Let-a-b-c-be-there-real-numbers-Prove-that-if-sin-a-sin-b-sin-c-2-cos-a-cos-b-cos-c-5-and-sin-a-sin-b-sin-c-3-2-cos-a-pi-6-cos-b-pi-6-cos-c-pi-6-0-

Question Number 221354 by Nicholas666 last updated on 31/May/25

$$ \\ $$$$\:\:\mathrm{Let}\:{a},{b},{c}\:\mathrm{be}\:\mathrm{there}\:\mathrm{real}\:\mathrm{numbers}, \\ $$$$\:\mathrm{Prove}\:\mathrm{that}\:\mathrm{if}; \\ $$$$\:\mathrm{sin}\:{a}\:+\:\mathrm{sin}\:{b}\:+\:\mathrm{sin}\:{c}\:\geqslant\:\mathrm{2}\:\:\Rightarrow\:\mathrm{cos}\:{a}\:+\:\mathrm{cos}\:{b}\:+\:\mathrm{cos}\:{c}\:\leqslant\:\sqrt{\mathrm{5}}\:\:\:\:\:\:\:\:\:\: \\ $$$$\mathrm{and}, \\ $$$$\:\mathrm{sin}\:{a}\:+\:\mathrm{sin}\:{b}\:+\:\mathrm{sin}\:{c}\:\geqslant\:\frac{\mathrm{3}}{\mathrm{2}}\:\Rightarrow\:\mathrm{cos}\left({a}−\pi/\mathrm{6}\right)\:+\:\mathrm{cos}\left({b}−\pi/\mathrm{6}\right)\:+\:\mathrm{cos}\left({c}−\pi/\mathrm{6}\right)\:\geqslant\:\mathrm{0}\:.\:\:\:\:\:\:\:\:\:\:\:\:\: \\ $$$$ \\ $$

Commented by vnm last updated on 31/May/25

$$…+\mathrm{cos}\left({b}−\pi/\mathrm{6}\right)+…\:? \\ $$

Commented by Nicholas666 last updated on 31/May/25

$$\mathrm{yes} \\ $$

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