Question

How do we write `rcos(theta-alpha)=p` in Cartesian form?

Answer 1

In Cartesian form, `rcos(theta-alpha)=p` can be written as `xcosalpha+ysinalpha=p`

Explanation:

The relation between polar coordinates `(r,theta)` and Cartesian coordinates `(x,y)` is given by

`x=rcostheta` and `y=rsintheta` i.e. `r^2=x^2+y^2`

Hence we can write `rcos(theta-alpha)=p` as

`r(costhetacosalpha+sinthetasinalpha)=p`

or `rcosthetacosalpha+rsinthetasinalpha=p`

or `xcosalpha+ysinalpha=p`

Additional Information `-` This is the equation of a line on which a perpendicular drawn from origin makes an angle `alpha` with `x`-axis and is of length `p` as shown below,