How to Find the Points of Intersection of Two Polar Curves: r sin2θ and r cos2θ

To find the points of intersection of the two polar curves r sin(2θ) and r cos(2θ), we set the two equations equal to each other:

sin(2θ) cos(2θ)

Next, we divide both sides by cos(2θ), assuming cos(2θ) ≠ 0 for θ not equal to 90 degrees (or π/2 radians):

tan(2θ) 1

The solutions to this equation are found when:

2θ frac{pi}{4} kπ, text{ where } k in mathbb{Z}

Dividing by 2, we get:

θ frac{pi}{8} frac{kπ}{2}

Next, we determine the corresponding values of r for these angles. We can substitute θ π/8 into either of the original equations. Let's use the first one:

r sin(2 cdot frac{pi}{8}) sin(frac{pi}{4}) frac{sqrt{2}}{2}

Now, we find other values of θ based on k: θ π/8, r sqrt{2}/2 θ 5π/8, r -sqrt{2}/2 (since r is negative, this point corresponds to θ 13π/8) θ 9π/8, r sqrt{2}/2

In summary, the points of intersection in polar coordinates are:

(sqrt{2}/2, π/8) (-sqrt{2}/2, 13π/8) (sqrt{2}/2, 9π/8)

In Cartesian coordinates, these translate to:

left(frac{sqrt{2}}{2} cdot cos(π/8) cdot frac{sqrt{2}}{2} cdot sin(π/8)right) left(-frac{sqrt{2}}{2} cdot cos(5π/8) cdot -frac{sqrt{2}}{2} cdot sin(5π/8)right) left(frac{sqrt{2}}{2} cdot cos(9π/8) cdot frac{sqrt{2}}{2} cdot sin(9π/8)right)

These coordinates represent the same Cartesian points but are reflected due to the periodic nature of trigonometric functions.

Understanding the Intersection in Polar Curves

The polar curve intersection is a fundamental concept in the study of trigonometric functions and polar coordinates. This intersection occurs where the two curves meet, and it is crucial for understanding the behavior of polar curves in various mathematical and scientific applications, such as physics and engineering.

The use of trigonometric functions, specifically sin and cos, allows us to explore the periodic nature of these curves. By setting the two equations equal, we can solve for θ and then determine the corresponding r values. The periodicity of the trigonometric functions ensures that we capture all points of intersection, not just the first.

Real-World Applications

The concept of finding the intersection of polar curves has practical applications in various fields:

In science and engineering, understanding the intersection points helps in analyzing complex systems, such as planetary orbits or electrical circuits. In architecture and design, the intersection points can be used to create aesthetically pleasing patterns and designs. In navigation and mapping, the intersection points can help in determining the precise location of geographical features.

By mastering the techniques to find the intersection points of polar curves, students and professionals can gain a deeper understanding of the underlying mathematical principles and apply them to solve real-world problems.