We know the reference angle is #30^@#, and from the unit circle, we know the coordinates for #30^@# are

#(sqrt3/2,1/2)#

Our angle, #150^@# is in the second quadrant, where cosine is negative and sine is positive. Unit circle coordinates are given by

#(costheta, sintheta)#

This means the coordinates for #150^@# are

#(-sqrt3/2,1/2)#

We know:

#color(blue)(cos150=-sqrt3/2)#

#color(darkblue)(sin150=1/2)#

#color(lime)(tantheta)=color(darkblue)(sintheta)/color(blue)(costheta)#

And from our definitions of trig functions:

#color(red)(cottheta)=1/color(lime)(tantheta)#

#color(darkviolet)(sectheta)=1/color(blue)(costheta)#

#color(orange)(csctheta)=1/color(darkblue)(sintheta)#

After plugging in the appropriate values (and rationalizing the denominator when necessary), we get

#color(lime)(tan150=-sqrt3/3)#

#color(red)(cot150=-sqrt3)#

#color(darkviolet)(sec150=(-2sqrt3)/3)#

#color(orange)(csc150=2)#

Hope this helps!

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