Question

Qual o calculo passoa passo do seno de 105

Answer 1

Lembre-se que, $\text{sen}~105^{\circ}=\text{sen}~75^{\circ}$, porque são ângulos suplementares.

Vamos utilizar $\text{sen}(a+b)=\text{sen}~a\cdot\text{cos}~b+\text{sen}~b\cdot\text{cos}~a$.

Como $75^{\circ}=30^{\circ}+45^{\circ}$, temos:

$\text{sen}~75^{\circ}=\text{sen}(30^{\circ}+45^{\circ})=\text{sen}~30^{\circ}\cdot\text{cos}~45^{\circ}+\text{sen}~45^{\circ}\cdot\text{cos}~30^{\circ}$.

Lembre-se que, $\text{sen}~30^{\circ}=\dfrac{1}{2}$, $\text{cos}~30^{\circ}=\dfrac{\sqrt{3}}{2}$ e $\text{sen}~45^{\circ}=\text{cos}~45^{\circ}=\dfrac{\sqrt{2}}{2}$.

Assim:

$\text{sen}~75^{\circ}=\dfrac{1}{2}\cdot\dfrac{\sqrt{2}}{2}+\dfrac{\sqrt{2}}{2}\cdot\dfrac{\sqrt{3}}{2}=\dfrac{\sqrt{2}+\sqrt{6}}{4}$

Portanto, $\text{sen}~105^{\circ}=\dfrac{\sqrt{2}+\sqrt{6}}{4}$.