Expert answers to "What will be the minimum value ofp2tan⁡90∘+q2tan⁡27∘+r2tan⁡63∘+s2tan⁡81∘Ifp+q+r+s=5where p,q,e,s..."

Nettie Potts

Answered question

2022-02-26

What will be the minimum value of

\frac{p^{2}}{{\tan 90}^{\circ}} + \frac{q^{2}}{{\tan 27}^{\circ}} + \frac{r^{2}}{{\tan 63}^{\circ}} + \frac{s^{2}}{{\tan 81}^{\circ}}

p + q + r + s = 5

where p,q,e,s are positive reals?

an2gi2m9gg

Beginner2022-02-27Added 9 answers

({\tan 9}^{\circ} + {\tan 27}^{\circ} + {\tan 63}^{\circ} + {\tan 81}^{\circ}) (\frac{p^{2}}{{\tan 9}^{\circ}} + \frac{q^{2}}{{\tan 27}^{\circ}} + \frac{r^{2}}{{\tan 63}^{\circ}} + \frac{s^{2}}{{\tan 81}^{\circ}}) \geq {(p + q + r + s)}^{2} = 5^{2}

From Cauchy–Schwarz
Hence

Then take

p = \frac{5 {\tan 9}^{\circ}}{{\tan 9}^{\circ} + {\tan 27}^{\circ} + {\tan 63}^{\circ} + {\tan 81}^{\circ}}

q = \frac{5 {\tan 27}^{\circ}}{{\tan 9}^{\circ} + {\tan 27}^{\circ} + {\tan 63}^{\circ} + {\tan 81}^{\circ}}

r = \frac{5 {\tan 63}^{\circ}}{{\tan 9}^{\circ} + {\tan 27}^{\circ} + {\tan 63}^{\circ} + {\tan 81}^{\circ}}

s = \frac{5 {\tan 81}^{\circ}}{{\tan 9}^{\circ} + {\tan 27}^{\circ} + {\tan 63}^{\circ} + {\tan 81}^{\circ}}

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