Problem 73 - Contemporary Precalculus

Question

Sketch the graph of the equation. $$r=\sin \theta+\cos \theta$$

Step-by-Step Solution

Answer

Question: Sketch the graph of the polar equation $$r = \sin \theta + \cos \theta$$ and describe its key features. Answer: The graph of the polar equation $$r = \sin \theta + \cos \theta$$ is a tilted kite shape with vertices at (1, 0), (0, 1), (-1, 0), and (0, -1). The function r increases from the point (1, 0) to (0, 1), then decreases to (-1, 0), continues decreasing to (0, -1), and finally increases back to (1, 0), completing one loop.

Step 1: Identify key points

Look for points where \(\theta\) is a multiple of \(\frac{\pi}{2}\), as these are important since sine and cosine functions take their maximum and minimum values at these points. Calculate the corresponding values of r for each of these points: $$r_0 = \sin(0) + \cos(0) = 0+1=1$$ $$r_{\frac{\pi}{2}} = \sin(\frac{\pi}{2}) + \cos(\frac{\pi}{2}) = 1+0=1$$ $$r_{\pi} = \sin(\pi) + \cos(\pi) = 0-1=-1$$ $$r_{\frac{3\pi}{2}} = \sin(\frac{3\pi}{2}) + \cos(\frac{3\pi}{2}) = -1+0=-1$$

Step 2: Determine the behavior of r between the key points

We need to determine whether r is increasing or decreasing between these points. To do this, check the behavior of the derivatives of sine and cosine functions between these points. Since the derivatives of sine and cosine do not depend on the value of r, we just need to analyze the sign of the sum of their derivatives: $$\frac{d}{d\theta}\sin\theta+\frac{d}{d\theta}\cos\theta=\cos\theta-\sin\theta$$ Now, identify the behavior between the key points: 1. \((0 < \theta < \frac{\pi}{2}):\) both \(\sin\theta\) and \(\cos\theta\) are positive, so the sum is positive, and r is increasing. 2. \((\frac{\pi}{2} < \theta < \pi):\) sine is positive and cosine is negative, and sine dominates; the sum is negative, and r is decreasing. 3. \((\pi < \theta < \frac{3\pi}{2}):\) both sine and cosine are negative, so the sum is negative, and r is decreasing. 4. \((\frac{3\pi}{2} < \theta < 2\pi):\) sine is negative and cosine is positive, and cosine dominates; the sum is positive, and r is increasing.

Step 3: Convert into Cartesian coordinates and plot the graph

Convert the polar coordinate points into Cartesian coordinates using the formulas: $$x = r \cos \theta$$ $$y = r \sin \theta$$ For key points: 1. \((\theta=0, r=1): x=1, y=0\) 2. \((\theta=\frac{\pi}{2}, r=1): x=0, y=1\) 3. \((\theta=\pi, r=-1): x=-1, y=0\) 4. \((\theta=\frac{3\pi}{2}, r=-1): x=0, y=-1\) Use these points to draw a smooth curve while keeping in mind the increasing/decreasing behavior of r established in step 2. The graph should start at (1,0), touch the point (0,1), then go to (-1,0), touch the point (0,-1) and finally end at (1,0) completing one loop. The graph should look like a tilted kite shape.

Key Concepts

Graphing Polar Equations

Polar equations provide a different way of representing graphs compared to the usual Cartesian system. In polar coordinates, each point on a graph can be expressed as \(r, \theta\), where \(r\) is the distance from the origin, and \(\theta\) is the angle from the positive x-axis.
This way of visualizing equations can reveal beautiful symmetrical patterns and shapes, often making complex relationships easier to understand. For example, the equation \(r = \sin \theta + \cos \theta\) results in a graph that initially might look complex but actually forms a simple `kite` or `diamond` shape when plotted.
Understanding how these equations behave as \(\theta\) changes allows us to predict how the graph will transform, enhancing our capability to sketch them accurately.

• Identify points where \(r\) is maximized or minimized.
• Analyze the behavior of \(r\) to determine how the graph curves.
• Trace the path in a stepwise manner for a complete figure.

Trigonometric Functions

Trigonometric functions \(\sin\) and \(\cos\) are fundamental elements in trigonometry that describe relationships in triangles. These functions oscillate and repeat over intervals, which can generate periodic behaviors in equations.
For the graph of \(r = \sin \theta + \cos \theta\), calculating \(r\) at key angles like 0, \(\frac{\pi}{2}\), \(\pi\), and \(\frac{3\pi}{2}\) helps highlight where these functions hit peaks and valleys.

• Key Property : The range for both \(\sin \theta\) and \(\cos \theta\) is from -1 to 1.
• Max & Min Points : Occur notably at angles that are multiples of \(\frac{\pi}{2}\).
• Behavior Analysis : Examining the derivatives like \(\cos \theta - \sin \theta\) helps predict if \(r\) increases or decreases.

Recognizing these patterns with trigonometric functions provides insight into the graph's symmetrical nature and makes sketching smoother.

Cartesian Coordinates

The Cartesian coordinate system is widely used and defines points through an x (horizontal) and y (vertical) axis on a plane. This system helps represent polar coordinates in another form, providing a direct method to sketch graphs by plotting exact points.
In converting from polar to Cartesian coordinates, we use:

• Formula : \(x = r \cos \theta\)
• Formula : \(y = r \sin \theta\)

These formulas transform the polar coordinates of key points into a format familiar to many, making it easier to plot them on standard graph paper.
Key points, such as (1, 0) or (0, 1), establish the building blocks of our graph. Recognizing these conversions allows for a more tangible understanding of the geometry and relation within polar graphs.

Converting Coordinates

Converting between polar and Cartesian coordinates is a vital skill in mathematics. This process involves transforming one representation of a point into another, each offering a unique perspective for problem-solving.

• Polar to Cartesian : Using formulas \(x = r \cos \theta\) and \(y = r \sin \theta\) allows precise plotting.
• Cartesian to Polar : Sometimes necessary to find \(r = \sqrt{x^2 + y^2}\) and \(\theta = \tan^{-1}(y/x)\).

Executing these conversions effectively requires attention to angle measurements, ensuring degrees are possibly converted to radians when necessary, and vice-versa. Such skills not only help in graphing but also enhance problem-solving abilities when dealing with geometrical transformations and function analysis.By mastering the ability to transition between coordinate systems, students can approach equations from multiple angles, providing deeper insight and greater flexibility in handling various mathematical challenges.