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Simple trigo (Posted on 2014-01-03) Difficulty: 2 of 5
Find the value of

(i)cos 36
(ii)tan 15

using geometry.

No Solution Yet Submitted by Danish Ahmed Khan    
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Solution No Subject | Comment 2 of 6 |

Draw a 36o-72o-72o triangle, with two long sides = 1, and the short side = x . 

Now, bisect one of the 72 angles forming two triangles:
a 36o-36o-108o isosceles triangle whose short sides are both x , and a smaller 36o-72o-72o isosceles triangle whose long  
sides are x , and short side is 1−x  

Since the two 36o-72o-72o triangles are proportional: <o:p></o:p>


solving      x2+x-1=0

and ignoring the negative answer

we get   x=(-1+sqrt(5))/2

a.k.a.     phi, the golden ratio<o:p></o:p>

Now bisect the obtuse angle in the 36o-36o-108o triangle to    get a right 
(36o-54o-90o) triangle, with a hypotenuse=(-1+sqrt(5))/2  and the long leg =1/2 .


cos 36 =1/(-1+sqrt(5))=(1+phi)/2  =.809 approximately.

  Posted by Ady TZIDON on 2014-01-03 15:46:14
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