Problem 43 - Algebra 2

Question

Find the exact value of each trigonometric function. $$ \cot \left(\frac{\pi}{3}\right) $$

Step-by-Step Solution

Answer

\( \cot\left(\frac{\pi}{3}\right) = \frac{\sqrt{3}}{3} \)

Step 1: Understand the Cotangent Function

The cotangent function is the reciprocal of the tangent function. Mathematically, it is expressed as \( \cot(\theta) = \frac{1}{\tan(\theta)} \). Therefore, to find \( \cot\left(\frac{\pi}{3}\right) \), we first need to determine \( \tan\left(\frac{\pi}{3}\right) \).

Step 2: Determine \( \tan\left(\frac{\pi}{3}\right) \)

The angle \( \frac{\pi}{3} \) radians corresponds to \( 60^\circ \). The tangent of \( \frac{\pi}{3} \) or \( 60^\circ \) is well-known: \( \tan\left(\frac{\pi}{3}\right) = \sqrt{3} \).

Step 3: Calculate \( \cot\left(\frac{\pi}{3}\right) \)

With \( \tan\left(\frac{\pi}{3}\right) = \sqrt{3} \), the cotangent is the reciprocal: \( \cot\left(\frac{\pi}{3}\right) = \frac{1}{\sqrt{3}} \). To make this expression rational, we multiply the numerator and denominator by \( \sqrt{3} \). Thus, \( \frac{1}{\sqrt{3}} \) becomes \( \frac{\sqrt{3}}{3} \).

Key Concepts

Cotangent Function

The cotangent function is an integral part of trigonometry. It's often represented by the symbol \( \cot \). As a trigonometric function, cotangent relates the angles of a right triangle to the lengths of its sides. Specifically, it is the ratio of the adjacent side length to the opposite side length when the angle is taken at the corner of the triangle. In other words, for a given angle \( \theta \), the cotangent function is defined as follows:

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

Since \( \cot(\theta) \) is the reciprocal of the tangent function \( \tan(\theta) \), understanding the tangent function's values allows us to find cotangent values easily. Knowing this identity helps if you need to compute the cotangent of common angles such as \( \frac{\pi}{3} \).

Tangent Function

The tangent function, denoted as \( \tan \), is a fundamental function in trigonometry. It expresses the ratio of the length of the opposite side to the length of the adjacent side in a right triangle for a given angle \( \theta \). Formally, we write:

• \( \tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} \)

The tangent function is particularly useful because it helps establish a circular trigonometric identity with cotangent. To solve problems involving tangent, knowing the special angles (like \( 0, \frac{\pi}{6}, \frac{\pi}{4}, \frac{\pi}{3}, \frac{\pi}{2} \), etc.) and their tangent values is invaluable. For example, \( \tan\left(\frac{\pi}{3}\right) = \sqrt{3} \). The tangent function also ties into the concept of periodicity, repeating every \( \pi \) radians, offering a consistency useful in trigonometric equations.

Reciprocal Identity

The reciprocal identity is a crucial concept in trigonometry that connects various trigonometric functions through their reciprocal relationships. Reciprocity is at the heart of understanding the interplay between different trigonometric functions.

• \( \cot(\theta) = \frac{1}{\tan(\theta)} \)
• \( \tan(\theta) = \frac{1}{\cot(\theta)} \), if \( \cot(\theta) eq 0 \)

These identities serve a pivotal role when you want to switch from one function form to another and simplify trigonometric expressions. It helps keep calculations straightforward, especially when dealing with special angles and the computations derived from them. By applying the reciprocal identity, you can effectively solve problems like finding \( \cot\left(\frac{\pi}{3}\right) \), starting from known \( \tan \) values.

Radian Measure

Radians are a way to measure angles, offering an alternative to degrees, which most are familiar with. In the context of trigonometry, radians provide a more natural way to work with angles, particularly because of how they relate to the unit circle. One revolution around a circle is \( 2\pi \) radians, corresponding to \( 360^\circ \).Conversion between degrees and radians is useful:

• \( 180^\circ = \pi \text{ radians} \)
• To convert degrees to radians: multiply by \( \frac{\pi}{180} \)
• To convert radians to degrees: multiply by \( \frac{180}{\pi} \)

Radian measure is often applied in calculus and advanced math due to its clean and direct relationship with \( \pi \). For example, knowing \( \frac{\pi}{3} \) radians equates to \( 60^\circ \) means we can utilize it directly in trigonometric functions like \( \tan \) and \( \cot \). The radian measure thus aids in understanding and solving trigonometric problems effectively.