Problem 31 - Calculus Early Transcendentals

Question

\(29-48\) Sketch the curve with the given polar equation. $$r=\sin \theta$$

Step-by-Step Solution

Answer

The curve is a semicircle on the upper half-plane, centered at \((0,\frac{1}{2})\) with radius \(\frac{1}{2}\).

Step 1: Understanding the Polar Equation

The given polar equation is \( r = \sin \theta \). This means the distance \( r \) from the origin to a point on the curve is expressed in terms of \( \theta \), the angle from the positive x-axis. Our goal is to sketch this curve.

Step 2: Identify Range of \( \,\sin \theta \)

The function \( \sin \theta \) oscillates between -1 and 1 for all values of \( \theta \). Thus, the range of \( r \) in the polar equation is from \(-1\) to \(1\). However, because \( r \) as a distance cannot be negative, we consider only \(0 \leq r \leq 1\) for meaningful points in polar coordinates.

Step 3: Plot Key Points for \( r \, = \, 0 \, \text{and} \, 1\)

First, evaluate \( r = \sin \theta \) at key angles. At \( \theta = 0\) and \(\theta = \pi\), \( r = \,\sin(0) = 0\). This means the curve passes through the origin when \( \theta \) is 0 or \( \pi \). At \( \theta = \frac{\pi}{2}\), \( r = 1\). Plot these points on the polar graph.

Step 4: Determine the Shape of the Curve

The curve \( r = \sin \theta \) is a circle. Since \( \sin \theta = 0\) at \( \theta = 0\) and \( \theta = \pi\), and \( \sin \theta = 1\) at \( \theta = \frac{\pi}{2}\), this describes the top half of a unit circle centered at \((0, \frac{1}{2})\) in polar coordinates.

Step 5: Sketch the Half-Circle

With the points and the range identified, sketch the circle from \( \theta = 0 \) to \( \theta = \pi\). It is important to note that it is not the whole circle, but the top half, spanning from the positive x-axis up to the negative x-axis.

Key Concepts

Polar Equations

Polar equations relate the distance from a point to the origin, known as the radial distance \(r\), to the angle \(\theta\), measured from the positive x-axis. Unlike Cartesian coordinates that use \(x\) and \(y\) to describe locations on a plane, polar coordinates use these two variables \(r\) and \(\theta\) to define points. In polar equations like \(r = \sin \theta\), \(r\) changes with the angle, creating different curves when graphed. This makes polar equations useful for modeling rotational symmetries and circular patterns. Understanding these different equations helps in sketching the various curves they define.

Unit Circle

The unit circle is a fundamental concept in trigonometry, centered at the origin of a coordinate plane with a radius of 1. It is defined by the equation \(x^2 + y^2 = 1\) in Cartesian coordinates. When considering its usage in polar coordinates, the circle is traversed by varying \(\theta\), where each angle corresponds to a position on the circle.

• When \(\theta = 0\), the point is at \( (1,0) \).
• As \(\theta\) increases to \(\pi/2\), the point moves to \( (0,1) \).
• By \(\theta = \pi\), it reaches \( (-1,0) \).
• Completing the circle at \(\theta = 2\pi\) returns to \( (1,0) \).

Many polar equations, like \(r = \sin \theta\), use parts or wholes of the unit circle to define their shapes.

Trigonometric Functions

In polar coordinates, trigonometric functions such as \(\sin\), \(\cos\), and \(\tan\) are crucial for defining relationships between angles and distances.

• The sine function, \(\sin \theta\), often correlates with vertical distances, important when constructing polar equations.
• These functions are periodic, meaning they repeat values in a consistent pattern. For example, \(\sin \theta\) goes through a full cycle every \(2\pi\) radians, or 360 degrees.
• In a polar equation like \(r = \sin \theta\), \(r\) is directly tied to the oscillatory nature of \(\sin \theta\), going from values at 0 up to 1 and back.

Graphing in Polar Coordinates

Graphing in polar coordinates involves plotting points based on their distance from the origin \(r\), and direction \(\theta\). This is different from the grid-like representation of Cartesian coordinates and can create unique images such as spirals, roses, and cardioids. When sketching \(r = \sin \theta\), for instance, knowing how \(r\) changes across certain angles is key.

• Since \(r = \sin \theta\) only takes non-negative values from 0 to 1, the curve is plotted from the origin outwards to the furthest distance at each \(\theta\), within these limits.
• The graph effectively becomes the top half of a unit circle, situated between the angles zero and \(\pi\), traced as \(\theta\) varies.
• It requires mapping each part of the function over specified ranges to create a coherent picture, reflecting how trigonometric output determines the radial points from the origin.

"}]}]} } } } } `` `` `` df `` `` `` To complete a graph in polar coordinates, understanding both the visual transformation and the mathematical translation from polar equations becomes essential to reveal the geometrical properties they represent.