MHB What Is the Minimum Cosine Value in Trigonometric Constraints?

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The discussion revolves around finding the minimum value of cos a given the constraints that the sum of cosines and sines of three angles equals one. The derived solution for the smallest possible value of cos a is -1/4(1 + √7). Participants confirm this result, indicating agreement on the calculation. The focus remains on the mathematical derivation and verification of the answer. The problem highlights the interplay between trigonometric identities and constraints.
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Here is this week's POTW:

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Let $a,\,b$ and $c$ be three real numbers such that $\cos a+\cos b+\cos c=1$ and $\sin a+\sin b+\sin c=1$.

Find the smallest possible value of $\cos a$.

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I get ##-\frac 14(1+\sqrt 7)##.
Hint:

Combine the two constraint equations into one complex one. Visualise the minimisation of cos a in the complex plane and consider what that means for the other two points.


Write ##\alpha=\cos(a)+i\sin(a)## etc.
##\alpha+\beta+\gamma=1+i##.
Minimising cos a means getting ##\alpha## as close as possible to ##-1##, which means getting ##\beta+\gamma## as close as possible to 2+i.
Since ##|\beta|\leq 1##, ##|\gamma|\leq 1## and ##|2+i|>2##, that implies ##\beta=\gamma##.
The rest is algebra.
 
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I got the same answer.


I looked at it as the vector equation ##\vec A + \vec B + \vec C = \vec R## where ##\vec A##, ##\vec B##, and ##\vec C## are unit vectors with directions ##a##, ##b##, and ##c## respectively, and ##\vec R = \hat i + \hat j##. When ##\vec A## points as far counterclockwise from the ##x##-axis as possible, ##\vec B## and ##\vec C## have to be parallel, so we end up with a triangle where we know the length of all three sides. Then the angle ##a## can be found using the law of cosines.
[\ispoiler]
 
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