February 26th, 2013, 05:00 AM  #1 
Senior Member Joined: Sep 2012 Posts: 201 Thanks: 1  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. 
February 26th, 2013, 05:44 AM  #2 
Global Moderator Joined: Nov 2009 From: Northwest Arkansas Posts: 2,766 Thanks: 4  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. 
February 26th, 2013, 05:54 AM  #3 
Global Moderator Joined: Dec 2006 Posts: 19,547 Thanks: 1752 
If x is acute, tan(x) is positive. tan²(x) = sin²(x)/cos²(x) = sin²(x)/(1  sin²(x)) = (3/4)/(1  (3/4)) = 3, so tan(x) = ?3. Alternatively, cot²(x) = csc²(x)  1 = 1/sin²(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 rightangled 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. 
February 26th, 2013, 08:05 AM  #4  
Math Team Joined: Dec 2006 From: Lexington, MA Posts: 3,267 Thanks: 407  Re: Trig identities Hello, taylor_1989_2012! Quote: Code: * * * * * _ 2 * * /3 * * * x * * * * * * adj  
March 4th, 2013, 02:05 AM  #5 
Senior Member Joined: Sep 2012 Posts: 201 Thanks: 1  Re: Trig identities
Thanks for all the replies, once again big help, and thanks again.


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