Question

Determine o valor de y= sen80 cos70 + sen460 cos470

Answer 1

Relações utilizadas:

$\begin{array}{rclc} \mathrm{sen}\left(\alpha+k\cdot 360^{\circ} \right )&=&\mathrm{sen\,}\alpha&\;\;\;\;\text{(i)}\\ \\ \cos\left(\alpha+k\cdot 360^{\circ} \right )&=&\cos\alpha&\;\;\;\;\text{(ii)}\\ \\ \mathrm{sen}\left(90^{\circ}-\alpha \right )&=&\cos \alpha&\;\;\;\;\text{(iii)}\\ \\ \cos\left(90^{\circ}-\alpha \right )&=&\mathrm{sen\,} \alpha&\;\;\;\;\text{(iv)}\\ \\ \mathrm{sen}\left(90^{\circ}+\alpha \right )&=&\cos \alpha&\;\;\;\;\text{(v)}\\ \\ \cos\left(90^{\circ}+\alpha \right )&=&-\mathrm{sen\,} \alpha&\;\;\;\;\text{(vi)}\\ \\ \end{array}$

(nas relações $\text{(i)}$ e $\text{(ii)}$, $k$ pode assumir qualquer valor inteiro)


Então

$y=\mathrm{sen\,}80^{\circ}\cos 70^{\circ}+\mathrm{sen\,}460^{\circ}\cos 470^{\circ}\\ \\ y=\mathrm{sen\,}80^{\circ}\cos 70^{\circ}+\mathrm{sen}\left(460^{\circ}-360^{\circ} \right )\cos\left(470^{\circ}-360^{\circ} \right )\\ \\ y=\mathrm{sen\,}80^{\circ}\cos 70^{\circ}+\mathrm{sen\,}100^{\circ}\cos 110^{\circ}\\ \\ y=\mathrm{sen}\left(90^{\circ}-10^{\circ} \right )\cos \left(90^{\circ}-20^{\circ} \right )+\mathrm{sen}\left(90^{\circ}+10^{\circ} \right )\cos \left(90^{\circ}+20^{\circ} \right )\\ \\ y=\cos10^{\circ}\mathrm{\,sen\,}20^{\circ}+\cos 10^{\circ}\left(-\mathrm{sen\,}20^{\circ}\right)\\ \\ y=\cos10^{\circ}\mathrm{\,sen\,}20^{\circ}-\cos 10^{\circ}\mathrm{\,sen\,}20^{\circ}\\ \\ \boxed{y=0}$