1. [tex]x[/tex] in quadrant IV means [tex]\sin x<0[/tex], so
[tex]\cos^2x+\sin^2x=1\implies\sin x=-\sqrt{1-\cos^2x}=-\dfrac{\sqrt3}2[/tex]
2. [tex]x[/tex] in quadrant I means [tex]\cos x>0[/tex]. Then
[tex]\cos x=\sqrt{1-\sin^2x}=\dfrac{\sqrt2}2[/tex]
3. [tex]x[/tex] in quadrant II means [tex]\sin x>0[/tex]. Then
[tex]\tan x=\dfrac{\sin x}{\cos x}=\dfrac{\sqrt{1-\cos^2x}}{-\frac12}=\dfrac{\frac{\sqrt3}2}{-\frac12}=-\sqrt3[/tex]
4. If [tex]\sin x=-1[/tex], then [tex]\cos x=0[/tex], so [tex]\tan x[/tex] is undefined.
