Problem 38 - Thomas Calculus

Question

Find the points of intersection of the pairs of curves in Exercises \(31-38\) . $$ r^{2}=\sqrt{2} \cos 2 \theta, \quad r^{2}=\sqrt{2} \sin 2 \theta $$

Step-by-Step Solution

Answer

Intersection points are \((1, \frac{\pi}{8})\), \((-1, \frac{9\pi}{8})\), \((1, \frac{5\pi}{8})\), and \((-1, \frac{13\pi}{8})\).

Step 1: Identify Given Equations

We are given two polar equations: 1. \( r^2 = \sqrt{2} \cos 2\theta \) 2. \( r^2 = \sqrt{2} \sin 2\theta \). We need to find the points \((r, \theta)\) where these curves intersect, meaning where both equations are satisfied simultaneously.

Step 2: Set Equations Equal

Since both expressions equal \( r^2 \), we set them equal to each other: \[ \sqrt{2} \cos 2\theta = \sqrt{2} \sin 2\theta \] Simplifying gives us:\[ \cos 2\theta = \sin 2\theta \]

Step 3: Solve for Angle

To solve \( \cos 2\theta = \sin 2\theta \), divide both sides by \( \cos 2\theta \):\[ \tan 2\theta = 1 \]The solutions to this are \(2\theta = \frac{\pi}{4} + n\pi\) for integers \(n\). Thus, the general solution for \(\theta\) is:\[ \theta = \frac{\pi}{8} + \frac{n\pi}{2} \]

Step 4: Find Corresponding r-values

Substitute \(\theta = \frac{\pi}{8}\) into one of the original equations to find \(r\):For \( r^2 = \sqrt{2} \cos 2\theta \):\[ r^2 = \sqrt{2} \cos \left(\frac{\pi}{4}\right) = \sqrt{2} \cdot \frac{\sqrt{2}}{2} = 1\]Therefore, \(r = \pm 1\), and similarly for any \(\theta\) following \(\frac{\pi}{8} + \frac{n\pi}{2}\).

Step 5: List Points of Intersection

From \(\theta = \frac{\pi}{8} + \frac{n\pi}{2}\) and \(r = \pm 1\), we find specific points:For \(n=0\), \(\theta = \frac{\pi}{8}\), points are \((1, \frac{\pi}{8})\) and \((-1, \frac{\pi}{8} + \pi)\).For \(n=1\), \(\theta = \frac{5\pi}{8}\), points are \((1, \frac{5\pi}{8})\) and \((-1, \frac{5\pi}{8} + \pi)\).Continue for other integer values of \(n\) as needed.

Key Concepts

Points of Intersection

In the world of polar coordinates, finding the points of intersection of two curves means discovering where the paths of the curves cross each other. When we have a set of polar equations, like in the exercise above, we're looking for these shared points. This involves setting the equations equal to each other since both can describe the same radius squared, as seen in this example.

Here’s a straightforward way to think about it:

• Each polar equation describes a circle or a spiral in terms of a radius and an angle.
• When these equations are set equal, we find the angles at which they intersect.
• Then, we substitute these angles back into the original equations to find the specific radial distances.

By following this process, every point where the curves cross each other is easily identified. Thus, for each angle found, multiple corresponding radial positions (like positive and negative values) can be plotted, giving us every intersection point.

Polar Equations

Polar equations are expressions that describe curves with respect to a central pole (usually the origin) using a combination of radius \(r\) and angle \(\theta\). Unlike Cartesian coordinates that use straight lines in grid form, polar coordinates map out positions using these radial lines and angular turns.

Some key ideas about polar equations include:

• Polar equations can describe unique curves, such as circles and spirals, that Cartesian equations may find complex or impossible to represent.
• The form \(r = f(\theta)\) shows a direct relationship between angle and radius, often involving trigonometric functions like sine or cosine.
• Using polar equations involves working with angles measured in radians, providing a seamless description of round and spiral forms.

This is why polar equations are invaluable for capturing the essence of curves that dance boldly around a central point, as opposed to sticking to rigid, linear paths.

Trigonometric Identities

Trigonometric identities are mathematical statements expressing relationships between trigonometric functions. These identities simplify complex trigonometric expressions to make equations easier to solve and understand. In the intersection of polar curves, identities like the tangent identity can be crucial.

Remember a few of these handy identities:

• The basic identity, \(\tan \theta = \frac{\sin \theta}{\cos \theta}\), which played a prominent role in finding intersection points by allowing us to solve equations involving \(\sin\) and \(\cos\).
• Pythagorean identities, such as \(\sin^2 \theta + \cos^2 \theta = 1\), help transition between these functions with ease.
• Double angle formulas, which transform expressions like \(\cos 2\theta\) or \(\sin 2\theta\), are particularly useful for aligning angles correctly to solve for intersections in polar forms.

By employing these identities, navigating through polar equations becomes significantly more manageable. With these tools, solving for intersecting points or simplifying intricate trigonometric expressions becomes straightforward and efficient.