Respuesta:
1.
En estos 3 ejercicios, ocupas Valuar, ambos lados de la expresión, con el valor del ángulo que te dan.
a. Con β=30° :
[tex]Cos(30)=\frac{\sqrt{3} }{2}\\\\Sen(30)=\frac{1}{2}\\\\Tan(30)= \frac{\sqrt{3} }{3}[/tex]
Sustituyendo estos valores en la ecuación:
[tex]\frac{\sqrt{3} }{2} =\frac{\frac{1}{2} }{\frac{\sqrt{3} }{3} }\\\\\frac{\sqrt{3} }{2} =\frac{(1)(3)}{(2)(\sqrt{3} )}\\\\\frac{\sqrt{3} }{2} =\frac{3}{2\sqrt{3} }[/tex]
Radicalizando:
[tex]\frac{\sqrt{3} }{2} =\frac{3}{2\sqrt{3} }\frac{\sqrt{3} }{\sqrt{3} } }\\\\\frac{\sqrt{3} }{2} =\frac{3\sqrt{3} }{2\sqrt{3} \sqrt{3} }\\\\\frac{\sqrt{3} }{2} =\frac{3\sqrt{3} }{2(3)}\\\\\frac{\sqrt{3} }{2} =\frac{\sqrt{3} }{2}[/tex]
b. Con β=140° :
[tex]Sec(140)=\frac{1}{Cos(140)}\\\\Tan(140)=\frac{Sen(140)}{Cos(140)} \\\\Sen(140)=Sen(140)[/tex]
Sustituyendo:
[tex]\frac{1}{Cos(140)}Sen(140)=\frac{Sen(140)}{Cos(140)} \\\\\frac{Sen(140)}{Cos(140)}=\frac{Sen(140)}{Cos(140)}[/tex]
c. Con β=210° :
[tex]Tan(210)=\frac{\sqrt{3} }{3} \\\\Cos(210)=-\frac{\sqrt{3} }{2} \\\\Sen(210)=-\frac{1}{2}[/tex]
Sustituyendo:
[tex](\frac{\sqrt{3} }{3} )(-\frac{\sqrt{3} }{2} )=-\frac{1}{2}\\\\-\frac{\sqrt{3}\sqrt{3} }{(3)(2)} =-\frac{1}{2}\\\\-\frac{3}{(3)(2)} =-\frac{1}{2}\\\\-\frac{1}{2} =-\frac{1}{2}[/tex]
2.
a. Recordando las identidades recíprocas para Cot, y Csc.:
[tex]Cot(\theta )=\frac{Cos(\theta)}{Sen(\theta)} \\\\Csc(\theta)=\frac{1}{Sen(\theta)}[/tex]
Sustituyendo:
[tex]\frac{Cot(\theta)}{Csc(\theta)}=\frac{\frac{Cos(\theta)}{Sen(\theta)} }{\frac{1}{Sen(\theta)}}\\\\ \frac{Cot(\theta)}{Csc(\theta)}=\frac{(Cos(\theta))(Sen(\theta))}{Sen(\theta)} \\\\ \frac{Cot(\theta)}{Csc(\theta)}=Cos(\theta)[/tex]
b. Recordando las identidades recíprocas para Tan, y Sec:
[tex]Tan(\theta)=\frac{Sen(\theta)}{Cos(\theta)}\\\\Sec(\theta)=\frac{1}{Cos(\theta)}[/tex]
Sustituyendo:
[tex]\frac{Tan(\theta)}{Sec(\theta)} =\frac{\frac{Sen(\theta)}{Cos(\theta)} }{\frac{1}{Cos(\theta)} }\\\\\frac{Tan(\theta)}{Sec(\theta)} =\frac{(Sen(\theta))(Cos(\theta))}{Cos(\theta)}\\\\\frac{Tan(\theta)}{Sec(\theta)}=Sen(\theta)[/tex]
Suerte!!!
