Problem 51 - 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. $$r \sin \left(\theta+\frac{\pi}{6}\right)=2$$

Step-by-Step Solution

Answer

The Cartesian form is \( \sqrt{3}y + x = 4 \), a straight line.

Step 1: Recall Polar-to-Cartesian Conversion Formulas

Polar coordinates are represented by \( r \) and \( \theta \). To convert polar equations to Cartesian coordinates, we use the formulas: \( x = r\cos(\theta) \) and \( y = r\sin(\theta) \). Additionally, there is \( r^2 = x^2 + y^2 \).

Step 2: Expand the Angle Using Trigonometric Identities

The given polar equation is \( r \sin\left(\theta + \frac{\pi}{6}\right) = 2 \). We use the angle addition identity for sine: \( \sin(a + b) = \sin a \cos b + \cos a \sin b \). So \( \sin\left(\theta + \frac{\pi}{6}\right) = \sin\theta\cdot\cos\left(\frac{\pi}{6}\right) + \cos\theta\cdot\sin\left(\frac{\pi}{6}\right) \).

Step 3: Simplify Using Exact Trigonometric Values

The values for \( \cos\left(\frac{\pi}{6}\right) \) and \( \sin\left(\frac{\pi}{6}\right) \) are \( \frac{\sqrt{3}}{2} \) and \( \frac{1}{2} \), respectively. Substitute in the identity: \( \sin\left(\theta + \frac{\pi}{6}\right) = \sin\theta\cdot\frac{\sqrt{3}}{2} + \cos\theta\cdot\frac{1}{2} \).

Step 4: Multiply Both Sides by \( r \)

Substitute back into the equation: \( r(\sin\theta\cdot\frac{\sqrt{3}}{2} + \cos\theta\cdot\frac{1}{2}) = 2 \). Distribute \( r \): \( r\sin\theta\cdot\frac{\sqrt{3}}{2} + r\cos\theta\cdot\frac{1}{2} = 2 \).

Step 5: Substitute Cartesian Equivalents

Recall \( y = r\sin\theta \) and \( x = r\cos\theta \). Substitute these into the equation: \( y\cdot\frac{\sqrt{3}}{2} + x\cdot\frac{1}{2} = 2 \).

Step 6: Solve for Cartesian Equation

Multiply through by 2 to clear fractions: \( \sqrt{3}y + x = 4 \). This is the Cartesian form of the given polar equation.

Step 7: Identify the Graph

The equation \( \sqrt{3}y + x = 4 \) is in the form of a linear equation \( Ax + By = C \), where \( A = 1 \), \( B = \sqrt{3} \), and \( C = 4 \). This represents a straight line.

Key Concepts

Trigonometric Identities

Trigonometric identities are fundamental tools in mathematics that simplify expressions involving trigonometric functions. These identities come in various forms but most notably include formulas for angle addition and subtraction. In the polar to Cartesian conversion exercise, we utilized the angle addition identity for sine:

• \(\sin(a + b) = \sin a \cos b + \cos a \sin b\)

This identity is pivotal in expanding the expression \( \sin\left(\theta + \frac{\pi}{6}\right) \). By breaking down this equation into simpler components, we can handle complex trigonometric equations more easily.
Using specific trigonometric values such as \(\cos\left(\frac{\pi}{6}\right) = \frac{\sqrt{3}}{2}\) and \(\sin\left(\frac{\pi}{6}\right) = \frac{1}{2}\) helps in further simplifying the equation. These exact values are commonly memorized for efficient calculations, and they stem from the geometry of special triangles, such as the 30-60-90 triangle.

Linear Equations

Linear equations are expressions that represent straight lines when graphed on a coordinate system. These equations are generally structured as \(Ax + By = C\), where \(A\), \(B\), and \(C\) are constants. In the Cartesian equation \( \sqrt{3}y + x = 4 \), obtained from the original polar equation, we identify it as a linear equation.
Linear equations are crucial in mathematics and various applications due to their straightforward nature. Solving for one variable can reveal specific properties, such as the slope and intercept of the line. In our example:

• The slope \(m\) is \(-\frac{1}{\sqrt{3}}\) when the equation is put into slope-intercept form.
• The y-intercept occurs where the line crosses the y-axis, providing key insight into how the line behaves in space.

Linear equations form the foundation for more complex algebraic structures and are vital to understanding functions in general.

Coordinate Systems

Coordinate systems allow us to describe the location of points in space using numbers. The most common systems are the Cartesian and polar coordinates, each having distinct uses and advantages. Cartesian coordinates, expressed as \( (x, y) \), provide a straightforward method of plotting points on a grid. Meanwhile, the polar system, which uses \( (r, \theta) \), is beneficial when dealing with circular or rotational contexts.
Converting from polar to Cartesian involves the conversion formulas:

• \( x = r \cos(\theta) \)
• \( y = r \sin(\theta) \)
• Additionally, the relation \( r^2 = x^2 + y^2 \) links these two systems.

This transformation captures how angles and radii relate to the grid-like interface of Cartesian coordinates. Understanding these systems is fundamental in math, as they offer different perspectives and can simplify problem-solving depending on the context of a problem.