Question

Brainly.com.br Qual a sua pergunta? ggomes6677 há 5 horas Matemática Ensino médio (secundário) +5 pts Respondido A) Qual o valor do arco duplo com sen 120° ? b) Qual o valor do arco duplo com cos 90° ? c) Qual o valor do arco duplo com tan 60° ? d) Sabendo que sen(x) =5/3 e cos(x) =2/3,determine os valores de sen(2x), cos(2x) e tan (2x).

Answer 1

Seno do arco duplo

$\large\boxed{\boxed{\boxed{\boxed{\mathsf{sen(2x)=2.sen(x).cos(x)}}}}}$

Cosseno do arco duplo

$\boxed{\boxed{\mathsf{cos(2x)=cos^2(x)-sen^2(x)}}}\\\boxed{\boxed{\mathsf{cos(2x)=1-2sen^2(x)}}}\\\boxed{\boxed{\mathsf{cos(2x)=2cos^2(x)-1}}}$

Tangente do arco duplo

$\large\boxed{\boxed{\boxed{\boxed{\mathsf{tg(2x)=\dfrac{2tg(x)}{1-tg^2(x)}}}}}}$

a)

$\mathsf{sen(120^{\circ})=sen(2.60^{\circ})}\\\mathsf{sen(120^{\circ})=2.sen(60^{\circ}).cos(60^{\circ})=2.\dfrac{\sqrt{3}}{2}.\dfrac{1}{2}=\boxed{\boxed{\boxed{\dfrac{\sqrt{3}}{2}}}}}$

b)

$\mathsf{cos(90^{\circ})=cos(2.45^{\circ})}\\\mathsf{cos(90^{\circ})=2cos^2(45^{\circ})-1=2.(\dfrac{\sqrt{2}}{2})^2-1}\\\mathsf{cos(90^{\circ})=2.\dfrac{2}{4}-1=1-1=0}$

c)

$\mathsf{tg(60^{\circ})=tg(2.30^{\circ})=\dfrac{2.tg(30^{\circ})}{1-tg^2(30^{\circ})}}\\\mathsf{tg(60^{\circ})=\dfrac{2.\frac{\sqrt{3}}{3}}{1-(\frac{\sqrt{3}}{3})^2}=\dfrac{\frac{2\sqrt{3}}{3}}{1-\frac{1}{3}}}\\\mathsf{tg(60^{\circ})=\dfrac{\frac{2\sqrt{3}}{3}}{\frac{2}{3}}=\dfrac{2\sqrt{3}}{2}=\sqrt{3}}$

d)

$\mathsf{sen(2x)=2.\dfrac{5}{3}.\dfrac{2}{3}=\boxed{\boxed{\boxed{\dfrac{20}{9}}}}}\\\mathsf{cos(2x)=2.(\dfrac{2}{3})^2-1=\dfrac{8}{9}-1=\boxed{\boxed{\boxed{-\dfrac{1}{9}}}}}\\\mathsf{tg(2x)=\dfrac{\frac{20}{9}}{-\frac{1}{9}}=-20}$