Problem 18 - Calculus

Question

Find the points where the two curves meet. \(r^{2}=\sin 2 \theta\) and \(r^{2}=\cos 2 \theta\)

Step-by-Step Solution

Answer

Points of intersection are at \( \theta = \frac{\pi}{8} + \frac{n\pi}{2} \) with corresponding \(r^2\).

Step 1: Set the Equations Equal

Since both curves are defined in terms of \(r^2\), start by setting the right-hand sides equal to each other: \(\sin 2\theta = \cos 2\theta\).

Step 2: Solve for \(\theta\)

Use the identity \(\tan \theta = \frac{\sin \theta}{\cos \theta}\) to write \(\tan 2\theta = 1\). This implies that \(2\theta = \frac{\pi}{4} + n\pi\), where \(n\) is an integer.

Step 3: Simplify for \(\theta\)

Divide the equation from Step 2 by 2 to solve for \(\theta\): \[ \theta = \frac{\pi}{8} + \frac{n\pi}{2} \].

Step 4: Find Corresponding \(r^2\) Values

Substitute the values of \(\theta\) back into one of the original equations, such as \(r^2 = \sin 2\theta\). For example, if \(\theta = \frac{\pi}{8}\):\[ r^2 = \sin\left(\frac{\pi}{4}\right) = \frac{\sqrt{2}}{2} \].

Step 5: Check Both Curves

Verify that the solutions satisfy both original equations. If \(\theta = \frac{\pi}{8}\) gives \(r^2 = \frac{\sqrt{2}}{2}\) for both \(\sin 2\theta\) and \(\cos 2\theta\), then they intersect at points corresponding to this \(\theta\).

Step 6: Determine Additional Intersection Points

Plug in the other solutions from \( \theta = \frac{\pi}{8} + \frac{n\pi}{2} \) (e.g., \(\theta = \frac{5\pi}{8}, \frac{9\pi}{8}, etc.\)) into the equations and find corresponding values of \(r^2\). If they are equal for both equations, that's an intersection point.

Key Concepts

Polar Coordinates

Polar coordinates are an alternative system to Cartesian coordinates for describing points in a plane. Instead of using x and y, it relies on r (the radial distance from the origin) and \(\theta\) (the angle from a reference direction, usually the positive x-axis). This system is particularly useful for problems involving circles or curves centered at the origin because it simplifies many mathematical expressions.
When working with polar coordinates, a point is described by the coordinate pair (r, \(\theta\)). The value of r can be positive or negative: positive if the direction is along \(\theta\)'s angle, and negative if it's in the opposite direction. Another unique aspect of polar coordinates is that the same point can be represented by multiple pairs due to the periodic nature of angles in circles.

• The angle \(\theta\) is measured in radians, with 360 degrees equal to \(2\pi\) radians.
• Common identities can convert between polar and Cartesian coordinates, such as \(x = r \cos \theta\) and \(y = r \sin \theta\).

Leveraging polar coordinates can simplify problems that are complex in Cartesian coordinates, especially when symmetry or angles are involved.

Trigonometric Identities

Trigonometric identities allow for the simplification and solving of equations involving trigonometric functions. These identities, such as the Pythagorean identity or angle addition formulas, are essential in solving problems involving polar curves and intersections.
In this exercise, the identity for tangent \(\tan \theta = \frac{\sin \theta}{\cos \theta}\) is used to equate \(\sin 2\theta = \cos 2\theta\) to \(\tan 2\theta = 1\). From this relationship, the solution proceeds by determining the specific angles \(\theta\) that satisfy this condition.

• Knowing angles where \(\tan \theta = 1\) helps us find solutions for \(\theta\) — typically these occur at 45 degrees (or \(\frac{\pi}{4}\) radians) plus full rotations \(n\pi\).
• Using double angle identities, such as \(\sin 2\theta = 2 \sin \theta \cos \theta\) and \(\cos 2\theta = \cos^2 \theta - \sin^2 \theta\), facilitate solving more complex problems.

Trigonometric identities are crucial for transforming and solving equations involving curves in polar coordinates.

Intersection Points

Finding intersection points of curves is often about solving simultaneous equations. For polar curves, as in this exercise, where we have \(r^2 = \sin 2\theta\) and \(r^2 = \cos 2\theta\), the task is to find when these two expressions yield the same \(r\) value for the same \(\theta\).
Setting the given equations equal (\(\sin 2\theta = \cos 2\theta\)) enables the search for common \(\theta\) values. By expressing \(\tan 2\theta = 1\), we determine that the solutions for \(2\theta\) are \(\frac{\pi}{4} + n\pi\), leading to \(\theta = \frac{\pi}{8} + \frac{n\pi}{2}\).

• Substitute these \(\theta\) values back into one of the original equations to find the corresponding \(r^2\) values.
• Verify each solution by checking if it satisfies both original curve equations. If both curves yield the same \(r^2\) for any \(\theta\), this confirms an intersection point.

Understanding intersection points can often involve checking multiple solutions or cycles, but it is key in determining where curves in polar coordinates overlap or cross each other.