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February 26th, 2013, 06:00 AM   #1
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Trig identities

So I have this question fairly simple, I would say, but my issue is I can't actual visualise what it is asking so to speak; I will explain. Question: Given that x is an acute, find tan x if .

So I first went back to the unit circle to try and give a better understanding. From this I get the 2 identities and . So this is where I am sort of winging it so to speak. Because I have a in the equation my tan would look like the , I then sub the identities in and solve; have checked the answer in back of the book, which gives the correct answer, my concern is do I have the right method for working the question out, or have I gone wrong and by some luck got the right answer?

Big thanks in advance.
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February 26th, 2013, 06:44 AM   #2
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Re: Trig identities

Using your reasoning, you should have that cos^2 = 1 - 3/4 = 1/4.

So tan^2 = sin^2/cos^2 = (3/4)/(1/4) = 3.

Tan(x) is therefore the square root of 3. Here is where the assumption that x is acute comes in.

Of course, knowing that x is acute is enough to conclude that sin(x) = sqrt(3)/2, whence tan(x) = sqrt(3) immediately.
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February 26th, 2013, 06:54 AM   #3
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If x is acute, tan(x) is positive.
tan(x) = sin(x)/cos(x) = sin(x)/(1 - sin(x)) = (3/4)/(1 - (3/4)) = 3, so tan(x) = ?3.

Alternatively, cot(x) = csc(x) - 1 = 1/sin(x) - 1 = 4/3 - 1 = 1/3, so cot(x) = 1/?3. Hence tan(x) = ?3.

For a third method, x must be the angle in a right-angled triangle with leg opposite angle x of length ?3, and with hypotenuse of length 2. By Pythagoras, the leg adjacent to the angle x has length 1. Hence tan(x) = ?3.
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February 26th, 2013, 09:05 AM   #4
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Re: Trig identities

Hello, taylor_1989_2012!

Quote:





Code:
                  *
                * *
              *   *  _
          2 *     * /3
          *       *
        * x       *
      *  *  *  *  *
           adj





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March 4th, 2013, 03:05 AM   #5
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Re: Trig identities

Thanks for all the replies, once again big help, and thanks again.
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