Question

X+y=4 Convert the equation to polar coordinates and simplify.

Answer 1

In polar coordinates, the equation x + y = 4 simplifies to
`\(r\cos(\theta) + r\sin(\theta)` = 4.

Explanation:

Converting the Cartesian equation x + y = 4 into polar coordinates involves substituting
`\(x = r\cos(\theta)\)` and
`\(y = r\sin(\theta)\)`. Starting with the original equation x + y = 4, we replace x and y with their respective polar coordinate forms to get
`\(r\cos(\theta) + r\sin(\theta)` = 4.

In this polar coordinate form, we can simplify by factoring out the common term r on the left side of the equation to obtain
`\(r(\cos(\theta) + \sin(\theta))` = 4. This represents the equation in a more compact and manageable form.

The equation
`\(r(\cos(\theta) + \sin(\theta))` = 4 can be interpreted geometrically as the sum of the projections of the point r onto the x and y axes, which collectively equals 4. It showcases a relationship between the distance of the point from the origin and the angles formed with the positive x axis.

Solving for r in terms of
`\(\theta\)` may involve further manipulation using trigonometric identities or expressing r explicitly to understand how the radius r changes with varying angles
`\(\theta\)` to fulfill the equation
`\(r(\cos(\theta) + \sin(\theta))`= 4. This transformation enables us to express the relationship between r and
`\(\theta\)` in polar coordinates for the given Cartesian equation.

The resulting equation in polar coordinates
`\(r\cos(\theta) + r\sin(\theta)` = 4 provides a concise representation of the relationship between the radius r and the angle
`\(\theta\)` in a polar system that satisfies the original Cartesian equation x + y = 4.