What Is the Minimum Cosine Value in Trigonometric Constraints?

[anemone]

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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[haruspex]

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.

 

[vela]

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]