Polar coordinates

• r is the distance of the point from the pole and $\theta$ is the angle measured anticlockwise between the initial line and the line connecting the point to the pole.
• x = r cos $\theta$ and y = r sin $\theta$ .
• r^2 = x^2 + y^2.
• tan $\theta$ = y/x.
• r = a represents a circle with centre 0 and radius a.
• r = a $\theta$ represents a spiral beginning at 0.
• $\theta$ = $\alpha$ represents a half line that makes an angle $\alpha$ with the initial line and passes through 0.
• Polar coordinates are given in the form (r, $\theta$ ).
• Equations of the form r = a cos(n $\theta$ ) or r = a sin (n $\theta$ ) will have n loops symmetrically arranged around the pole.
• Curves of the form r = a (p + q cos $\theta$ ), where p = 0 will be circular.
• Curves of the form r = a (p + q cos $\theta$ ), where p = q, will be a cardioid (have a dimple that touches 0).
• Curves of the form r = a (p + q cos $\theta$ ), where p/q is more or equal than 2, will be egg shaped.
• Curves of the form r = a (p + q cos $\theta$ ), where p/q is between 1 and 2, will have a dimple that doesn't reach to 0.
• To find the area enclosed by a polar curve and the half lines $\theta$ = a and $\theta$ = b, do 1/2 $\int_a^b$ r^2 d$\theta$ .
• A tangent will be parallel to the initial line when dy/d$\theta$ = 0.
• A tangent will be perpendicular to the initial line when dx/d$\theta$ = 0.