Question

Convert the polar equation to rectangular form and sketch its graph. $$ r=\cot \theta \csc \theta $$

Short answer

The rectangular form is \( y^2 = x \), a parabola opening rightwards.

Step-by-step solution

Understand the Given Polar Equation

The polar equation given is \( r = \cot \theta \csc \theta \). Our goal is to convert this into rectangular form, which uses \(x\) and \(y\) instead of \(r\) and \(\theta\).

Use Trigonometric Identities

Recall that \( \cot \theta = \frac{1}{\tan \theta} = \frac{\cos \theta}{\sin \theta} \) and \( \csc \theta = \frac{1}{\sin \theta} \). Use these identities to rewrite the given equation: \( r = \frac{\cos \theta}{\sin^2 \theta} \).

Express Using Rectangular Coordinates

We know the relationships: \( x = r \cos \theta \), \( y = r \sin \theta \), and \( r^2 = x^2 + y^2 \). We need to express \( \frac{\cos \theta}{\sin^2 \theta} \) using \(x\) and \(y\). Substitute: \( r^2 = x^2 + y^2 \) therefore \( \sin \theta = \frac{y}{r} \) and \( \cos \theta = \frac{x}{r} \). Thus, \( \frac{\cos \theta}{\sin^2 \theta} = \frac{\frac{x}{r}}{\left(\frac{y}{r}\right)^2} = \frac{x}{r} \cdot \frac{r^2}{y^2} = x \cdot \frac{r}{y^2} \).

Simplify to Rectangular Form

We have \( r = x \cdot \frac{r}{y^2} \). Since \( r eq 0 \), we can multiply both sides by \( y^2 \) leading to \( r y^2 = x r \). Dividing by \( r \) (assuming \( r eq 0 \)), gives \( y^2 = x \), which is already in rectangular form.

Sketch the Graph

The rectangular equation \( y^2 = x \) represents a parabola opening to the right. The vertex of the parabola is at the origin (0,0), and it is symmetric with respect to the x-axis.

Key concepts

Trigonometric Identities

To tackle converting from polar to rectangular coordinates, we need a friendly understanding of trigonometric identities. These identities are mathematical tools that assist in simplifying and transforming equations. In dealing with the given exercise, two main identities are employed:

• The cotangent identity: \( \cot \theta = \frac{1}{\tan \theta} = \frac{\cos \theta}{\sin \theta} \)
• The cosecant identity: \( \csc \theta = \frac{1}{\sin \theta} \)

Together, these identities help recast the polar equation with familiar rectangular forms like \( x \) and \( y \). By using these identities, we can express \( r = \cot \theta \csc \theta \) as \( \frac{\cos \theta}{\sin^2 \theta} \), paving the way for the next steps where we substitute in terms of \( x \) and \( y \).
Never underestimate the power of these identities; they are foundational to understanding how polar and rectangular forms interrelate.

Rectangular Coordinates

Rectangular coordinates, known as Cartesian coordinates, express points in the plane using pairs \( (x, y) \). These coordinates are incredibly useful for recognizing and graphing shapes in our everyday geometric environment. When converting from polar to rectangular coordinates, the key lies in understanding the relationships between \( r \), \( \theta \), and \( x \), \( y \).

• We know that \( x = r \cos \theta \) and \( y = r \sin \theta \).
• Together with \( r^2 = x^2 + y^2 \), these equations are the stepping stones to transition from one form to another.

In the transformation process, replacing \( \sin \theta = \frac{y}{r} \) and \( \cos \theta = \frac{x}{r} \) bridges polar terms into the rectangular arena. The exercise simplifies to \( y^2 = x \), exemplifying a common conversion to a rectangular expression.

Graphing Polar Equations

Graphing polar equations typically results in visually intricate figures and can be quite different from graphing using rectangular coordinates. The exercise demonstrates the conversion of a polar equation to a rectangular form through the transformation \( y^2 = x \).
In rectangular plotting, this particular equation represents a parabola opening towards the right, with its symmetry axis being the x-axis and its vertex sitting cleanly at the point (0,0).

• This symmetrical aspect is crucial as it indicates that each pair of identical y-values traces out a mirror-image along the x-axis.
• Understanding this symmetry is key when sketching these graphs manually or using graphing tools.

The intimate link between the equations in different coordinate systems (polar and rectangular) presents a robust way to visualize complex relationships, giving students a clear graphical insight into algebraic forms.