Answer:
[tex]\cos x=-\dfrac{\sqrt{21}}{5}\\ \\\tan x =-\dfrac{2}{\sqrt{21}}\\ \\\cot x=-\dfrac{\sqrt{21}}{2}[/tex]
Step-by-step explanation:
Given
[tex]\sin x=\dfrac{2}{5}\\ \\\cos x<0[/tex]
Use formula:
[tex]\cos^2x+\sin ^2x=1\\ \\\cos^2x+\left(\dfrac{2}{5}\right)^2=1\\ \\\cos^2x=1-\dfrac{4}{25}\\ \\cos^2x=\dfrac{21}{25}\\ \\\cos x=\pm \dfrac{\sqrt{21}}{5}[/tex]
Since [tex]\cos x<0,[/tex] then
[tex]\cos x=-\dfrac{\sqrt{21}}{5}[/tex]
By definition,
[tex]\tan x =\dfrac{\sin x}{\cos x}=\dfrac{\frac{2}{5}}{-\frac{\sqrt{21}}{5}}=-\dfrac{2}{\sqrt{21}}\\ \\ \\\cot x=\dfrac{1}{\tan x}=\dfrac{1}{-\frac{2}{\sqrt{21}}}=-\dfrac{\sqrt{21}}{2}[/tex]
