Upper-bound on sum of two vectors

Question

I have two vectors $a$ and $b$. The resultant of the two vectors is denoted as $c = a+b$. I wish to find the upper bound of $\mid c \mid$ in terms of $\mid a \mid$. Suppose $\phi$ is angle between $a$ and $b$. If $\phi$ is acute and $ \mid a\mid > \frac{\mid b \mid}{2\cos \phi}$, from law of cosines, we have ${\mid c\mid}^{2} = {\mid a \mid}^2+ {\mid b \mid}^2 - 2 \mid a\mid \mid b \mid \cos(\phi)$. Using law of cosines we can upper-bound $\mid c\mid$ as $ {\mid c \mid} < \mid a \mid $. Can we find a similar condition to upper bound $\mid c \mid$, when $\phi$ is obtuse?