Problem 15 - Precalculus

Question

Convert to rectangular form. $$r \cos \left(\theta-\frac{\pi}{6}\right)=2$$

Step-by-Step Solution

Answer

The rectangular form is \(\sqrt{3}x + y = 4\).

Step 1: Understand the Rectangular and Polar Forms

The problem involves converting from polar to rectangular form. In polar form, a point is represented as \((r, \theta)\), where \(r\) is the distance from the origin, and \(\theta\) is the angle from the positive x-axis. In rectangular form, it's represented as \((x, y)\), using the equations: \(x = r \cos \theta\) and \(y = r \sin \theta\).

Step 2: Expand the Given Expression

The given expression is \(r \cos \left(\theta - \frac{\pi}{6}\right) = 2\). Using the angle subtraction identity, \(\cos(a-b) = \cos a \cos b + \sin a \sin b\), we expand it to: \[\cos \theta \cos \frac{\pi}{6} + \sin \theta \sin \frac{\pi}{6} = 2/r\].

Step 3: Substitute Trigonometric Values

Substitute the known trigonometric values: \(\cos \frac{\pi}{6} = \frac{\sqrt{3}}{2}\) and \(\sin \frac{\pi}{6} = \frac{1}{2}\). This gives:\[r \left( \cos \theta \times \frac{\sqrt{3}}{2} + \sin \theta \times \frac{1}{2} \right) = 2\].

Step 4: Convert to Rectangular Coordinates

In polar to rectangular conversion, \(r \cos \theta = x\) and \(r \sin \theta = y\). Substituting these gives us:\[\frac{\sqrt{3}}{2} x + \frac{1}{2} y = 2\].

Step 5: Multiply by 2 to Simplify the Equation

By multiplying the entire equation by 2, we eliminate fractions:\[\sqrt{3} x + y = 4\.\] This is the rectangular form of the equation.

Key Concepts

Rectangular Coordinates

Rectangular coordinates are a fundamental part of representing points in a two-dimensional plane. They are often described as ordered pairs of numbers:

• \((x, y)\), where \(x\) is the horizontal coordinate, representing the distance along the x-axis.
• \(y\), represents the vertical coordinate, measuring the distance along the y-axis.

Together, these coordinates provide a precise location on the Cartesian plane. One common use of rectangular coordinates is in converting polar coordinates, which describe a point in terms of its distance from the origin (radius \(r\)) and its angle (\(\theta\)) with the positive x-axis. The conversion is done using the formulas:

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

These formulas help translate polar expressions into a form that highlights the numeric position of a point in space.In practical applications, rectangular coordinates are widely used in fields such as physics, engineering, and computer graphics to model, simulate, and analyze spatial relationships. For the given expression, writing it in rectangular form involves calculating specific numeric representations \((x, y)\), making the point of interest more tangible in Euclidean space.

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that hold true for all values of the included variables. These identities serve as important tools in mathematics,
allowing us to simplify expressions and solve equations involving trigonometric functions. Some of the most commonly used identities include:

• Pythagorean Identity: \(\sin^2\theta + \cos^2\theta = 1\)
• Reciprocal Identities: \(\sin \theta = \frac{1}{\csc \theta}\), \(\cos \theta = \frac{1}{\sec \theta}\), and \(\tan \theta = \frac{1}{\cot \theta}\)
• Angle Sum and Difference Identities: such as \(\cos(a \pm b) = \cos a \cos b \mp \sin a \sin b\)

In the context of converting the given polar equation,
we used the angle subtraction identity specifically for cosine, expanded as:\[\cos(a - b) = \cos a \cos b + \sin a \sin b\]This identity allows us to break down trigonometric expressions into sums or differences of products, which can be further simplified or reconfigured, making them easier to work with. Such identities are not only theoretical constructs, but they also have practical utility in a wide range of applications, from solving trigonometric equations to simplifying complex expressions in calculus and beyond.

Angle Subtraction Identity

The angle subtraction identity is a valuable formula in trigonometry that simplifies calculations involving the differences between angles.
This identity is derived from the cosine and sine angle-sum formulas, and it comes in two forms:

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

For the specific case at hand, the cosine version of the angle subtraction identity is utilized.
This application helps rewrite expressions where one angle is subtracted from another, allowing for easier evaluation by expressing the result in terms of the individual trigonometric components for each angle. Using known trigonometric values, specifically \( \cos \frac{\pi}{6} = \frac{\sqrt{3}}{2} \) and \( \sin \frac{\pi}{6} = \frac{1}{2} \), the problem becomes solvable with routine algebraic manipulation. These identities not only assist in solving equations but also further understanding of angle relationships, crucial for deeper studies in mathematics and physics. The systematic breakdown of seemingly complex trigonometric expressions showcases their elegance and utility in simplifying and solving real-world problems.