Question

$$\text {Sketch the following sets of points.}$$ $$\\{(r, \theta): r=3\\}$$

Short answer

Answer: The corresponding equation of a circle in rectangular coordinates is \(x^2 + y^2 = 9\).

Step-by-step solution

Convert Polar Equation to Rectangular Form

To convert the polar equation \(r = 3\) into rectangular coordinates, we can use the relations between polar and rectangular coordinates: \(x = r\cos(\theta)\) and \(y = r\sin(\theta)\). Since \(r = 3\), we have \(x = 3\cos(\theta)\) and \(y = 3\sin(\theta)\).

Express Rectangular Relations as Equation of a Circle

Now we want to express these relations as the equation of a circle with a certain radius and center. We can start by finding \(x^2\) and \(y^2\): \begin{align*} x^2 &= (3\cos(\theta))^2 = 9\cos^2(\theta),\\ y^2 &= (3\sin(\theta))^2 = 9\sin^2(\theta). \end{align*} Summing \(x^2\) and \(y^2\), we obtain: \[x^2 + y^2 = 9\cos^2(\theta) + 9\sin^2(\theta)=9(\cos^2(\theta) + \sin^2(\theta)).\] By applying the Pythagorean identity \(\cos^2(\theta) + \sin^2(\theta) = 1\), we arrive at the equation of the circle: \[x^2 + y^2 = 9.\]

Sketch the Circle

Now we will sketch the circle described by the equation \(x^2 + y^2 = 9\). This circle has a radius of \(3\) (since \(9 = 3^2\)) and is centered at the origin \((0, 0)\). Start drawing by placing the center of the circle at the origin, and then draw the circle with a radius of \(3\) around this point. Finally, label the circle with its polar equation, which is \(r = 3\).

Key concepts

Conversion to Rectangular Coordinates

The process of converting polar coordinates to rectangular coordinates is a fundamental concept in mathematics. Polar coordinates are represented as \((r, \theta)\), where \(r\) is the radius or distance from the origin, and \(\theta\) is the angle in radians from the positive x-axis. To convert these into rectangular coordinates \((x, y)\), we use the following formulas:

• \(x = r \cos(\theta)\)
• \(y = r \sin(\theta)\)

In the given exercise, the polar equation is \(r = 3\). By substituting \(r = 3\) into the expressions above, we find that \(x = 3 \cos(\theta)\) and \(y = 3 \sin(\theta)\).
This establishes a set of rectangular coordinates that traces out a path as \(\theta\) varies from 0 to \(2\pi\), effectively describing a circle. The use of trigonometric functions allows us to capture the circular motion in a rectangular system.

Equation of a Circle

The equation of a circle is a crucial concept in coordinate geometry and is closely tied with the conversion from polar to rectangular coordinates. When we have a polar equation like \(r = 3\), it describes a circle, which we can also express in rectangular form.
Once we have \(x = 3 \cos(\theta)\) and \(y = 3 \sin(\theta)\), the next step is to find \(x^2\) and \(y^2\) by squaring these expressions:

• \(x^2 = (3 \cos(\theta))^2 = 9 \cos^2(\theta)\)
• \(y^2 = (3 \sin(\theta))^2 = 9 \sin^2(\theta)\)

Adding these squared terms gives us:
\[ x^2 + y^2 = 9 \cos^2(\theta) + 9 \sin^2(\theta) \] This is simplified using the identity for circles and trigonometry, leading to:
\[ x^2 + y^2 = 9(\cos^2(\theta) + \sin^2(\theta)) \] On applying the Pythagorean identity, we confirm that \(\cos^2(\theta) + \sin^2(\theta) = 1\), resulting in:
\[ x^2 + y^2 = 9 \]
This shows us that the equation defines a circle centered at the origin with a radius of 3, which aligns perfectly with the original polar equation \(r = 3\).

Pythagorean Identity

The Pythagorean identity is a simple yet powerful tool in trigonometry. It expresses a fundamental property of trigonometric functions and states that \(\cos^2(\theta) + \sin^2(\theta) = 1\) for any angle \(\theta\). This identity is essential for converting polar equations to rectangular form, especially when dealing with circles.
In the context of the given exercise, after expressing \(x^2 = 9 \cos^2(\theta)\) and \(y^2 = 9 \sin^2(\theta)\), we sum these to get:
\[ x^2 + y^2 = 9 \cos^2(\theta) + 9 \sin^2(\theta) \] The Pythagorean identity helps simplify this to:
\[ x^2 + y^2 = 9(\cos^2(\theta) + \sin^2(\theta)) = 9 \times 1 = 9 \]
This transformation shows that the original polar equation represents a circle, simplifying the understanding and sketching of the desired graph. It elegantly bridges the relationship between the polar and rectangular systems. By embracing the beauty of this identity, students can solve complex conversions and enhance their comprehension of coordinate systems.