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 June 14th, 2010, 04:45 PM #1 Newbie   Joined: Jun 2010 Posts: 14 Thanks: 0 Trigonometry, finding angle when only one side length known Hi everyone, I run into following problem: Figure illustrates Triangles ABC and BDC where C is a right angle, AC=17cm and DC=6cm. Given angle BDC = 2x and angle BAD = x, find x. I cannot find solution, I have tried Pythagorean theorem cot^2(x)+1=cosec^2(x) and I ended up with 289/BC + BC = BC^4+34BC^2+289 where I have no clue how to continue that. I also tried BC = 17tan(x) = 6tan2(x) and again didn't find solution. Can you please help me?
 June 14th, 2010, 05:15 PM #2 Senior Member     Joined: Feb 2010 Posts: 706 Thanks: 141 Re: Trigonometry, finding angle when only one side length known The answer is x = 28.47213442457299779465217033046743894005698630301 79190093760352286477414715453273837029511364458721 6528734401286968... Would it help to know that AD = DB = 11 ?
 June 14th, 2010, 05:36 PM #3 Newbie   Joined: Jun 2010 Posts: 14 Thanks: 0 Re: Trigonometry, finding angle when only one side length known it certainly does help, but I still don't understand why is that (AD=DB)?
 June 14th, 2010, 06:36 PM #4 Global Moderator   Joined: Dec 2006 Posts: 20,753 Thanks: 2136 Angle ABD = 2x - x = x.
June 15th, 2010, 06:40 AM   #5
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Re: Trigonometry, finding angle when only one side length known

Hello, lesterxburnham!

Quote:
 Figure illustrates Triangles ABC and BDC where C is a right angle. AC = 17cm and DC = 6cm. Given angle BDC = 2x and angle BAD = x. Find x.

$\text{We have: }\,\angle BDC \,=\,2x\;\;\Rightarrow\;\;\angle BDA \,=\,180\,-\,2x$

$\text{Then: }\,\angle ABD \:=\:180^o\,-\,x\,-\,(180^o\,-\,2x} \:=\:x$

$\text{Hence, }\Delta ABD\text{ is isosceles: }\:AD \,=\,BD\,=\,11$

$\text{In right triangle }BCD:\;\cos\,2x \:=\:\frac{6}{11}$

[color=beige]. . [/color]$\text{Hence: }\:2x \:=\:\cos^{-1}\left(\frac{6}{11}\right) \;=\;56.9442685^o$

$\text{Therefore: }\:x \;\approx\;28.5^o$

 June 15th, 2010, 09:31 AM #6 Newbie   Joined: Jun 2010 Posts: 14 Thanks: 0 Re: Trigonometry, finding angle when only one side length known thank you so much for explanation, soroban and thank you all for help!

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