How to divide sin?

  1. 1. The problem statement, all variables and given/known data
    Its a word problem but I will just state everything here in simple form.

    Given
    ------------
    V initial = 100ft/s
    r=300ft
    r= 1/32 V^2 Sin 2theta

    Unknown
    ---------
    Solve for theta

    3. The attempt at a solution
    300=1/32 (100)^2 sin 2theta
    300= 1/32 (10000) sin 2theta
    300= 312.5 sin 2theta
    300/312.5= sin 2theta
    .96 = sin 2theta

    I know the next step would be to divide by sin but how???
     
  2. jcsd
  3. Mark44

    Staff: Mentor

    "sin" is not a number, and is not multiplying 2theta. Sine is a function, so your line at the end is similar to .96 = f(2x).

    If you have a variable that is the argument to a function, what can you do to get at the function's argument?
     
  4. You have 2θ there. That is really like saying Sin(θ+θ) isn't it?
     
  5. tiny-tim

    tiny-tim 26,054
    Science Advisor
    Homework Helper

    (have a theta: θ and try using the X2 tag just above the Reply box :wink:)
    nooo … the next step is to use sine tables (or the arcsin button on your calculator) :smile:
     
  6. hmm..

    f(2x)=.96
    2x=.96
    x=.48??? -.-grrr

    I'm having a little trouble =/ I can't seem to get the inverse ..

    What I did before was I asked myself:
    sin of what? = .96 <-- plugged it in calculator as sin of inverse =/
    2θ = 1.287
    θ = .6435

    But I want to figure our whats going on in between.
    Thanks but I want to really understand the steps in between and not just get the answer through the calculator.

    And is there a button here for theta?
     
  7. Mark44

    Staff: Mentor

    f is a function. What happened to it? You can't just ignore it.
     
  8. rock.freak667

    rock.freak667 6,221
    Homework Helper

    If you have sinx = A, then x = sin-1(A).
     
  9. I know but I didn't know what to do, can you help me?

    I'm used to things such as:
    f(x) = 2x+4
    y = 2x+4
    y/2 -2 = x
    f-1x = 1/2y - 2

    but when it comes to

    f(2x) = .96 then I'm confused on what to do.
     
  10. Isn't sin2[tex]\theta[/tex] a trigonometric Identity that becomes 2 sin [tex]\theta[/tex] cos[tex]\theta[/tex]?
     
  11. Mark44

    Staff: Mentor

    In your problem, the function is the sine function, and your equation is:
    sin(2x) = .96

    (I'm using x instead of theta.)
    The thing to do is to apply the inverse sin function to both sides.

    sin-1(sin(2x)) = sin-1(.96)

    ==> 2x = sin-1(.96)
    ==> x = (1/2)sin-1(.96)

    This will give you one value for x, but it might be that your problem calls for other solutions. If so, you will need to use some of the ideas from trig to get the other solutions.
     
  12. Mark44

    Staff: Mentor

    But in this problem, that's the wrong way to go. Making this replacement turns the problem into [itex]2 sin \theta cos\theta = .96[/itex]

    and this doesn't get you closer to a solution. Using inverse functions does.
     
  13. Thanks, but please bare with me, my goal is to understand everything in mathematics as much as I can. I hope I am not being a bother but I am not satisfied with knowing that I should simply multiply by the inverse on both sides.

    My question is, how does sin-1 cancel with sin when you multiply them together? It doesn't sound mathematically logical? I need the steps in between. :blushing:
     
  14. Mark44

    Staff: Mentor

    No, you're not being a bother. If you don't understand something it's better to keep asking questions until things are clear.

    One thing that you need to understand here is that the operation is NOT multiplication. This is something you have not been clear on since your first post in this thread.

    sin(2x) is sometimes written as sin 2x. In either form it is NOT sin times 2x - it's sin of 2x. Similar to what I wrote earlier - f(2x) is not f times 2x. It's f OF 2x, where f was the name of some unspecified function.
    They're not being multiplied. What is happening is that I am forming a composite function. Whenever you have a function that has an inverse, applying them together in either order gives you the identity function, the function that leaves its argument completely unchanged.

    For example, if f(x) = 2x + 3, then the inverse of this function is f-1(x) = (x - 3)/2.

    f(f-1(2) = 2 and f-1(f(0)) = 0.
    You can verify these statements by using the formulas for the functions.

    In a similar way sin(sin-1(x) = x, but there are some restrictions of the values of x that are allowed. Also, in the opposite order, sin-1(sin(x)) = x, and there are some restrictions here, as well. In the first equation, x has to be between -1 and +1, inclusive. In the second equation, x has to be between -pi/2 and +pi/2 if you're working in radians, or between -90 deg and +90 deg, if you're working in degrees.
     
  15. Yes, sin-1(sin(theta) ) = theta
     
  16. eumyang

    eumyang 1,347
    Homework Helper

    That's not always true. Did you not read Mark44's post above?

    For instance, what is
    [tex]\sin^{-1} (\sin 3\pi/4)[/tex]
    ? Hint: it's not 3π/4.
     
  17. Hurkyl

    Hurkyl 16,090
    Staff Emeritus
    Science Advisor
    Gold Member

    Noooooooo! :frown: You have to apply methods for solving trigonometric equations! Pretending it's an ordinary invertible operation you can simply "undo" is incorrect!
     
  18. Hurkyl

    Hurkyl 16,090
    Staff Emeritus
    Science Advisor
    Gold Member

    Do they still teach English students how to diagram sentences? If you've seen that, there's a similar thing for mathematical expressions.

    The diagram (which can called a "parse tree") for the expression 2x+1 would be:
    Code (Text):

         +
        / \
       /   \
      *     1
     / \
    2   x
     
    Note this reflects how it's computed -- if you substitute x=4 and computed, you would
    • Replace x with 4
    • Compute 2*4 and replace that little part of the tree with the result:
      Code (Text):

           +
          / \
         /   \
        8     1
       
    • Compute 8+1, and replace the tree with 9


    Functions, such as sin are like + and * and other operations. The diagram for the expression
    sin 2x​
    looks like
    Code (Text):
     sin
      |
      *
     / \
    2   x
    And for
    arcsin sin 2x​
    it would be
    Code (Text):
    arcsin
      |
     sin
      |
      *
     / \
    2   x
    I hope this helps you understand how to read mathematical expressions.
     
  19. Oops... forgot about the range of theta this IS true for, as noted by Mark44.
     
  20. Mark44

    Staff: Mentor

    AFAIK, they don't, and more's the pity. It's been out of the curriculum in the U.S. for so long, that I would guess that most high school English teachers here have never been exposed to this concept.
     
  21. Really? They still do it in our school.:bugeye:
     
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