Question

A Cartesian and a polar graph of \(r=f(\theta)\) are given in the figures. Mark the points on the polar graph that correspond to the points shown on the Cartesian graph. $$r=1-2 \sin 3 \theta$$

Short answer

Answer: To determine which points on the Cartesian graph correspond to the points on the polar graph, follow these steps: 1. Convert the polar equation \(r = 1 - 2\sin{3\theta}\) to Cartesian coordinates. 2. Identify the points on the Cartesian graph and note their coordinates. 3. Calculate the corresponding polar coordinates for each Cartesian point. 4. Check if any points on the polar graph have the same coordinates as the points on the Cartesian graph. 5. Mark the corresponding points on the polar graph that have the same coordinates as those on the Cartesian graph.

Step-by-step solution

Convert the polar equation to Cartesian coordinates

To convert the polar equation \(r = 1 - 2\sin{3\theta}\) to Cartesian coordinates, we will use the following conversion formulas: \(x = r\cos{\theta}\) and \(y = r\sin{\theta}\). First, find the expressions for \(x\) and \(y\) in terms of \(\theta\): \(x = (1 - 2\sin{3\theta})\cos{\theta}\), \(y = (1 - 2\sin{3\theta})\sin{\theta}\).

Identify the points on the Cartesian graph

Now, we need to identify the specific points on the Cartesian graph that are given. Note down their coordinates \((x_i, y_i)\) for each point.

Calculate the corresponding polar coordinates for each Cartesian point

For each Cartesian point \((x_i, y_i)\), calculate the corresponding polar coordinates \((r_i, \theta_i)\). To calculate \(r_i\) and \(\theta_i\), use the formulas: \(r_i = \sqrt{x_i^2 + y_i^2}\), and \(\theta_i = \arctan{\frac{y_i}{x_i}}\) (pay attention to the quadrant of the point to determine the correct angle).

Find the points that have the same coordinates in both graphs

With the polar coordinates \((r_i, \theta_i)\) of the given points on the Cartesian graph, check whether any points on the polar graph have the same coordinates. To do this, substitute the calculated polar coordinates into the polar equation, \(r = 1 - 2\sin{3\theta}\), and see if they satisfy the equation: \(r_i = 1 - 2\sin{3\theta_i}\). If the equation holds true, then the point \((r_i, \theta_i)\) is the same as \((x_i, y_i)\) on the Cartesian graph.

Mark the corresponding points on the polar graph

Finally, mark the points that have been found to have the same coordinates in both graphs. These are the points on the polar graph that correspond to the points shown on the Cartesian graph.

Key concepts

Cartesian Coordinates

Cartesian coordinates are a fundamental system used to locate points in a plane using ordered pairs \(x, y\). This system operates using two perpendicular axes: the x-axis (horizontal) and the y-axis (vertical). Each point in the plane is represented as a distance from these two axes.

Here's a simple way to understand Cartesian coordinates:

• X-coordinate (abscissa): Measures the distance horizontally from the y-axis.
• Y-coordinate (ordinate): Measures the distance vertically from the x-axis.

By understanding these components, you can locate any position within the Cartesian plane. In the context of converting polar equations to Cartesian coordinates, you use the relations \( x = r \cos{\theta} \) and \( y = r \sin{\theta} \).

These formulas allow translation between polar and Cartesian systems, reflecting aspects such as direction and distance from a fixed point, the origin.

Coordinate Conversion

Coordinate conversion is the process of translating coordinates from one system to another, which is crucial for interpreting graphs between polar and Cartesian systems.

In polar coordinates, a point is defined by the radius \(r\) and the angle \(\theta\). Conversion becomes necessary when you have a polar equation such as \(r = 1 - 2 \sin 3\theta \) and you need the equivalent representation in Cartesian form.

Here's a quick guide:

• To find the Cartesian equivalent: Use \(x = r \cos{\theta}\) and \(y = r \sin{\theta}\).
• To find the Polar equivalent from Cartesian: Use \(r = \sqrt{x^2 + y^2}\) and \(\theta = \arctan{\frac{y}{x}}\). Remember to consider the quadrant to adjust \(\theta\).

The conversion highlights the benefits of each system. While Cartesian coordinates are great for straight lines and grids, polar coordinates simplify expressions involving circles and angles.

Trigonometric Functions

Trigonometric functions play a vital role when dealing with polar coordinates, as they relate angles to the lengths of sides in right triangles. This relationship is helpful in many applications, including coordinate conversion.

Key functions include:

• Sine (\(\sin\)): Opposite side over hypotenuse. Used in the equation \( y = r \sin{\theta} \).
• Cosine (\(\cos\)): Adjacent side over hypotenuse. Used in the equation \( x = r \cos{\theta} \).
• Tangent (\(\tan\)): Opposite side over adjacent side. Important for finding angles \(\theta\) using \(\theta = \arctan{\frac{y}{x}} \).

These functions transform coordinate relationships by connecting geometric interpretations with algebraic expressions.

Using these trigonometric relations is essential for converting coordinates and understanding the physical meaning of equations like \(r = 1 - 2 \sin 3\theta \). They build the bridge between the angle representation and the x-y coordinate framework.