Problem 15 - Essential Calculus

Question

Find the points of intersection of the graphs of the equations. $$ \begin{array}{l} r=4-5 \sin \theta \\ r=3 \sin \theta \end{array} $$

Step-by-Step Solution

Answer

The points of intersection are at at (\(\frac{3}{2}, \frac{\pi}{6}\)) and (\(\frac{3}{2}, \frac{5\pi}{6}\)).

Step 1: Set Equations Equal

Start by setting the two given equations equal to each other and solve for \(\theta\): \[4 - 5 \sin \theta = 3 \sin \theta\]

Step 2: Simplify and Solve for 'sin \(\theta\)'

Rearrange the equation to isolate 'sin \(\theta\)': \[5 \sin \theta + 3 \sin \theta = 4 \] \[\sin \theta = \frac{4}{8} = \frac{1}{2}\]

Step 3: Find values of \(\theta\)

Now that we have \(\sin \theta = \frac{1}{2}\), we can find \(\theta\) using knowledge of common angles. We get two values as sin is positive in first and second quadrant. So, \(\theta = \frac{\pi}{6}, \frac{5\pi}{6}\).

Step 4: Substitute \(\theta\) into original equations

Substitute values of \(\theta\) into the original equations to get the corresponding \(r\) values: For \(\theta = \frac{\pi}{6}\), substitute into \(r=3 \sin \theta\), we get \(r=3 \sin \frac{\pi}{6} = \frac{3}{2}\). Similarly, for \(\theta = \frac{5\pi}{6}\), \(r=3 \sin \frac{5\pi}{6} = \frac{3}{2}\).

Key Concepts

Intersection of Graphs

The intersection of graphs refers to the points where two or more graphs meet or cross each other. In the context of polar coordinates, these intersections are expressed in terms of radius, \(r\), and angle, \(\theta\). To find these intersection points, we often set the equations for the graphs equal to each other and solve for \(\theta\), which gives us the angle at which the intersections occur.
Once \(\theta\) is known, we substitute it back into the original polar equations to determine the corresponding \(r\) values, thereby finding the points of intersection in polar form. This process involves simplifying the equations to find angles common to both graphs.
When dealing with curves like circles and limacons in polar coordinates, identifying intersection points helps us understand their spatial relationships, and solve graph-related problems in calculus and geometry.

Trigonometric Equations

Trigonometric equations involve trigonometric functions like sine, cosine, and tangent and can be solved by finding values of the variable that make the equation true. In our exercise, we have the equation \(4 - 5 \sin \theta = 3 \sin \theta\), which is a trigonometric equation in terms of \(\sin \theta\).
To solve it:

• We first rearrange terms to isolate \(\sin \theta\) — combining \(4\) and \(-5 \sin \theta\) with \(3 \sin \theta\).
• This results in \(8 \sin \theta = 4\).
• Simplifying further gives us \(\sin \theta = \frac{1}{2}\).

Known solutions for \(\sin \theta = \frac{1}{2}\) are \(\theta = \frac{\pi}{6}\) and \(\theta = \frac{5\pi}{6}\), since sine is positive in the first and second quadrants. Recognizing these standard angles is crucial for solving trigonometric equations efficiently.

Radian Measure

Radian measure is a method for measuring angles based on the radius of a circle. It's an essential concept in trigonometry and calculus. A radian represents the angle created when the arc length is equal to the radius. In other words, there are \(2\pi\) radians in a full circle, equivalent to \(360\) degrees.
Using radian measure allows for a simpler integration and differentiation of trigonometric functions compared to degrees. In the context of the problem, angles \(\theta = \frac{\pi}{6}\) and \(\theta = \frac{5\pi}{6}\) are used to solve the trigonometric equation.
For beginners, it might be helpful to memorize some key radian-to-degree conversions:

• \(\frac{\pi}{6} = 30^\circ\)
• \(\frac{\pi}{3} = 60^\circ\)
• \(\frac{\pi}{2} = 90^\circ\)
• \(\pi = 180^\circ\)

Understanding radians boosts comprehension and ease when applying trigonometric concepts in mathematical problems.