Problem 52 - Thomas Calculus

Question

Replace the polar equations in Exercises \(27-52\) with equivalent Cartesian Replace the polar equations in Exercises \(27-52\) with equivalent Cartesian equations. Then describe or identify the graph. $$r \sin \left(\frac{2 \pi}{3}-\theta\right)=5$$

Step-by-Step Solution

Answer

The Cartesian equation is \(\sqrt{3}x + y = 10\), which is a line.

Step 1: Rewrite the Polar Equation

The given polar equation is \( r \sin\left(\frac{2\pi}{3}-\theta\right)=5 \). This equation requires simplification using known trigonometric identities.

Step 2: Apply Trigonometric Identity

We use the angle subtraction identity for sine: \(\sin(a-b) = \sin a \cos b - \cos a \sin b. \)So, in our equation: \(a=\frac{2\pi}{3}\) and \(b=\theta\), we get:\[\sin\left(\frac{2\pi}{3}-\theta\right) = \sin\frac{2\pi}{3}\cos\theta - \cos\frac{2\pi}{3}\sin\theta.\]We know \(\sin\frac{2\pi}{3} = \frac{\sqrt{3}}{2}\) and \(\cos\frac{2\pi}{3} = -\frac{1}{2}\). Substitute these values into your equation.

Step 3: Substitute Values and Simplify

Substituting the values, the equation becomes:\[r\left(\frac{\sqrt{3}}{2}\cos\theta + \frac{1}{2}\sin\theta\right) = 5.\] Upon rearranging it:\[\frac{\sqrt{3}}{2}r\cos\theta + \frac{1}{2}r\sin\theta = 5.\]

Step 4: Convert to Cartesian Coordinates

Cartesian coordinates relate to polar coordinates by \(x = r\cos\theta\) and \(y = r\sin\theta\). Substitute these into the equation:\[\frac{\sqrt{3}}{2}x + \frac{1}{2}y = 5.\]Simplifying gives:\[x\sqrt{3} + y = 10.\]

Step 5: Identify the Graph

The equation \(x\sqrt{3} + y = 10\) is a linear equation in terms of \(x\) and \(y\). This represents a straight line with a slope of \(-\sqrt{3}\) and a y-intercept of \(10\).

Key Concepts

Polar to Cartesian Conversion

When working with equations, sometimes we need to change from polar to Cartesian coordinates. Polar coordinates use a radius \(r\) and an angle \(\theta\) to describe a point in the plane. On the other hand, Cartesian coordinates use \(x\) and \(y\) positions. Knowing how to convert between these systems is crucial.

The basic formulas for conversion are:

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

To convert our equation from the problem, we replace \(r\cos\theta\) and \(r\sin\theta\) with \(x\) and \(y\). This helps translate the equation into the more familiar form we can graph or manipulate using Cartesian coordinates.

For example, in our exercise, we initially deal with the polar form \(r \sin\left(\frac{2\pi}{3}-\theta\right)=5\). By using trigonometric identities and substituting the conversion formulas, we transform it into \(x \sqrt{3} + y = 10\), which is a linear equation in Cartesian form.

Trigonometric Identities

Trigonometric identities are fundamental tools used to simplify expressions and solve equations involving angles. In our exercise, we utilized the angle subtraction identity for sine:

\[\sin(a-b) = \sin a \cos b - \cos a \sin b.\]
This identity helps break down complex trigonometric expressions into simpler components based on known angle values.

In the specific problem given, we have:

• \(a = \frac{2\pi}{3}\)
• \(b = \theta\)

Using this identity, the expression \(\sin\left(\frac{2\pi}{3}-\theta\right)\) transforms into a manageable form involving terms \(\frac{\sqrt{3}}{2}\) and \(\frac{1}{2}\). This substitution lets us rewrite and work with the equation effectively, paving the way for converting into Cartesian form.

Mastering these identities is vital as they often appear in calculus, physics, and engineering problems, where trigonometric simplifications are needed.

Linear Equations

Linear equations describe straight lines in a Cartesian plane, expressed in the form \(ax + by = c\). They are one of the simplest and most fundamental types of equations you'll encounter in algebra.

In the converted equation from our exercise, \(x \sqrt{3} + y = 10\), we recognize it as a linear equation. Here:

• \(a = \sqrt{3}\)
• \(b = 1\)
• \(c = 10\)

The equation directly tells us about the slope and position of the line on the graph. Specifically, the slope, which is \(-\frac{\sqrt{3}}{1}\), informs us how steep the line is. The y-intercept of 10 indicates where this line crosses the y-axis.

Linear equations are widely used because they model a constant rate of change. Recognizing them in different forms, like the one derived from polar coordinates, is an essential skill in math and related fields.