Problem 32 - College Trigonometry

Question

In Exercises \(31-40\), sketch the region in the \(x y\) -plane described by the given set. $$ \\{(r, \theta) \mid 0 \leq r \leq 4 \sin (\theta), 0 \leq \theta \leq \pi\\} $$

Step-by-Step Solution

Answer

The region is the upper semicircle centered at (0, 2) with radius 2.

Step 1: Understand the inequalities

The inequality given is \(0 \leq r \leq 4\sin(\theta)\) and \(0 \leq \theta \leq \pi\). This describes a region in polar coordinates. \(r\) is the radius from the origin and \(\theta\) is the angle from the positive x-axis measured counterclockwise.

Step 2: Identify the shape described by the inequality

The inequality \(r = 4\sin(\theta)\) defines a circle. In polar coordinates, this can be converted to the Cartesian equation by substituting \(r = \sqrt{x^2 + y^2}\) and \(\sin(\theta) = \frac{y}{r}\), which simplifies to \(x^2 + (y-2)^2 = 4\). This is a circle centered at \((0, 2)\) in the \(xy\)-plane with a radius of 2.

Step 3: Examine the limits of \(\theta\)

The limits \(0 \leq \theta \leq \pi\) correspond to angles that cover the upper half of the plane (from the positive x-axis to the negative x-axis). This means we are considering only the top half of the circle described by the equation \(r = 4\sin(\theta)\).

Step 4: Translate the polar region into the Cartesian plane

The region of interest is inside the circle \(x^2 + (y-2)^2 = 4\) (from Step 2) and is restricted to \(0 \leq \theta \leq \pi\), which corresponds to the top half of the circle. Thus, the sketch should depict only the upper semicircle above the x-axis and within the circle centered at \((0, 2)\) with radius 2.

Step 5: Sketch the described region

Draw a circle centered at \((0, 2)\) with a radius of 2 and shade only the upper half above the x-axis, which is bounded by \(\theta = 0\) (along the positive x-axis) and \(\theta = \pi\) (along the negative x-axis).

Key Concepts

Cartesian Coordinates

The Cartesian coordinate system allows us to pinpoint the exact location of a point in two dimensions using two numbers, usually referred to as \(x\) and \(y\). This system is created by a horizontal axis called the x-axis and a vertical axis called the y-axis. Where these two axes meet is known as the origin, which has the coordinates \((0, 0)\).

Here’s why Cartesian coordinates are useful:

• They allow us to describe geometric shapes and lines using equations.
• In the case of our problem, the conversion from polar to Cartesian helps us see that the expression \(r = 4\sin(\theta)\) results in a circle equation of \(x^2 + (y-2)^2 = 4\) in the Cartesian plane.
• Understanding the transformation process from polar to Cartesian coordinates deepens comprehension of mathematical analysis and geometry.

In this exercise, using the Cartesian system, we identify the circle's center at \((0, 2)\) with a radius of 2. The area we want to describe is the top half of this circle.

Circle Equations

Circle equations in mathematics traditionally have the form \((x - h)^2 + (y - k)^2 = r^2\), where \((h, k)\) are the coordinates of the circle's center, and \(r\) is the radius.

For our exercise, the transformation of the polar equation \(r = 4\sin(\theta)\) into its Cartesian form results in the equation \(x^2 + (y-2)^2 = 4\). This transformation is done using the relationships:

• \( r = \sqrt{x^2 + y^2} \)
• \(\sin(\theta) = \frac{y}{r}\)

By converting the circle's equation into its Cartesian form, we understand that:

• The circle is centered at \((0, 2)\), which signifies its vertical displacement upwards by 2 units from the origin.
• The radius of the circle is 2, as indicated by \(r^2 = 4\).

This interpretation allows us to visualize and solve the region described in the subsequent graphing exercise with greater clarity.

Graphing Polar Regions

Graphing regions in polar coordinates involves understanding how the values of \(r\) and \(\theta\) define a curve. Polar coordinates use a radius and an angle to determine a point's position.

For the equation \(r = 4\sin(\theta)\), the region described is part of a circle in the polar plane. Let's break this down further:

• The value \(\theta\) ranges from \(0\) to \(\pi\), which means we only consider the angles from the positive x-axis to the negative x-axis, covering the upper half of the plane.
• \(r = 4\sin(\theta)\) essentially describes points for radiuses that change with \(\theta\), forming the top hemisphere of the circle.

To sketch the region:

• Start with the circle centered at \((0, 2)\), then shade the portion above the x-axis, representing \(0 \leq \theta \leq \pi\).
• This results in a semicircle from \(\theta = 0\) to \(\theta = \pi\).
• The radius at points in this region goes from \(0\) at the axes to \(2\) at the top of the circle.

The primary goal of graphing in polar regions is to pinpoint areas that fulfill given conditions, aiding us in visual analysis and interpretations.