Find-the-minimum-value-of-sin-x-cos-x-tan-x-cot-x-sec-x-csc-x-for-real-numbers-x-

Question Number 135792 by benjo_mathlover last updated on 16/Mar/21

F i n d t h e m i n i m u m v a l u e

o f ∣ \sin x + \cos x + \tan x + \cot x + \sec x + \csc x ∣

f o r r e a l n u m b e r s x .

Answered by MJS_new last updated on 16/Mar/21

x = t + \frac{5 π}{4} leads to

f (t) = \frac{2 \sqrt{2} \cos^{3} t + \sqrt{2} \cos t - 2}{1 - {2 \cos}^{2} t}

this has no real zeros

f^{'} (t) = - \frac{4 (2 \sqrt{2} \cos^{3} t + {2 \cos}^{2} t - 3 \sqrt{2} \cos t + 1) \sin t}{{(\sqrt{2} + 2 \cos t)}^{2} (\sqrt{2} - 3 \cos t)}

this has zeros

\sin t = 0

\cos t = \frac{\sqrt{2}}{2}

\cos t = 1 - \frac{\sqrt{2}}{2} [\cos t = - 1 - \frac{\sqrt{2}}{2} rejected]

testing all of these we get

\min ∣ f (t) ∣ = - 1 + 2 \sqrt{2}

Answered by liberty last updated on 16/Mar/21

s e t {\begin{cases} \sin x = a \\ \cos x = b \end{cases}

w e w a n t t o m i n i m i z e

L =∣ a + b + \frac{a}{b} + \frac{b}{a} + \frac{1}{a} + \frac{1}{b} ∣

= ∣ \frac{a b (a + b) + a^{2} + b^{2} + a + b}{a b} ∣

w h e r e a^{2} + b^{2} = 1 . l e t a + b = c

w e h a v e c^{2} = {(a + b)}^{2} = 1 + 2 a b

s o 2 a b = c^{2} - 1 . c o n s i d e r

c = \sin x + \cos x = \sqrt{2} \sin (x + \frac{π}{4})

s o t h e r a n g e o f c i s t h e i n t e r v a l

[- \sqrt{2}, \sqrt{2}] . C o n s e q u e n t l y

i t s u f f i c e s t o f i n d t h e m i n i m u m

o f L (c) =∣ \frac{c (c^{2} - 1) + 2 (c + 1)}{c^{2} - 1} ∣

L (c) =∣ c + \frac{2}{c - 1} ∣, f o r c i n t h e

i n t e r v a l [- \sqrt{2}, \sqrt{2}], t a k i n g

d e r i v a t i v e \frac{d L}{d c} = 1 - \frac{2}{{(c - 1)}^{2}} = 0

w e g e t c = 1 \pm \sqrt{2} . T e s t i n g f o r

c = 1 - \sqrt{2} \Rightarrow L =∣ 1 - \sqrt{2} + \frac{2}{1 - \sqrt{2} - 1} ∣

= ∣ 1 - \sqrt{2} - \sqrt{2} ∣= 2 \sqrt{2} - 1 (m i n)

f o r c = 1 + \sqrt{2} \to r e j e c t e d

f o r m a x v a l u e i f c = \sqrt{2}

w e g e t L =∣ \sqrt{2} + \frac{2}{\sqrt{2} - 1} ∣ = ∣ \sqrt{2} + 2 \sqrt{2} + 2 ∣= 3 \sqrt{2} + 2

Commented by liberty last updated on 16/Mar/21

Commented by liberty last updated on 16/Mar/21

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