How do I find the polar equation of this cartesian equation?

Question

Find the equation in polar coordinated of the line through the origin with slope (1)/(5)

 

theta = 

Answer 1

This problem involves converting a function from a cartesian form to polar form.

Step 0: Cartesian Form vs. Polar Form

Points can be plotted on a 2D plane in two ways, in cartesian form and in polar form.

Cartesian form describes a point's horizontal and vertical position. Each point in the cartesian plane has a horizontal coordinate 'x' and a vertical coordinate 'y', forming a pair (x,y).

Polar form describes the distance from the origin and the angle from the positive horizontal axis. Each point in the polar plane has a radius component 'r' and an angular component θ, forming a pair (r,θ).

Step 1: Conversion Equations

There are two equations that can take us from cartesian form to polar form.

r = √(x²+y²) and θ = arctan(y / x)

There are two equations that can take us from polar form to cartesian form.

x = rcos(θ) and y = rsin(θ)

Step 2: Equation of the Line in Cartesian Form

Let's use algebra to figure out what the cartesian form of the line is. We are told that the line passes through the origin and has a slope of 1/5. That means that the y-intercept 'b' is equal to 0 and the slope 'm' is equal to 1/5. Using the slope-intercept form of a line, the cartesian form of the line is this.

y = (1/5)x

Step 3: Converting From Cartesian Form to Polar Form

Now let's convert this to polar form. Let's look at our conversion equations and the line equation.

r = √(x²+y²) and θ = arctan(y / x)

y = (1/5)x

Notice that if we divide both sides of the line equation by x, we get this.

y / x = 1/5

Let's substitute that into the θ equation.

θ = arctan(1/5)

By converting from the cartesian form equation to polar form, we find that every point on the line satisfies θ = arctan(1/5), representing a straight line through the origin at that constant angle.