Question

For the following exercises, sketch the graph of each conic. $$ r(2+\sin \theta)=4 $$

Short answer

The graph is an ellipse centered at the pole, elongated vertically.

Step-by-step solution

Rewrite the Equation in Standard Form

Given the polar equation \(r(2+\sin\theta) = 4\), first solve for \(r\) in terms of \(\theta\). Divide both sides by \((2 + \sin\theta)\) to get:\[r = \frac{4}{2 + \sin\theta}.\]This is now in the standard form for polar equations.

Identify the Type of Conic

To identify the type of conic section, compare the equation \(r = \frac{4}{2 + \sin\theta}\) with the standard polar form of a conic \(r = \frac{ed}{1 + e\sin\theta}\) or \(r = \frac{ed}{1 + e\cos\theta}\). Recognize that the form is similar to \(r = \frac{ed}{1 - e\sin\theta}\), indicating an ellipse since the constant term 2 serves as \((1 - e)\).

Sketch the Graph of the Ellipse

Use the information derived from Step 2 to sketch the graph of the ellipse. Since the equation indicates the ellipse is centered at the pole and symmetric around the horizontal axis due to the \(\sin\theta\) term, plot points for different angles to depict its shape, making sure it is elongated along the direction of sine, which is vertically oriented around the y-axis.

Key concepts

Conic Sections

Conic sections are curves obtained by intersecting a cone with a plane. These curves have various shapes: a circle, an ellipse, a parabola, or a hyperbola. Each shape has unique properties and can be described using different equations. In polar coordinates, conic sections have specific equation forms that relate the radius \(r\) to the angle \(\theta\). Understanding the general form of conic equations helps in identifying the conic type.

• Circle: All points equidistant from a central point.
• Ellipse: All points where the sum of distances from two foci is constant.
• Parabola: All points equidistant from a single focus and a directrix.
• Hyperbola: All points where the difference of distances from two foci is constant.

The equation \(r(2+\sin \theta)=4\) needs to be converted to identify the conic, which is generally accomplished by rewriting it in a recognizable standard form.

Ellipse

An ellipse is a conic section that looks like an elongated circle. In polar coordinates, its standard equation can look similar to \(r = \frac{ed}{1 + e \sin \theta}\) or \(r = \frac{ed}{1 + e \cos \theta}\), where \(e\) is the eccentricity. If \(0 < e < 1\), the conic is an ellipse. An ellipse maintains symmetry about its principal axis, which is determined by the trigonometric function involved in the equation.

• The principal axis is aligned with the feature in the equation. If \(\sin\) is present, it's vertical; for \(\cos\), horizontal.
• When sketching, trace the shape carefully around its axes, noting the wider reach along one than the other.

In the exercise, the form \(r = \frac{4}{2 + \sin \theta}\) reveals an ellipse with its symmetric feature associated with the sine term, indicating a vertical elongation.

Graph Sketching

Graph sketching of polar equations requires understanding the behavior of the equation as angle \(\theta\) varies. By transforming the equation \(r = \frac{4}{2 + \sin \theta}\), we begin to visualize how the ellipse appears on the polar graph. It is beneficial to calculate values of \(r\) for specific \(\theta\) values.

• Use angles like \(0, \frac{\pi}{2}, \pi, \frac{3\pi}{2},\) and full rotations back to \(2\pi\).
• Note how the radius \(r\) changes, helping mark important points on the graph.
• Check symmetry: since \(\sin\theta\) is vertical, expect pointing upwards or downwards elongation.

Once key points are plotted, the full structure of the ellipse emerges, completing the sketch.

Trigonometric Functions

Trigonometric functions \(\sin\) and \(\cos\) play a vital role in polar coordinates by influencing the orientation of conic sections. The function chosen in the polar equation decides the axis along which the conic section is stretched.

• Sinusoidal Influence: If \(\sin\theta\) appears in a conic's equation, it affects the vertical orientation in the sketch, dictating ellipse elongations or other adjustments.
• Cosine Impact: Conversely, \(\cos\theta\) guides horizontal orientation, impacting overall graph positioning.
• Amplitude and Period: Although the periodic properties of \(\sin\) and \(\cos\) don't change the type of conic, they change how it appears when drawn.

In relation to the given ellipse, \(\sin\theta\)'s presence in \(r = \frac{4}{2 + \sin \theta}\) ensures the vertical symmetry in the graph's orientation.