To graph this easily, we can convert it to rectangular form.
In order to convert, we need to the cosine angle difference formula:
#cos(color(red)A-color(blue)B)=coscolor(red)Acoscolor(blue)B+sincolor(red)Asincolor(blue)B#
Knowing that #rsintheta=y# and #rcostheta=x#, we can convert:
#1=rcos(theta-pi/6)#
#1=r(costhetacoscolor(black)(pi/6)+sinthetasincolor(black)(pi/6))#
#1=r(costheta*sqrt3/2+sintheta*1/2)#
#1=rcostheta*sqrt3/2+rsintheta*1/2#
#1=x*sqrt3/2+y*1/2#
#1-x*sqrt3/2=y*1/2#
#2-x*sqrt3=y#
#y=2-x*sqrt3#
#y=-sqrt3 x+2#
Now we can graph this linear equation like any other line.
An easy strategy would be to solve for the #x#- and #y#-intercepts, then connect the dots.
The #x#-intecept occurs when #y=0#, so:
#color(white)=>y=-sqrt3 x+2#
#=>0=-sqrt3 x+2#
#color(white)=>sqrt3 x=2#
#color(white)=>x=2/sqrt3#
#color(white)=>x=(2sqrt3)/3#
This means that the #x#-intercept is at #((2sqrt3)/3,0)#. Call this point #A#. The #y#-intercept occurs when #x=0#, so:
#color(white)=>y=-sqrt3 x+2#
#=>y=-sqrt3 *0+2#
#color(white)=>y=2#
This means that the #y#-intercept occurs at #(0,2)#. Call this point #B#. Now that we have our two points, we can graph the line:
That's it. Hope this helped!
