Question

In Exercises 13-20, find the points of intersection of the graphs of the equations. $$ \begin{array}{l} r=1+\cos \theta \\ r=1-\sin \theta \end{array} $$

Short answer

The points of intersection are \((1 - \sqrt{2}/2 , \frac{3\pi}{4})\) and \((1 + \sqrt{2}/2 , \frac{7\pi}{4})\)

Step-by-step solution

Setting Equations Equal

For the two polar graphs \(r = 1 + \cos \theta\) and \(r = 1 - \sin \theta\) to intersect, their r-values must be equal at the same \(\theta\). This means we set the equations equal: \(1 + \cos \theta = 1 - \sin \theta\).

Rearrange the Equation

Next, isolate the trigonometric functions on either side of the equation: \(\cos \theta + \sin \theta = 0.\)

Solve the Equation

This equation is equivalent to \(\tan \theta = -1\). It has solutions \(\theta = \frac{3\pi}{4}, \frac{7\pi}{4}\) within the interval \([0, 2\pi]\)

Find the Corresponding Points

By substituting these \(\theta\)-values back into either of the original equations, we can find the corresponding r-values. When \(\theta = \frac{3\pi}{4}\), \(r = 1 + \cos \frac{3\pi}{4} = 1 - \sqrt{2}/2 = 1 - \sqrt{2}/2\). When \(\theta = \frac{7\pi}{4}\), \(r = 1 + \cos \frac{7\pi}{4} = 1 + \sqrt{2}/2 = 1 + \sqrt{2}/2\)

Key concepts

Polar Coordinates

Polar coordinates are a two-dimensional coordinate system that specify the location of a point based on its distance from a reference point called the pole (similar to the origin in Cartesian coordinates) and the angle from a reference direction, usually the positive x-axis. A point in polar coordinates is represented as \( (r, \theta) \), where \( r \) is the radial distance and \( \theta \) is the polar angle.

In the context of the given exercise, we are interested in determining where two polar graphs intersect, which involves finding the set of polar coordinates that satisfy two different polar equations simultaneously. The method in the solution involves equating the two values for \( r \) and solving for \( \theta \) to find the points of intersection.

Trigonometric Equations

Trigonometric equations are mathematical expressions that involve trigonometric functions like sine, cosine, and tangent. Solving such equations usually requires the application of various algebraic and trigonometric identities. Understanding the periodic nature and the fundamental properties of these functions is crucial in finding all possible solutions within a given interval.

In the exercise provided, we are given the trigonometric equation \( 1 + \cos \theta = 1 - \sin \theta \) derived from setting the two polar equations equal to each other. The task is then to manipulate and solve the equation to find all the values of \( \theta \) within the specified range. Central to solving such equations is the knowledge of trigonometric relationships and the unit circle.

Solving Trigonometric Identities

To solve trigonometric identities, one must often manipulate the given equation using algebraic techniques and known trigonometric identities to make the equation more solvable. This may involve factoring, combining like terms, or using identities such as the Pythagorean identity.

For instance, in the provided step-by-step solution, the trigonometric equation \( \cos \theta + \sin \theta = 0 \) was rewritten as \( \tan \theta = -1 \) by dividing both sides by \( \cos \theta \) (assuming \( \cos \theta eq 0 \) to avoid division by zero). This step made use of the fact that \( \tan \theta = \sin \theta / \cos \theta \). Recognizing such relationships and knowing when to apply them is integral to solving trigonometric identities.

Polar Equations Intersection Points

Finding the points of intersection between two polar graphs involves determining the sets of \( (r, \theta) \) coordinates that satisfy both equations simultaneously. This is typically achieved by setting the two equations for \( r \) equal to each other and solving for the angle \( \theta \) first. Once the angles of intersection are found, these angles are used to calculate the corresponding radial distances to acquire the full polar coordinates of the intersection points.

In our exercise, after finding the angles of intersection as \( \frac{3\pi}{4} \) and \( \frac{7\pi}{4} \) by solving the trigonometric equation, we plug these angles back into the original equations to get the corresponding \( r \) values. This method yields the complete intersection points of the polar equations, providing us with the exact locations where the two graphs cross paths on the polar coordinate plane.