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Math Help - Trigonometric Equations Help Please!

  1. #1
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    Trigonometric Equations Help Please!

    Please can you tell me how to solve the following equations for values of x between (0.2 π) {π is pie}

    a) tan (4x) = 1

    b) sin(x-2π) + sin (x - 2π) =

    c) 2 cos (x + ) = -1

    d) sin x + cos x = 3/2
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  2. #2
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    Well when you want to isolate a variable that's inside a trig function, you just take the inverse trig function.

    For instance, with the first problem:

    \tan(4x)=1.

    We take \tan^{-1} of both sides (sometimes, \tan^{-1} is written and pronounced \arctan). Get:

    \tan^{-1}(\tan(4x)) = \tan^{-1}(1), so resolving the left side gives:

    4x = \tan^{-1}(1).

    We now have to figure out what \tan^{-1}(1) is. Well it's like asking "what angle in a right triangle gives a ratio of opposite side over adjacent side equal to 1?" It should be obvious that a 45-45-90 triangle passes this test, so \tan^{-1}(1) resolves to 45^o, or \frac{\pi}{4}. However, there may be more answers. Notice that \frac{5\pi}{4} works as well. In fact, for any integer n, \tan\left(n\pi + \frac{\pi}{4}\right) = 1, so \tan^{-1}(1) could be any of these.

    We take this info and go back to our equation:

    4x = \tan^{-1}(1). So:

    4x = n\pi + \frac{\pi}{4}. Then:

    x = \frac{1}{4}\left(n\pi + \frac{\pi}{4}\right).

    All right, but you were given a restriction of possible values, 0 < x < 2\pi.

    So we want to figure out the only values of n that make our solution valid.  0 <  \frac{1}{4}\left(n\pi + \frac{\pi}{4}\right) < 2\pi,
     0 < n\pi + \frac{\pi}{4} < 8\pi,

     -\frac{\pi}{4} < n\pi < 8\pi - \frac{\pi}{4},

     -\frac{1}{4} < n < 8 - \frac{1}{4}.

    So n could only be 0,1,2,3,4,5,6, or 7.
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  3. #3
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    Thanks for your help. I will try to work out the others.
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  4. #4
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    How do we known that 45 degrees is pi over 4?
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  5. #5
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    Talking

    Quote Originally Posted by bigfoot View Post
    How do we known that 45 degrees is pi over 4?
    How does ``2\pi" relate to ``180^{\circ}"?

    Dividing by 8, how then does ``\frac{\pi}{4}" relate to ``45^{\circ}"?

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