Question

Find the points of intersection of the graphs of the equations. $$ \begin{array}{l} r=3+\sin \theta \\ r=2 \csc \theta \end{array} $$

Short answer

The point of intersection of the graphs of the equations \(r = 3 + \sin \theta \) and \(r = 2 \csc \theta \) is (4, \(\pi/2\)).

Step-by-step solution

Convert the Equation

First, we need to convert the second equation, \(r = 2 \csc \theta \), into a similar trigonometric form like the first one. Because \(\csc \theta = 1/\sin \theta\), the equation can be rewritten as \(r = 2 / \sin \theta\).

Equating the Two Equations

Next, we equate the two equations to find \(\theta\): \(3 + \sin \theta = 2 / \sin \theta.\) Here, we are assuming that \(\sin \theta \neq 0\), because \(\sin \theta = 0\) leads to undefined values in the second equation.

Solve for Theta

Multiply the equations by \(\sin \theta\) to remove the denominator in the right-hand side: \(3\sin \theta + \sin^2 \theta = 2.\) Rearrange the terms to get a quadratic equation: \(\sin^2 \theta + 3\sin \theta - 2 = 0\). This quadratic equation can be factored: \((\sin \theta - 1)(\sin \theta + 2) = 0\). Therefore, the solutions are \(\sin \theta = 1\) and \(\sin \theta = -2\). Since \(-1 \leq \sin \theta \leq 1\), \(\sin \theta = -2\) has no solutions.

Find the Intersection Point(s)

Finally, substitute \(\sin \theta = 1\) into the original equations to find the intersection point(s). The intersection point is (r, \(\theta\)) = (4, \(\pi/2\)).

Key concepts

Trigonometric Identities

Understanding trigonometric identities is crucial when dealing with trigonometric equations, especially when we are asked to find points of intersection. A trigonometric identity is an equality that holds for all values of the variable for which both sides of the equality are defined. Identities can simplify complex trigonometric expressions and help us to solve equations.

Common trigonometric identities include reciprocal identities like \(\frac{1}{\text{sin} \theta} = \text{csc} \theta\), ratio identities like \(\text{tan} \theta = \frac{\text{sin} \theta}{\text{cos} \theta}\), and Pythagorean identities such as \(\text{sin}^2 \theta + \text{cos}^2 \theta = 1\). In our exercise, recognizing that \(\text{csc} \theta\) is the reciprocal of \(\text{sin} \theta\) was essential in converting one of the polar equations so that both equations involved \(\text{sin} \theta\), enabling us to solve for the points of intersection.

Solving Quadratic Equations

Solving quadratic equations is a staple in algebraic manipulation, and it is often required in various fields of mathematics, including trigonometry. A quadratic equation is typically written in the form \(ax^2 + bx + c = 0\). There are multiple methods to solve such equations: factoring, completing the square, and using the quadratic formula.

When we encounter a quadratic trigonometric equation like \(\text{sin}^2 \theta + 3\text{sin} \theta - 2 = 0\), we treat \(\text{sin} \theta\) as a variable and apply the same methods. In the given exercise, the equation factored neatly into \((\text{sin} \theta - 1)(\text{sin} \theta + 2) = 0\), leading to potential solutions for \(\text{sin} \theta\). However, it's important to remember the domain restrictions of trigonometric functions, as not all solutions may be valid in the context of trigonometry.

Graphing Polar Equations

Graphing polar equations is different from graphing Cartesian equations because it requires a different coordinate system. In the polar coordinate system, points are determined by a distance from the origin, denoted as \(r\), and an angle \(\theta\) from the positive x-axis.

An equation such as \(r=3+\sin \theta\) represents a curve in the polar coordinate system. When graphing polar equations, we plot points for various values of \(\theta\) and connect these points to create the curve. It's important to recognize patterns, such as symmetry and periodicity, which are common in trigonometric functions.

To find the points of intersection between two polar equations, we set them equal to each other, as we did in this exercise. After algebraic manipulations, we ultimately plot the intersection points on the polar grid. The ability to graph polar equations aids in visualizing solutions and understanding the relationship between different trigonometric curves.