Problem 76 - Algebra and Trigonometry Real Mathematics, Real People

Question

Convert the polar equation to rectangular form. $$r=-3 \cos 2 \theta$$

Step-by-Step Solution

Answer

The rectangular coordinates of the equation are \(x = -3 \cos\theta \cos 2\theta\) and \(y = -3 \sin\theta \cos 2\theta\).

Step 1: Substitution of rectangular coordinates

Substitute \(-3\cos 2\theta\) for \(r\) in the formulas \(x = r\cos\theta\) and \(y = r\sin\theta\).

Step 2: Determine x-coordinate

Substitute \(-3 \cos 2\theta\) for \(r\) in the formula \(x = r\cos\theta\) to get \(x = -3 \cos 2\theta \cos\theta\).

Step 3: Determine y-coordinate

Substitute \(-3 \cos 2\theta\) for \(r\) in the formula \(y = r\sin\theta\) to get \(y = -3 \cos 2\theta \sin\theta\).

Step 4: Simplify

After simplification, we obtain the final rectangular form of the equation: \(x = -3 \cos\theta \cos 2\theta\) and \(y = -3 \sin\theta \cos 2\theta\).

Key Concepts

Polar Coordinates

Polar coordinates are a bit different from the typical rectangular coordinates you might be used to. Instead of describing a point using horizontal and vertical distances (like x and y), polar coordinates use a distance from a central point (known as the origin) and an angle from a reference direction, usually the positive x-axis.

• In polar coordinates, each point is represented as \((r, \theta)\) where \(r\) is the radius or the distance from the origin.
• \(\theta\) is the angle in radians or degrees, often measured counterclockwise from the positive x-axis.

Using these coordinates can be especially powerful for graphing circles, spirals, and other shapes where radius and angle provide a more natural description.

Rectangular Coordinates

Rectangular coordinates, also known as Cartesian coordinates, are used to describe the position of points on a plane based on their horizontal (x) and vertical (y) distances from an origin point, typically where the x and y axes intersect. These coordinates form a grid that can easily represent shapes and plotting points.

• The x-coordinate measures how far left or right a point is from the origin.
• The y-coordinate indicates how far up or down a point is from the origin.

When converting from polar to rectangular coordinates, expressions involving \(r\) and \(\theta\) in polar form are transformed into functions of \(x\) and \(y\). This involves substituting \(r \cos \theta\) for \(x\) and \(r \sin \theta\) for \(y\).

Trigonometric Identities

Trigonometric identities are fundamental relationships between trigonometric functions that hold true for all angle measures. These identities can simplify the conversion between different coordinate systems or solve trigonometric equations.
Some common identities include:

• \(\sin^2 \theta + \cos^2 \theta = 1\)
• \(\tan \theta = \frac{\sin \theta}{\cos \theta}\)

For converting polar equations like \(r = -3 \cos 2\theta\) into rectangular form, knowing identities like \(\cos 2\theta = \cos^2 \theta - \sin^2 \theta\) can be particularly helpful. These identities allow us to express the function in terms of \(\cos \theta\) and \(\sin \theta\), which are directly related to \(x\) and \(y\) in rectangular coordinates. Thus, they are critical tools for simplifying the expressions involved in the conversion.