Question

calcule os valores de cotangente,secante e cossecante de 1050 graus

Answer 1

Vamos começar reduzindo ao 1° quadrante.

$1050^\circ~=~360^\circ+360^\circ+330^\circ\\\\\\\boxed{1050~\cong~330^\circ}$

Agora podemos calcular o que é pedido:

$cotg(1050^\circ)~=~cotg(330^\circ)\\\\\\330^\circ~est\acute{a}~no~4^o~quadrante,~logo~a~cotg~ser\acute{a}~negativa.\\cotg(330^\circ)~\acute{e}~igual,~em~modulo~a~cotg(360^\circ-330^\circ).\\\\\\cotg(1050^\circ)~=~-cotg(360^\circ-330^\circ)\\\\\\cotg(1050^\circ)~=~-cotg(30^\circ)\\\\\\cotg(1050^\circ)~=~-(tg(30^\circ))^{-1}\\\\\\cotg(1050^\circ)~=~-\left(\frac{3}{\sqrt{3}}\right)\\\\\\cotg(1050^\circ)~=~-\frac{3}{\sqrt{3}}~.~\frac{\sqrt{3}}{\sqrt{3}}$

$\boxed{cotg(1050^\circ)~=~-\sqrt{3}}$

$sec(1050^\circ)~=~sec(330^\circ)\\\\\\330^\circ~est\acute{a}~no~4^o~quadrante,~logo~a~sec~ser\acute{a}~positiva.\\sec(330^\circ)~\acute{e}~igual,~em~modulo~a~sec(360^\circ-330^\circ).\\\\\\sec(1050^\circ)~=~sec(360^\circ-330^\circ)\\\\\\sec(1050^\circ)~=~sec(30^\circ)\\\\\\sec(1050^\circ)~=~(cos(30^\circ))^{-1}\\\\\\sec(1050^\circ)~=~\left(\frac{2}{\sqrt{3}}\right)\\\\\\sec(1050^\circ)~=~\frac{2}{\sqrt{3}}~.~\frac{\sqrt{3}}{\sqrt{3}}$

$\boxed{sec(1050^\circ)~=~\frac{2\sqrt{3}}{3}}$

$cossec(1050^\circ)~=~cossec(330^\circ)\\\\\\330^\circ~est\acute{a}~no~4^o~quadrante,~logo~a~cossec~ser\acute{a}~negativa.\\cossec(330^\circ)~\acute{e}~igual,~em~modulo~a~cossec(360^\circ-330^\circ).\\\\\\cossec(1050^\circ)~=~-cossec(360^\circ-330^\circ)\\\\\\cossec(1050^\circ)~=~-cossec(30^\circ)\\\\\\cossec(1050^\circ)~=~-(sen(30^\circ))^{-1}\\\\\\cossec(1050^\circ)~=~-\left(\frac{2}{1}\right)\\\\\\\boxed{cossec(1050^\circ)~=~-2}$