ...editing... In post #8, had I kept the next term when I solved for ## \Delta ##, I would have had the exact solution (given by Svein) without even knowing that it was the exact answer...
     
    Last edited: Jul 3, 2016
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook

Have something to add?
Draft saved Draft deleted



Similar Discussions: Smallest number whose sin(x) in radian and degrees is equal?
  1. Radians or Degrees (Replies: 14)

Loading...