Question

Find the points of intersection of the polar graphs. \(r=\sin (3 \theta)\) and \(r=\cos (3 \theta)\) on \([0, \pi]\)

Short answer

Intersect at \((\frac{\sqrt{2}}{2}, \frac{\pi}{12})\), \((-\frac{\sqrt{2}}{2}, \frac{5\pi}{12})\), and \((\frac{\sqrt{2}}{2}, \frac{3\pi}{4})\).

Step-by-step solution

Set the Polar Equations Equal

To find the points of intersection, we need to set the two polar equations equal to each other:\[ \sin(3\theta) = \cos(3\theta) \]

Solve the Trigonometric Equation

Now, solve for \(\theta\) by dividing both sides by \(\cos(3\theta)\) (assuming \(\cos(3\theta) eq 0\)):\[ \tan(3\theta) = 1 \]Hence, \(3\theta = \frac{\pi}{4} + k\pi\), where \(k\) is an integer.

Determine the Values for θ in the Interval

Solve for \(\theta\) within the interval \([0, \pi]\):- When \(k = 0\), \(\theta = \frac{\pi}{12}\)- When \(k = 1\), \(\theta = \frac{5\pi}{12}\)- When \(k = 2\), \(\theta = \frac{9\pi}{12} = \frac{3\pi}{4}\)Thus, the solutions for \(\theta\) in the interval \([0, \pi]\) are \(\frac{\pi}{12}\), \(\frac{5\pi}{12}\), and \(\frac{3\pi}{4}\).

Find Corresponding r-values for each θ

With \(\theta\) found, substitute back into either polar equation to find \(r\):- For \(\theta = \frac{\pi}{12}\), \(r = \sin(3 \times \frac{\pi}{12}) = \sin\frac{\pi}{4} = \frac{\sqrt{2}}{2}\)- For \(\theta = \frac{5\pi}{12}\), \(r = \sin(3 \times \frac{5\pi}{12}) = \sin\frac{5\pi}{4} = -\frac{\sqrt{2}}{2}\)- For \(\theta = \frac{3\pi}{4}\), \(r = \sin(3 \times \frac{3\pi}{4}) = \sin\frac{9\pi}{4} = \frac{\sqrt{2}}{2}\)

List the Points of Intersection

The points of intersection in polar coordinates are \[ \left(\frac{\sqrt{2}}{2}, \frac{\pi}{12}\right), \left(-\frac{\sqrt{2}}{2}, \frac{5\pi}{12}\right), \left(\frac{\sqrt{2}}{2}, \frac{3\pi}{4}\right) \]

Key concepts

Points of Intersection

When dealing with polar equations, finding the points of intersection essentially means pinpointing where the two graphs overlap. In this exercise, we explore the intersection of two specific polar graphs given by the equations \(r = \sin(3\theta)\) and \(r = \cos(3\theta)\). To find these intersections:

• Set the two equations equal: \( \sin(3\theta) = \cos(3\theta) \). This gives you a condition that must be true for at least some values of \(\theta\).
• Solve this equation for specific angles \(\theta\) that satisfy it—it’s like solving a puzzle to find where the curves meet.

The result is a set of \(\theta\) values which indicate the angles at which the polar curves intersect. For this problem, it happens at three distinct angles: \(\theta = \frac{\pi}{12}\), \(\theta = \frac{5\pi}{12}\), and \(\theta = \frac{3\pi}{4}\). Picking the correct intervals and understanding how these angles relate to the radial distance \(r\) provides the complete point of intersection.

Trigonometric Equations

Trigonometric equations involve the functions sine, cosine, and tangent among others, and are fundamental in solving problems involving angles and circular patterns. In this scenario, the equation \(\tan(3\theta) = 1\) arises when solving the trigonometric equation \(\sin(3\theta) = \cos(3\theta)\). Here's how it works:

• By dividing both sides by \(\cos(3\theta)\), assuming \(\cos(3\theta) eq 0\), the equation transforms into \(\tan(3\theta) = 1\).
• This equation implies that the tangent of \(3\theta\) hits its standard angle values where tangent is equal to one, at specific angles given by \(3\theta = \frac{\pi}{4} + k\pi\).
• After simplifying, \(\theta\) values are found in the context of our polar interval, \([0, \pi]\), achieving the intersections.

Understanding how these functions work and transforming them into simpler forms helps unravel the conditions needed for intersections, solving for the specific values of \(\theta\) that satisfy the equation.

Polar Graphs

Polar graphs visually represent equations in a coordinate system that uses radius and angles instead of the usual x and y-axis. They provide a distinct method to depict curves like spirals and flowers that are often difficult to represent with Cartesian coordinates. For graphs given by \(r = \sin(3\theta)\) and \(r = \cos(3\theta)\):

• Understand that \(r\) represents the distance from the origin, and \(\theta\) is the angle from the polar axis, typically measured from the positive x-axis in a counter-clockwise direction.
• These specific functions create rose curves, fascinating cylindrical shapes, with multiple symmetrical petals dependent on the multiplicative factor of \(\theta\).
• Tracking how these curves overlap helps identify intersection points, adding visual intuition to the numeric solutions obtained from solving equations.

To truly grasp the elegance of polar graphs, it can be beneficial to graph these equations by hand or use digital graphing tools to see how they intersect. This visualization aids in understanding complex trigonometric and polar relationships that are common in higher mathematics.