A Cauchy Constraint on $\cos A$

★ 7/10trigonometry Completing the Square Cauchy-Schwarz inequality vectors

Problem

Triangle $ABC$ has $AB = AC = 1$ . A moving point $P$ in its plane satisfies

\overrightarrow{AP} = \lambda\overrightarrow{AB} + 2\mu\overrightarrow{AC} \quad (\lambda, \mu \in \mathbb{R}), \qquad |AP| = 1.

If $\lambda + \mu \leqslant \dfrac{2\sqrt{10}}{5}$ always holds, find the minimum value of $\cos A$ .

Answer

Solution

Difficulty	7/10
Topics	trigonometry, Completing the Square, Cauchy-Schwarz, inequality, vectors

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