Problem 3 - Precalculus with Limits

Question

Fill in the blank to complete the trigonometric identity. \( \dfrac{1}{\tan u} \)= ________

Step-by-Step Solution

Answer

\(1/ \tan u = \cot u\)

Step 1: Recall the definition of tangent

The first step is to recall the definition of the trigonometric function tangent. The function \(\tan u\) is defined as the ratio between sine and cosine of the same angle. Therefore, \(\tan u = \sin u / \cos u\).

Step 2: Write the reciprocal

Now we want to find what \(1/ \tan u\) equals to. Since \(\tan u\) equals to \(\sin u / \cos u\), the reciprocal of \(\tan u\) is simply \(\cos u / \sin u\), which is \(1/ \tan u\).

Step 3: Fill in the blank

Now that we have found that \(1/ \tan u = \cos u / \sin u\), we can fill in the blank in the original equation. Thus, the completed equation is \(1/ \tan u = \cot u\).

Key Concepts

Tangent Function

The tangent function is one of the six fundamental trigonometric functions. It is defined as the ratio of the opposite side to the adjacent side in a right-angled triangle, related to a specific acute angle. The formula for the tangent of angle \(u\) is given by:
\[ \tan u = \frac{\text{opposite side}}{\text{adjacent side}} \]
When considering the unit circle representation, where the length of the hypotenuse is always 1, \( \tan u \) is also represented as the ratio of the y-coordinate (sine) to the x-coordinate (cosine) of a point on the unit circle. This gives a clearer mathematical expression:
\[ \tan u = \frac{\text{sin u}}{\text{cos u}} \]
This ratio helps determine the slope of the line that represents the angle in the coordinate plane. The tangent function exhibits periodic behavior, repeating its values every \( \text{π} \) radians or 180 degrees, and it has infinitely many asymptotes where it is undefined, specifically at \( \frac{\text{π}}{2} + k\text{π} \), where \(k\) is an integer.

Reciprocal Trigonometric Functions

In trigonometry, reciprocal functions play a significant role as they relate to the primary trigonometric functions. The three main reciprocal trigonometric functions are the cosecant \(\text{csc} u\), secant \(\text{sec} u\), and cotangent \(\text{cot} u\). They are defined as follows:
\[ \text{csc} u = \frac {1}{\text{sin} u} \]
\[ \text{sec} u = \frac {1}{\text{cos} u} \]
\[ \text{cot} u = \frac {1}{\text{tan} u} \] or \[ \text{cot} u = \frac {\text{cos} u}{\text{sin} u} \]
Understanding these reciprocal identities is crucial for solving various trigonometric problems, especially when simplifying expressions or solving equations. They are also essential when working with trigonometric proofs where the relationship between reciprocal and original trigonometric functions are utilized to demonstrate equivalences.

Trigonometric Functions Ratios

Ratios of trigonometric functions often appear in mathematical expressions and are a key concept in trigonometry. They compare two of the six trigonometric function values of the same angle. Some of the most commonly used trigonometric function ratios include:

• \( \frac{\text{sin} u}{\text{cos} u} = \tan u \)
• \( \frac{\text{cos} u}{\text{sin} u} = \text{cot} u \)
• \( \frac{1}{\text{sin} u} = \text{csc} u \)
• \( \frac{1}{\text{cos} u} = \text{sec} u \)

These ratios provide fundamental connections between the trigonometric functions that can be applied in simplification, integration, differentiation, and during the process of finding limits in calculus. It is important for students to understand the interrelationships between these functions to facilitate the solving of complex trigonometric problems.