Problem 132 - A Graphical Approach to Precalculus with Limits

Question

Work each problem. Using a method similar to the one given in this section showing that \(\frac{\sin \theta}{\cos \theta}=\tan \theta,\) show that \(\frac{\cos \theta}{\sin \theta}=\cot \theta\)

Step-by-Step Solution

Answer

\( \frac{\cos \theta}{\sin \theta} = \cot \theta \) is verified using the reciprocal of tangent.

Step 1: Understanding the Problem

We need to show that the expression \( \frac{\cos \theta}{\sin \theta} \) is equal to \( \cot \theta \). This requires us to recall the definitions and properties of trigonometric functions, particularly cotangent.

Step 2: Recalling the Cotangent Definition

The cotangent function is defined as \( \cot \theta = \frac{1}{\tan \theta} \). Since \( \tan \theta = \frac{\sin \theta}{\cos \theta} \), substituting this into the definition of cotangent gives us \( \cot \theta = \frac{1}{\frac{\sin \theta}{\cos \theta}} \).

Step 3: Simplification Using Reciprocal

The previous expression \( \cot \theta = \frac{1}{\frac{\sin \theta}{\cos \theta}} \) can be simplified by taking the reciprocal, which leads to \( \cot \theta = \frac{\cos \theta}{\sin \theta} \).

Step 4: Conclusion

By simplifying the reciprocal of tangent, we have shown that \( \frac{\cos \theta}{\sin \theta} = \cot \theta \). The definitions and properties of trigonometric functions help us understand and verify this identity.

Key Concepts

Trigonometric Functions

Trigonometric functions are fundamental in mathematics, mainly used to relate the angles of triangles to the lengths of their sides. These functions are commonly used to model periodic phenomena, such as sound and light waves, among many others.
The primary trigonometric functions include:

• Sine (\( \sin \theta \)
• Cosine (\( \cos \theta \)
• Tangent (\( \tan \theta \)

These functions are essential in trigonometry and have numerous applications in various fields like physics, engineering, and computer science.
Each function represents a specific ratio of sides in a right-angled triangle, related to a particular angle \( \theta \). For example, \( \sin \theta \) is the ratio of the opposite side to the hypotenuse, while \( \cos \theta \) is the ratio of the adjacent side to the hypotenuse. Understanding these foundational concepts is critical for grasping more complex identities involving trigonometric functions.

Cotangent

The cotangent, commonly denoted as \( \cot \theta \), is one of the lesser-known members of the trigonometric functions family. But it's just as important as its more famous relatives.
The cotangent of an angle \( \theta \) is defined as the reciprocal of the tangent function. In simpler terms, it is the ratio of the cosine of the angle to the sine of the angle:
\[ \cot \theta = \frac{\cos \theta}{\sin \theta} \]This definition is pivotal for solving many trigonometric equations and identities.
It provides a way to look at relationships between angles in a unique manner different from the primary functions. Understanding cotangent also helps in comprehending the trigonometric unit circle, where understanding the reference angles and their properties become necessary. Knowing about cotangent is crucial because it highlights the fundamental reciprocal relationship between sine and cosine, further broadening one's understanding of circular functions.

Reciprocal Identities

Trigonometric identities are equations involving trigonometric functions that hold true for all values of the occurring variables. Reciprocal identities are a specific set of these identities where each primary trigonometric function has a corresponding reciprocal function.
These identities are extremely helpful for simplifying trigonometric expressions and solving trigonometric equations.
Some of the basic reciprocal identities include:

• Secant function: \( \sec \theta = \frac{1}{\cos \theta} \)
• Cosecant function: \( \csc \theta = \frac{1}{\sin \theta} \)
• Cotangent function: \( \cot \theta = \frac{1}{\tan \theta} \)

Reciprocal identities help the transition from a complex expression to a more manageable form, enabling easier computation and deriving results more swiftly.
Using these reciprocal relationships, as seen in the identity \( \cot \theta = \frac{\cos \theta}{\sin \theta} \), it becomes easy to understand how fractions and reciprocals function within the context of trigonometric equations. Mastering these can simplify the process of solving complex mathematical problems.