Problem 30 - A Graphical Approach to Precalculus with Limits

Question

For each expression, (a) give the exact value and (b) if the exact value is irrational, use your calculator to support your answer in part (a) by finding a decimal approximation. $$\cot \frac{\pi}{3}$$

Step-by-Step Solution

Answer

The exact value of \( \cot \frac{\pi}{3} \) is \( \frac{\sqrt{3}}{3} \), and its decimal approximation is approximately 0.57735.

Step 1: Determine Cotangent Definition

The cotangent of an angle is the reciprocal of the tangent of that angle. That is, \( \cot \theta = \frac{1}{\tan \theta} \). We need to first find \( \tan \frac{\pi}{3} \).

Step 2: Find Tangent of \( \frac{\pi}{3} \)

From trigonometric identities, we know that \( \tan \frac{\pi}{3} = \sqrt{3} \).

Step 3: Calculate Cotangent Value

Using the formula for cotangent, calculate \( \cot \frac{\pi}{3} = \frac{1}{\tan \frac{\pi}{3}} = \frac{1}{\sqrt{3}} \).

Step 4: Rationalize the Denominator

Rationalizing the denominator involves multiplying the numerator and the denominator by \( \sqrt{3} \). Therefore, \( \frac{1}{\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}} = \frac{\sqrt{3}}{3} \).

Step 5: Provide Decimal Approximation

Using a calculator, we find the decimal approximation of \( \frac{\sqrt{3}}{3} \approx 0.57735 \) to five decimal places.

Key Concepts

Cotangent

Cotangent is a fundamental concept in trigonometry. It is one of the six basic trigonometric functions. It is derived from the tangent function. Specifically, the cotangent of an angle in a right triangle is defined as the ratio of the adjacent side length to the opposite side length. This can be expressed as:

• \( \cot \theta = \frac{1}{\tan \theta} \)

This means cotangent is the reciprocal of the tangent function. When you have a specific angle, like \( \frac{\pi}{3} \), the cotangent can be simplified by first finding the tangent of that angle and then taking its reciprocal.
For \( \frac{\pi}{3} \), we use known trigonometric values: \( \tan \frac{\pi}{3} = \sqrt{3} \). Therefore, \( \cot \frac{\pi}{3} = \frac{1}{\sqrt{3}} \).
This is often used in various calculations in geometry, physics, and engineering to solve problems involving right triangles and modeling cyclic phenomena.

Decimal Approximation

Decimal approximation is a useful mathematical tool that helps us work with irrational numbers. Irrational numbers are those that cannot be expressed as simple fractions and have non-repeating, non-ending decimal parts.

• A prime example would be \( \sqrt{3} \), which appears in the expression \( \cot \frac{\pi}{3} = \frac{\sqrt{3}}{3} \).

To approximate irrational numbers, we use a calculator to get decimal values that are close enough for practical purposes. For example, the decimal approximation of \( \frac{\sqrt{3}}{3} \) is approximately 0.57735.
This kind of approximation is fundamental in fields where accurate but not exact calculations are deemed sufficient, like in statistics, engineering simulations, and numerical analysis.

Rationalization

Rationalization is a technique used to eliminate radicals from the denominator of a fraction. Having a radical in the denominator is not usually preferred, so mathematicians developed a way to simplify such expressions.
The process works by multiplying both the numerator and the denominator by a radical that will cancel out the radical in the denominator.

• For instance, to rationalize \( \frac{1}{\sqrt{3}} \), multiply by \( \frac{\sqrt{3}}{\sqrt{3}} \) to obtain \( \frac{\sqrt{3}}{3} \).

This process makes figures easier to handle and interpret in many branches of science and engineering. Beyond calculation simplification, rationalization also helps in achieving better precision in both manual calculations and when using computational tools.