MHB Show Cos A = 1/12 in Square ABCD

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To demonstrate that cos a = 1/12 in square ABCD, the law of cosines is suggested as a key tool. The approach involves applying the law of cosines twice: first to triangle AEC to determine side EC, and then to triangle CED to calculate cos a. This method leverages the relationships between supplementary angles to facilitate the calculations. The discussion indicates that this two-step application is essential for reaching the desired result. Ultimately, the law of cosines provides a structured way to solve for cos a in this geometric context.
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View attachment 467In square ABCD need to show $cos a = \frac{1}{12}$
 

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While I haven't worked this problem yet, I am thinking the law of cosines may help, along with the relationship between supplementary angles.
 
I gave it some thought, and I believe applying law of cosines twice will do the trick. First you apply to the triangle $AEC$ to find side $EC$, and then apply it to the triangle $CED$ to find $\cos a$.
 

Thread 'Trigonometry problem of interest'

Seemingly by some mathematical coincidence, a hexagon of sides 2,2,7,7, 11, and 11 can be inscribed in a circle of radius 7. The other day I saw a math problem on line, which they said came from a Polish Olympiad, where you compute the length x of the 3rd side which is the same as the radius, so that the sides of length 2,x, and 11 are inscribed on the arc of a semi-circle. The law of cosines applied twice gives the answer for x of exactly 7, but the arithmetic is so complex that the...
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