Problem 195 - Precalculus

Question

For the following exercises, find the exact value of each expression. $$ \cot \frac{\pi}{3} $$

Step-by-Step Solution

Answer

\( \cot \frac{\pi}{3} = \frac{\sqrt{3}}{3} \).

Step 1: Understand the Cotangent Function

The cotangent function is the reciprocal of the tangent function. Therefore, \( \cot \theta = \frac{1}{\tan \theta} \). To find \( \cot \frac{\pi}{3} \), we will first determine \( \tan \frac{\pi}{3} \).

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

For the angle \( \frac{\pi}{3} \), we know that the tangent is given by the ratio of the sine and cosine functions. Therefore, \( \tan \frac{\pi}{3} = \frac{\sin \frac{\pi}{3}}{\cos \frac{\pi}{3}} \). Since \( \sin \frac{\pi}{3} = \frac{\sqrt{3}}{2} \) and \( \cos \frac{\pi}{3} = \frac{1}{2} \), we have \( \tan \frac{\pi}{3} = \frac{\frac{\sqrt{3}}{2}}{\frac{1}{2}} = \sqrt{3} \).

Step 3: Calculate the Cotangent

Now that we have \( \tan \frac{\pi}{3} = \sqrt{3} \), we can find \( \cot \frac{\pi}{3} \) as the reciprocal of \( \tan \frac{\pi}{3} \). Therefore, \( \cot \frac{\pi}{3} = \frac{1}{\sqrt{3}} \).

Step 4: Simplify the Cotangent Value

Rationalize the denominator to simplify \( \frac{1}{\sqrt{3}} \). Multiplying the numerator and the denominator by \( \sqrt{3} \), we get \( \frac{\sqrt{3}}{3} \). Thus, \( \cot \frac{\pi}{3} = \frac{\sqrt{3}}{3} \).

Key Concepts

Trigonometric Functions

Trigonometric functions are fundamental in mathematics, representing the relationships between angles and side lengths in right triangles. They are critical in fields such as physics, engineering, and computer science. The key trigonometric functions include sine (\( \sin \)), cosine (\( \cos \)), and tangent (\( \tan \)), each defining a specific ratio of a triangle's sides.

To understand cotangent (\( \cot \)), which is the focus of our exercise, it's essential first to grasp tangent. Tangent represents the ratio of the opposite side to the adjacent side in a right triangle and is given as the sine over cosine:

• \( \tan \theta = \frac{\sin \theta}{\cos \theta} \)

Cotangent is then defined as the reciprocal of tangent, illuminating how these functions interrelate to provide comprehensive insights into trigonometric relationships.

Tangent

Tangent is one of the primary trigonometric functions, often used for calculating angles and distances. Specifically, in right triangles, it expresses the ratio of the opposite side to the adjacent side, which can be extremely useful for problems involving angles and slope.

For angles on the unit circle, such as \( \frac{\pi}{3} \), the tangent value is determined using sine and cosine. In our exercise, we calculated:

• \( \tan \frac{\pi}{3} = \frac{\sin \frac{\pi}{3}}{\cos \frac{\pi}{3}} = \frac{\frac{\sqrt{3}}{2}}{\frac{1}{2}} = \sqrt{3} \)

This computation is essential for proceeding to calculate the cotangent because it forms the basis of understanding its reciprocal nature.

Reciprocal Identities

Reciprocal identities are key concepts in trigonometry and are essential tools for simplifying expressions and solving equations. These identities involve pairs of trigonometric functions that are reciprocals of each other, such as tangent and cotangent.

In essence:

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

This reciprocal relationship was used to convert the valued tangent of \( \frac{\pi}{3} \):

• Since \( \tan \frac{\pi}{3} = \sqrt{3} \), this means \( \cot \frac{\pi}{3} = \frac{1}{\sqrt{3}} \)

Reciprocal identities simplify trigonometric expressions and allow for easier manipulation of equations, particularly when functions like sine, cosine, and tangent need to be expressed in alternative forms.

Rationalizing Denominators

Rationalizing denominators is a mathematical technique used to remove the square root or irrational number from the denominator of a fraction. This method creates numerically simpler expressions to work with.

In our problem, after finding \( \cot \frac{\pi}{3} = \frac{1}{\sqrt{3}} \), we applied this technique due to the square root in the denominator. The process involved multiplying both the numerator and the denominator by \( \sqrt{3} \):

• \( \frac{1}{\sqrt{3}} \cdot \frac{\sqrt{3}}{\sqrt{3}} = \frac{\sqrt{3}}{3} \)

By rationalizing the denominator, the expression \( \frac{\sqrt{3}}{3} \) is derived, which is often preferred for solutions due to its simplified and more aesthetically pleasing form, especially important in educational and formal settings.