Problem 50 - Algebra and Trigonometry Real Mathematics, Real People

Question

Convert the rectangular equation to polar form. Assume \(a<0\) $$y=-\sqrt{3} x$$

Step-by-Step Solution

Answer

The polar form of the given rectangular equation is \(θ = -π/3\) when \(a < 0\).

Step 1: Identifying the Equation and Variables

The given equation is \(y = -\sqrt{3} * x\). We have two variables 'x' and 'y' present in this equation.

Step 2: Applying the Conversion Formula

Now we replace 'x' and 'y' to convert in polar co-ordinates. Remember the equivalents x = r cos θ and y = r sin θ, so we can rewrite the given equation in the form \(r sin θ = -\sqrt{3} * r cos θ\) or \(sin θ / cos θ = -\sqrt{3}\).

Step 3: Simplification of the Equation

Simplify the equation by expressing the equation in form of θ. The ratio \(sin θ / cos θ\) is same as tangent i.e., \(tan θ\). So, the equation becomes \(tan θ = -\sqrt{3}\). Now this is our polar co-ordinate equation. This equation is valid for a<0. For a > 0, the equation becomes \(\tan θ = √3\) or \(θ = π/3\). But for a < 0 as per assumption, \(\tan θ = -√3\) or \(θ = -π/3\)

Key Concepts

Rectangular to Polar Conversion

Rectangular to polar conversion is a process that helps in transforming coordinates of a point from the Cartesian plane (or rectangular form) to the polar plane. In the Cartesian coordinate system, points are identified using two coordinates: x and y, which represent the horizontal and vertical distances respectively, from the origin.

The polar coordinate system, however, determines points through a radius, denoted as 'r', and an angle from a reference direction, \'θ\', measured in radians or degrees. The conversion relies on the fundamental relationships given by:

• \( x = r \cos θ \)
• \( y = r \sin θ \)

In the provided exercise, the equation \( y = -\sqrt{3}x \) needs to be expressed in terms of 'r' and 'θ'. This involves substituting for 'x' and 'y' using the above relationships and simplifying to obtain a polar equation.

Trigonometric Ratios

Trigonometric ratios are crucial in connecting rectangular and polar systems. These ratios come from the sides of a right triangle relative to an angle θ. The three primary trigonometric ratios are:

• Sine (sin), which is the ratio of the opposite side to the hypotenuse.
• Cosine (cos), which is the ratio of the adjacent side to the hypotenuse.
• Tangent (tan), which is the ratio of the opposite side to the adjacent side.

These ratios are represented as:

• \( \sin θ = \frac{\text{opposite}}{\text{hypotenuse}} \)
• \( \cos θ = \frac{\text{adjacent}}{\text{hypotenuse}} \)
• \( \tan θ = \frac{\text{opposite}}{\text{adjacent}} \)

By leveraging these ratios, rectangular coordinates can be seamlessly transformed into the polar system, simplifying the analysis of complex geometrical and physical problems.

Tangent Function

The tangent function, \( \tan \), is one of the most significant trigonometric ratios, especially in converting equations. It's defined as \( \tan θ = \frac{\sin θ}{\cos θ} \). This ratio represents the slope of the line in polar coordinates.

In the context of the exercise, after substituting the rectangular coordinates with polar equivalents, the expression \( \frac{\sin θ}{\cos θ} = -\sqrt{3} \) emerges. Here, the tangent function plays a pivotal role in matching this ratio to a corresponding angle, leading us to the solution.

Solving for \( θ \) gives the polar angle describing the line. For \( \tan θ = -\sqrt{3} \), \( θ \) equals \( -\frac{\pi}{3} \), recognizing that tangent periodicity returns an angle within its curve.

Polar Equations

Polar equations uniquely describe lines, curves, and shapes using polar coordinates. In the example provided, converting \( y = -\sqrt{3}x \) results in a polar equation derived using the tangent function, specifying the angle \'θ\'.

For a given situation where the parameter \( a < 0 \), the solution \( \tan θ = -\sqrt{3} \) corresponds to a distinct angle \( \theta = -\frac{\pi}{3} \). This indicates a line with a specific angular orientation in the polar plane.

Polar equations are exceptionally useful for describing phenomena with inherent symmetry around a point, like circular or spiral patterns, which are otherwise cumbersome in the rectangular form. Converting these equations enhances visualization and comprehension of geometric orientations and motions.