Máx. e Mín. de expressão trigonométrica.

por Betoneira de Natal Dom 13 Mar 2022, 14:02

Encontre o valor máximo e mínimo da seguinte expressão:

Máx. e Mín. de expressão trigonométrica. Svg+xml;base64,<?xml version='1.0' encoding='UTF-8'?>
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Resp.: Máximo: 13
Mínimo: -13

Como posso fazer essa relação?
No meu gabarito, ele diz a respeito de relacionar o coeficiente q está multiplicando o sen.x e o cos.x como lados de um triângulo pitagórico.
Porém, como posso afirmar que eles são realmente lados de um triângulo pitagórico?

por **Elcioschin** Dom 13 Mar 2022, 14:09

Leia: https://pir2.forumeiros.com/t150465-o-truque-do-triangulo-retangulo

Quaisquer dois valores reais podem ser catetos de um triângulo retângulo!

y = 5.senx + 12.cosx --> a² = 5² + 12² ---> a = 13

Seja cosθ = 5/13 e senθ = 12/13 --->

y = 13.[(5/13).senx + (12/13).cosx]

y = 13.(senx.cosθ + senθ.cosx) ---> y = 13.sen(x + θ)

Valor máximo ---> x + θ = 90º ---> y(máx) = 13
Valor mínimo ---> x + θ = 270º ---> y(mín) = -13

por **Giovana Martins** Dom 13 Mar 2022, 14:30

Outro jeito:

[latex]\\\mathrm{Desigualdade\ de\ Cauchy-Schwarz:}[/latex]

[latex]\\\\\mathrm{\left ( \alpha _1^2+\alpha _2^2+...+\alpha _n^2 \right )\left ( \beta _1^2+\beta _2^2+...+\beta _n^2 \right )\geq \left ( \alpha _1\beta _1+\alpha _2\beta _2+...+\alpha _n\beta _n \right )^2}[/latex]

[latex]\\\\\mathrm{Para\ o\ nosso\ caso:\ \left ( \alpha _1^2+\alpha _2^2 \right )\left ( \beta _1^2+\beta _2^2 \right )\geq \left ( \alpha _1\beta _1+\alpha _2\beta _2\right )^2}[/latex]

[latex]\\\\ \mathrm{Sendo\ f(x)= 12cos(x)+5sin(x):}[/latex]

[latex]\\\\ \mathrm{\left [ (12)^2+(5)^2 \right ]\left [\underset{1}{ \underbrace{\mathrm{cos^2(x)+sin^2(x)}}} \right ]\geq \left [ 12cos(x)+5sin(x) \right ]^2}[/latex]

[latex]\\\\\mathrm{169\geq \left [ 12cos(x)+5sin(x) \right ]^2\to |12cos(x)+5sin(x)|\leq 13}[/latex]

[latex]\\\\\mathrm{ -13\leq 12cos(x)+5sin(x)\leq 13\to -13\leq f(x)\leq 13}[/latex]

[latex]\\\\\mathrm{\therefore f(x)=\pm 13\to\left\{\begin{matrix}\mathrm{ f_{min}=-13}\\ \mathrm{f_{max}=13} \end{matrix}\right.\ }[/latex]

por **Elcioschin** Dom 13 Mar 2022, 17:28

Uma terceira solução usando derivadas:

E(x) = 5.senx + 12.cosx ---> Derivando:

E'(x) = 5.cosx + 12.(- senx) ---> E'(x) = 5.cosx - 12.senx

Para termos valores máximos e mínimos devemos ter E'(x) = 0 --->

5.cosx - 12.senx = 0 ---> 5.cosx = 12.senx ---> 25.cos²x = 144.sen²x ---> 25.(1 - sen²x) = 144.sen²x --->

25 = 169.sen²x ---> sen²x = 25/169 --> senx = ± 5/13 ---> cosx = ± 12/13

Basta agora calcular os valores máximo e mínimo.

por Betoneira de Natal Dom 13 Mar 2022, 18:23

Muito obrigado pela ajuda, vou estudar as três formas!

Obrigado Elcio e Gi!

por **Giovana Martins** Dom 13 Mar 2022, 18:28

Uma outra solução a partir da Substituição de Weierstrass.

[latex]\\\mathrm{Seja\ t=tan\left ( \frac{x}{2} \right ),t\in \mathbb{R}.\ Portanto, tem-se:sin\left ( \frac{x}{2} \right )=\frac{t}{\sqrt{1+t^2}}\ e\ cos\left ( \frac{x}{2} \right )=\frac{1}{\sqrt{1+t^2}}}\\\\\mathrm{De\ f(x)=asin(x)+bcos(x)\to f(x)=\frac{at+b}{\sqrt{1+t^2}}\to [f(x)]^2=\frac{a^2t^2+2abt+b^2}{1+t^2}}\\\\\mathrm{a^2t^2+2abt+b^2=[f(x)]^2+[f(x)]^2t^2\to^2\left \{ a^2-[f(x)]^2 \right \}+2abt+b^2-[f(x)]^2=0}\\\\\mathrm{Sendo\ t\in \mathbb{R}\ \therefore \ \Delta \geq 0\leftrightarrow a^2b^2-\left \{ a^2-[f(x)]^2 \right \}\left \{ b^2-[f(x)]^2 \right \}\geq 0\leftrightarrow [f(x)]^2 \left \{ a^2+b^2-[f(x)]^2 \right \}\geq 0}\\\\\mathrm{Logo, [f(x)]^2\geq 0, \forall\ x\in\mathbb{R}. \ Por\ outro\ lado,\left | f(x) \right |\leq \sqrt{a^2+b^2}\to -\sqrt{a^2+b^2}\leq f(x)\leq \sqrt{a^2+b^2}}\\\\\mathrm{Sendo\ a=5\ e\ b=12,tem-se:-\sqrt{(5)^2+(12)^2}\leq f(x)\leq \sqrt{(5)^2+(12)^2}\to -13\leq f(x)\leq 13}[/latex]

Se eu não esqueci de nenhuma restrição ao longo da resolução, creio que assim também seja possível de resolver. Não é uma ideia minha, mas não lembro onde eu vi isso para dar os créditos (tem muito tempo que eu vi isso. Não me lembro bem se é assim mesmo). Eu só lembrei disso, porque o Método da Substituição de Weierstrass é relativamente comum para resolver umas integrais mais chatinhas.