<s>[INST] If $a$ and $b$ are cathets a right triangle whose hypotenuse is $1$ determine the highest value of $2a + b$ Can some one help me out on where to go?
If $a$ and $b$ are cathets a right triangle whose hypotenuse is $1$ determine the highest value of $2a + b$ ?
 [/INST] Let the two sides be $\sin(x)$ and $\cos(x)$.
So we want to maximize $2 \cos(x) + \sin(x)$.
We can write (2,1) in polar form as 
$$
2 = R\cos(y),~ 1 = R\sin(y)~~~~\Rightarrow R = \sqrt{5}$$
Hence
$$
2 \cos(x) + \sin(x) = R (\cos(x) \cos(y) + \sin(x) \sin(y)) =\sqrt{5} \cos(x-y)$$
We really do not need to find $x$ or $y$ since
$$
\text{Maximum value of } \sqrt{5} \cos(x-y) = \sqrt{5}$$
Hence answer is $\sqrt{5}$
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