How do you graph the polar equation #1=rcos(theta-pi/6)#?

Question

Zoe Austin · Accepted Answer

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!

Aaliyah Abdullah · Answer

undefined To graph the polar equation $1 = r \cos(\theta - \frac{\pi}{6})$, follow these steps:

1. Recognize that the equation represents a cardioid, which is a type of polar curve.
2. The parameter $r$ represents the distance from the origin to a point on the curve, and $\theta$ represents the angle between the initial ray and the line connecting the origin to the point.
3. Rewrite the equation in terms of $r$ to isolate it: $r = \frac{1}{\cos(\theta - \frac{\pi}{6})}$.
4. Plot points by selecting various values of $\theta$, calculating the corresponding values of $r$, and converting them to Cartesian coordinates using the formulas $x = r \cos(\theta)$ and $y = r \sin(\theta)$.
5. Plot the points on a polar grid or convert them to rectangular coordinates and plot them on a Cartesian grid.
6. Connect the points to form the curve of the cardioid.