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Trig help

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  1. Aug 7, 2016 #1
    1. The problem statement, all variables and given/known data
    Find all numbers x ∈ [0, 2π] satisfying tan x = cos x. Your answers should be expressed in radians, rounded to 4 decimal places. Show all your working.

    [You will need to use a scientific calculator that has buttons such as sin−1 or arcsin so as to be able to find the angles for which the sin function attains given values.

    3. The attempt at a solution

    I don't know where to start I have had a look at the unit circle but cant see anywhere where they are equal
     
  2. jcsd
  3. Aug 7, 2016 #2
    The hint they gives you suggests that you should be solving your equation for sin x. What is tan x in terms of sin x and cos x? If you substitute for tan x in your equation, what do you get?
     
  4. Aug 7, 2016 #3

    Charles Link

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    Suggestion: Express ## \tan{x}=\sin{x}/\cos{x} ##. Also ## \ ## ## \cos^2{x}=1-\sin^2{x} ##. Much simpler to work with the ## sin{x} ## than to graph ## y= \tan{x} ## and ## y= \cos{x} ##. A graphical solution might be a good check for the solution.
     
  5. Aug 7, 2016 #4
    tan x = sinx/cosx

    which means

    sinx/cosx=cosx

    sinx=cos^2x

    Is this what you mean?
     
  6. Aug 7, 2016 #5
    Yes. Now, what is cos^2 x in terms of sin^2x?
     
  7. Aug 7, 2016 #6
    that means

    sinx=1-sin^2x
     
  8. Aug 7, 2016 #7
    Good. Now solve this quadratic equation for sin x using the quadratic formula.
     
  9. Aug 7, 2016 #8

    SammyS

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    Gold Member

    It's clearer to write cos^2(x). Even better to use the Superscript feature, X2, to give cos2 x.
     
  10. Aug 7, 2016 #9
    does that mean

    sin x = (-1 +sqrt5)/2 or (1 +sqrt5)/2
     
  11. Aug 7, 2016 #10
    One of these roots is >1. As a decimal, what is the other root. What angles does your calculator say that this corresponds to on the interval between x = 0 and x = 2pi?
     
  12. Aug 7, 2016 #11
    0.618 which means arcsin 0.618 = 0.6662

    is this my answer?
     
  13. Aug 7, 2016 #12

    Charles Link

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    If I may offer a hint here=.666 radians is about 40 degrees. Is that the only place where sin(x)=.618?
     
  14. Aug 7, 2016 #13
    Not good enough. There is another angle on the interval that has this same value of sine.
     
  15. Aug 7, 2016 #14
    it would be in the second quadrant but how do i calculate it
     
  16. Aug 7, 2016 #15
    Using the sine of the difference between two angles formula, what is ##\sin(\pi -\theta)##?
     
    Last edited: Aug 7, 2016
  17. Aug 8, 2016 #16
    does this mean that the other answer would be 3.8078?
     
  18. Aug 8, 2016 #17

    Mark44

    Staff: Mentor

    No. It looks like you added your first answer to ##\pi##, rather than subtracting it from ##\pi##.
     
  19. Aug 8, 2016 #18
    so the answer would be 2.48 radians
     
  20. Aug 8, 2016 #19

    Mark44

    Staff: Mentor

    You can check both your answers by substituting them in your equation: ##\tan(x) = \cos(x)##. The left and right sides should be equal for those two numbers. Your calculator should be in radian mode, though.
     
  21. Aug 8, 2016 #20

    Mark44

    Staff: Mentor

    Since you have rounded your answers (or at least haven't written all the digits shown on your calculator), the left and right sides of the equation will only be close, not exactly the same.
     
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