Question

a soma das soluções da equação trigonométrica √3. cos x + sen x =√3, no intervalo [0,2π] é:

Answer 1

A soma das soluções da equação trigonométrica √3. cos (x) + sen(x) =√3

, no intervalo [0,2π] é:

Explicação passo-a-passo:

sen²(x) + cos²(x) = 1

sen(x) = √3 - √3*cos(x)

sen²(x) = (√3 - √3*cos(x))²

sen²(x) = 3 - 6cos(x) + 3cos²(x)

3 - 6cos(x) + 3cos²(x)  + cos²(x) = 1

2 - 6cos(x) + 4cos²(x) = 0

1 - 3cos(x) + 2cos²(x) = 0

delta

Δ² = 9 - 8 = 1, Δ = 1

cos(x) = (3 + 1)/4 = 1

cos(x) = (3 - 1)/4 = 1/2

se cos(x) = 1 ,  x1 = 0, x2 = 2π

se cos(x) = 1/2, x3 = π/3, x4 = 5π/3

1 - 3cos(x) + 2cos²(x) = 0

1 - 3cos(0) + 2cos²(0) = 0

1 - 3 + 2 = 0

1 - 3cos(2π) + 2cos²(2π) = 0

1 - 3 + 2 = 0

1 - 3cos(π/3) + 2cos²(π/3) = 0

1 -  3/2 + 2/4 = 4/4 - 6/4 + 2/4 = 0

√3*cos (x) + sen(x) = √3

√3* cos (5π/3) + sen(5π/3) = √3

√3/2 - √3/2 = 0 e nao √3

as soluções são

S = (0, π/3, 2π)

a soma

S = π/3 + 2π = π/3 + 6π/3

S = 7π/3

Answer 2

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$\boxed{\begin{array}{l}\sf \sqrt{3}cos(x)+sen(x)=\sqrt{3}\\\sf sen(x)=\sqrt{3}-\sqrt{3}cos(x)\\\sf cos^2(x)+sen^2(x)=1\\\sf cos^2(x)+(\sqrt{3}-\sqrt{3}cos(x))^2=1\\\sf cos^2(x)+3-6cos(x)+3cos^2(x)-1=0\\\sf 4cos^2(x)-6cos(x)+2=0\div2\\\sf 2cos^2(x)-3cos(x)+1=0\\\sf\Delta=9-8=1\\\sf cos(x)=\dfrac{3\pm1}{4}\begin{cases}\sf cos(x)=1\\\sf cos(x)=\dfrac{1}{2}\end{cases}\end{array}}$

$\boxed{\begin{array}{l}\sf cos(x)=1\implies x=0~~ou~~x=2\pi\\\sf cos(x)=\dfrac{1}{2}\implies x=\dfrac{\pi}{3}~~ou~~x=\dfrac{5\pi}{3}\end{array}}$

$\underline{\rm verificac_{\!\!,}\tilde ao\!:}$

$\boxed{\begin{array}{l}\sf x=0:~~\sqrt{3}cos(0)+sen(0)=\sqrt{3}\cdot1+0=\sqrt{3}\\\sf x=0~\acute e~soluc_{\!\!,}\tilde ao.\\\sf x=2\pi:~~\sqrt{3}cos(2\pi)+sen(2\pi)=\sqrt{3}\cdot1+0=\sqrt{3}\\\sf x=2\pi~\acute e~soluc_{\!\!,}\tilde ao.\\\sf x=\dfrac{\pi}{3}:~~\sqrt{3}cos\bigg(\dfrac{\pi}{3}\bigg)+sen\bigg(\dfrac{\pi}{3}\bigg)=\sqrt{3}\cdot\dfrac{1}{2}+\dfrac{\sqrt{3}}{2}\\\sf=\dfrac{2\sqrt{3}}{2}=\sqrt{3}\\\sf x=\dfrac{\pi}{3}~\acute e~soluc_{\!\!,}\tilde ao\end{array}}$

$\boxed{\begin{array}{l}\sf x=\dfrac{5\pi}{3}:~~\sqrt{3}\cdot cos\bigg(\dfrac{5\pi}{3}\bigg)+sen\bigg(\dfrac{5\pi}{3}\bigg)\\\sf=\sqrt{3}\cdot\dfrac{1}{2}-\dfrac{\sqrt{3}}{2}=0\\\sf x=\dfrac{5\pi}{3}~n\tilde ao~\acute e~soluc_{\!\!,}\tilde ao.\end{array}}$

$\large\boxed{\begin{array}{l}\sf S=\bigg\{0,2\pi,\dfrac{\pi}{3}\bigg\}\\\sf soma=0+2\pi+\dfrac{\pi}{3}\\\sf soma=\dfrac{6\pi}{3}+\dfrac{\pi}{3}\\\\\sf soma=\dfrac{7\pi}{3}\end{array}}$