Question

Show that $$ \begin{aligned} &x=h+r \cos \theta\\\ &y=k+r \sin \theta\\\ &\text { represents the equation of a circle. } \end{aligned} $$

Short answer

The given parametric equations represent a circle centered at \\( (h, k) \\) with radius \\( r \\).

Step-by-step solution

Identify the Circle Equation

The standard equation of a circle in the Cartesian plane with center \(h, k\) and radius \(r\) is given by \( (x - h)^2 + (y - k)^2 = r^2 \). Our goal is to transform the given parametric equations into this form.

Substitute Parametric Equations

The given parametric equations are \( x = h + r \cos \theta \) and \( y = k + r \sin \theta \). Substitute these into the circle equation: \[(x - h)^2 + (y - k)^2 = (r \cos \theta)^2 + (r \sin \theta)^2.\]

Simplify Each Term

Simplify the squared terms \( (r \cos \theta)^2 + (r \sin \theta)^2 \): \( r^2 \cos^2 \theta + r^2 \sin^2 \theta = r^2 (\cos^2 \theta + \sin^2 \theta). \) Use the identity \( \cos^2 \theta + \sin^2 \theta = 1 \) to further simplify to \( r^2.\)

Conclusion

After simplifying, we have \((x-h)^2 + (y-k)^2 = r^2\), confirming that the given parametric equations \(x = h + r \cos \theta \) and \(y = k + r \sin \theta \) indeed represent the equation of a circle with center \( (h, k) \) and radius \( r \).

Key concepts

Parametric Equations

Parametric equations provide us with a way to define coordinates using a single variable, often referred to as a parameter. In the context of circles, these equations are used to express the x and y coordinates based on angles instead of relying directly on Cartesian coordinates. This is extremely useful because it provides a direct representation of a point's location on a circle as it relates to an angle measure:

• The parametric equations for a circle with center \((h, k)\) and radius \(r\) are given by \(x = h + r \cos \theta\) and \(y = k + r \sin \theta\).
• Here, \(\theta\) is the parameter, usually representing the angle in radians.
• As \(\theta\) varies from \(0\) to \(2\pi\), the equations trace out the entire circumference of the circle.

By expressing the coordinates in terms of \(\theta\), parametric equations link circles clearly to angles, allowing us to derive other useful information like arc length and sector areas.

Cartesian Plane

The Cartesian plane is a 2D plane defined by two axes that are perpendicular to each other. These axes are labeled as the x-axis and the y-axis. Together, they allow us to plot points and geometric shapes using pairs of numbers, known as coordinates.

• Each point in the Cartesian plane can be identified by an ordered pair, \((x, y)\), which represents its horizontal and vertical positions.
• The standard equation for a circle in the Cartesian plane with center \((h, k)\) and radius \(r\) is \((x - h)^2 + (y - k)^2 = r^2\).
• This equation is derived from the Pythagorean theorem, describing the set of all points that are exactly \(r\) units away from the center \((h, k)\).

On the Cartesian plane, each pair of parametric equations can correspond to a specific shape or a path traced by a point, making visualization of movements and positions intuitive.

Trigonometric Identities

Trigonometric identities are fundamental relationships between trigonometric functions that hold true for all angle values. They play a crucial role in simplifying expressions and solving equations involving trigonometric functions. One of the key identities used here is:

• \(\cos^2 \theta + \sin^2 \theta = 1\)

This identity is essential in understanding why the parametric equations \(x = h + r \cos \theta\) and \(y = k + r \sin \theta\) can be rewritten as the equation of a circle in the Cartesian plane.

• When simplified using the identity, the equation \((r \cos \theta)^2 + (r \sin \theta)^2\) becomes \(r^2 (\cos^2 \theta + \sin^2 \theta) = r^2\), leading precisely to the circle's standard form in Cartesian coordinates.
• This simplification confirms the completeness of the circle as being traced by the parametric equations.

Trigonometric identities help bridge the gap between parametric forms and Cartesian forms, making complex mathematical concepts easier to manage and solve.