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Math Help - Evaluating an expression.

  1. #1
    Junior Member madmax29's Avatar
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    Evaluating an expression.

    i have;

    \frac{(sec^2x - tan x)}{e^x}, how do i calulate this to get 0.46??? if x = \pi/4

    using; \displaystyle \sec{x} = \frac{1}{\cos{x}};

    \frac{( \frac{(2)}{cos 2x +1} - tan x)}{e^x} whats next?
    Last edited by mr fantastic; November 12th 2010 at 05:01 PM. Reason: Moved from another thread.
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  2. #2
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    mr fantastic's Avatar
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    Quote Originally Posted by madmax29 View Post
    on a similar topic; i have;

    \frac{(sec^2x - tan x)}{e^x}, how do i calulate this to get 0.46??? if x = \pi/4

    using; \displaystyle \sec{x} = \frac{1}{\cos{x}};

    \frac{( \frac{(2)}{cos 2x +1} - tan x)}{e^x} whats next?
    Does the question ask you to find an answer correct to two decimal places? Then you substitute x = pi/4 into the expression and evaluate it using a calculator.

    (Note: The exact value of each trig function should be known to you but are not required if an approximate answer is asked for).

    I have no idea why you would be trying to re-write the expression.
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  3. #3
    Junior Member madmax29's Avatar
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    I can seem to get the answer the book requires, when i put the numbers in to the formula?

    I know that;
    \displaystyle \sec{x} = \frac{1}{\cos{x}}

    but what is sec^2(x)
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  4. #4
    Junior Member madmax29's Avatar
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    Quote Originally Posted by madmax29 View Post
    I can seem to get the answer the book requires, when i put the numbers in to the formula?

    I know that;
    \displaystyle \sec{x} = \frac{1}{\cos{x}}

    but what is sec^2(x)
    ITs ok, i think ive got it now!

    sec^2(x)= \displaystyle \frac{1}{\cos{x}^2}

    therefore does;

    sec^2(x^2)= \displaystyle \frac{2}{\cos{2(x^2)}+1} ???
    Last edited by madmax29; November 13th 2010 at 02:29 AM.
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  5. #5
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    No, it's nothing like that. since sec(x)= \frac{1}{cos(x)}, sec^2(x)= \left(\frac{1}{cos(x)}\right)^2= \frac{1}{cos^2(x)}. You square the entire function, not just "x".

    sec^2(x)- tan(x)= \frac{1}{cos^2(x)}- \frac{sin(x)}{cos(x)}= \frac{1- sin(x)cos(x)}{cos^2(x)}

    mr fantastic suggested that you put the numbers into a calculator. Is the problem that your calculator does not have a "sec" key? If so then sec(x)= \frac{1}{cos(x)} means that you can use the "cos" key followed by the "1/x" key:
    If x= \pi/4 then cos(x)= 0.70710678118654752440084436210485 and the reciprocal (1/x) of that is 1.4142135623730950488016887242097. Pressing the " x^2" key now gives 2. That should be no surprise. One of the "standard" values of cosine you should memorize is cos(\pi/4)= \frac{\sqrt{2}}{2}. Inverting that gives sec(\pi/4)= \frac{2}{\sqrt{2}}= \sqrt{2} and, squaring, 2.

    Of course, since \pi/4= \pi/2- \pi/4, sin(\pi/4)= cos(\pi/4)= \frac{\sqrt{2}}{2} also so that tan(\pi/4)= sin(\pi/4)/cos(\pi/4)= \left(\sqrt{2}/2}\right)/\left(\sqrt{2}/2\right)= 1. sec^2(\pi/4)- tan(\pi/4)= 2- 1= 1.

    Now, what is e^{\pi/4}? What is \frac{1}{e^{\pi/4}}?
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  6. #6
    Junior Member madmax29's Avatar
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    so from all of this we gather;

    1. sec(x)= \frac{1}{cos(x)}

    2. sec(x^2)= \frac{1}{cos(x^2)}

    3. sec^2(x)= \left(\frac{1}{cos(x)}\right)^2= \frac{1}{cos^2(x)}

    4. sec^2(x^2)= \left(\frac{1}{cos(x^2)}\right)^2= \frac{1}{cos^2(x^2)}

    Are these statements true?
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  7. #7
    Super Member TheChaz's Avatar
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    Quote Originally Posted by madmax29 View Post
    so from all of this we gather;

    1. sec(x)= \frac{1}{cos(x)}

    2. sec(x^2)= \frac{1}{cos(x^2)}

    3. sec^2(x)= \left(\frac{1}{cos(x)}\right)^2= \frac{1}{cos^2(x)}

    4. sec^2(x^2)= \left(\frac{1}{cos(x^2)}\right)^2= \frac{1}{cos^2(x^2)}

    Are these statements true?
    Yes, these statements are true.
    But #4 isn't related (as far as I can tell) to the original problem.
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  8. #8
    Junior Member madmax29's Avatar
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    Thanks,

    ...thats, ok, i need the last one for a similar question, but i didnt feel that i needed to post the whole thing up,

    Many thanks for your help
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