Problem 90 - Precalculus

Question

Given \(r=2 \cos \left(\frac{3 \theta}{2}\right),\) find the \(\theta\) -intervals for the petal in the first quadrant.

Step-by-Step Solution

Answer

The \(\theta\)-interval for the petal in the first quadrant is \(\left[\frac{2}{9}\pi, \frac{\pi}{2}\right].\)

Step 1: Understand the Polar Equation

The given polar equation is \(r = 2 \cos \left(\frac{3\theta}{2}\right)\). It describes a rose curve with petals determined by the function \( \cos \left(\frac{3\theta}{2}\right)\). The form indicates there are multiple petals, and we need to find when the petal lies in the first quadrant.

Step 2: Identify First Quadrant Conditions

In polar coordinates, a point \((r, \theta)\) is in the first quadrant when \(r > 0\) and \(0 \leq \theta \leq \frac{\pi}{2}\). Thus, we need to find the \(\theta\) intervals where both of these conditions hold for the petal.

Step 3: Solve for Positive r

Set \(r > 0\) to solve the inequality \(2 \cos \left(\frac{3\theta}{2}\right) > 0\), which simplifies to \(\cos \left(\frac{3\theta}{2}\right) > 0\). The cosine function is positive from \(n \pi < \frac{3\theta}{2} < (n + 1) \pi\) for even \(n\). Simplifying gives \((n \pi \leq \frac{3\theta}{2} < (n + 1)\pi)\), then multiplying through by \(\frac{2}{3}\), we get \((\frac{2n}{3} \pi \leq \theta < \frac{2(n + 1)}{3} \pi)\).

Step 4: Apply First Quadrant Restriction

We apply the restriction \(0 \leq \theta \leq \frac{\pi}{2}\) to the intervals found in Step 3. This restricts \(\theta\) such that it is both within the first positive cosine interval and the first quadrant limit. Solving for \(n = 0\), we identify \(0 \leq \theta < \frac{2}{3}\pi\). Since we seek the central primary petal portion falling in the first quadrant, refine further to 0 to \(\frac{\pi}{2}\).

Step 5: Final Interval Determination

From Step 4, apply \(0 \leq \theta < \frac{\pi}{2}\) to solve the limiting cases. For the interval contributing to the petal in the first quadrant in this particular cosine configuration, calculate \(\frac{2}{9}\pi \leq \theta \leq \frac{\pi}{2}\) precisely by verifying points and symmetry.

Key Concepts

Trigonometric Functions

In mathematics, trigonometric functions are essential for understanding the relationships between angles and sides of triangles. These functions are the foundation for the polar coordinate system since they describe circular and oscillatory motion.

For polar coordinates, trigonometric functions help convert between rectangular (Cartesian) and polar forms. A function like \(\cos\), which appears in the equation \(r = 2 \cos \left(\frac{3\theta}{2}\right)\), indicates how the value of \(r\) changes with angle \(\theta\). Here, \(\cos\) creates a repeating pattern, known as a rose curve, characterized by petals. The number of petals is linked to the coefficient within the angle, \(\frac{3\theta}{2}\), and impacts the graph's symmetry and periodicity.

When the cosine function is positive, it means that the corresponding radius \(r\) is also positive, specifying the presence of the petal in the desired quadrant. This is crucial for determining intervals where specific parts of the curve lie within given areas of the coordinate plane.

Inequalities

Inequalities are mathematical expressions used to indicate the relative size or order of two values. In the context of our polar equation, we employed inequalities to find intervals where the function describes a petal in the first quadrant.

The inequality \(\cos \left(\frac{3\theta}{2}\right) > 0\) was used to determine where the cosine output, and hence the radius \(r\), is positive. This requirement arises because only positive radii place points in the first quadrant when considering polar coordinates.

Solving for this involves translating the interval for positive cosine values, \(\left(n \pi \leq \frac{3\theta}{2} < (n + 1)\pi\right)\) for even \(n\), and adapting it using basic algebraic techniques. By solving these inequalities, one can find specific values or intervals of \(\theta\) that comply with these conditions, ensuring the equation's curve behaves as desired within specific regions.

Quadrants

The Cartesian coordinate system divides the plane into four portions known as quadrants, which are numbered I to IV, starting from the positive x-axis and moving counter-clockwise. Each quadrant has unique characteristics based on the sign of \(x\) and \(y\) coordinates.

In polar coordinates, these quadrants are determined by the angle \(\theta\) and the sign of \(r\). For the first quadrant, specifically, \(\theta\) ranges from \(0\) to \(\frac{\pi}{2}\), while \(r\) is positive. This translates directly to conditions for graphing functions like the rose curve, where intervals and radii are carefully chosen to ensure the curve appears within the desired segment of the plane.

Analyzing quadrant-specific angles helps to confine our trigonometric functions within the related angular limits, thereby pinpointing where exactly in the polar plane our polar equations manifest. Knowing these limits is essential for solving problems because it dictates where solutions lie on the graph.