Problem 58 - Algebra and Trigonometry

Question

Write the first expression in terms of the second if the terminal point determined by \(t\) is in the given quadrant. \(\csc t, \cot t ; \quad\) quadrant III

Step-by-Step Solution

Answer

\( \csc t = \sqrt{1 + \cot^2 t} \) in Quadrant III.

Step 1: Recall the Trigonometric Identities

We need to express \( \csc t \) in terms of \( \cot t \). First, recall the trigonometric identities related to these functions. We know that \( \csc t = \frac{1}{\sin t} \) and \( \cot t = \frac{\cos t}{\sin t} \).

Step 2: Understand the Quadrant III Conditions

In Quadrant III, both sine and cosine are negative. Therefore, \( \sin t < 0 \) and \( \cos t < 0 \). We have \( \cot t = \frac{\cos t}{\sin t} = \frac{a}{b} \), where both \(a\) and \(b\) are negative, making \( \cot t > 0 \), as the division of two negative numbers is positive.

Step 3: Express cosecant in terms of cotangent

Given that \( \cot t = \frac{\cos t}{\sin t} \), solve for \( \sin t \): \( \sin t = \frac{\cos t}{\cot t} \). Substitute this back into the identity for cosecant: \[ \csc t = \frac{1}{\sin t} = \frac{\cot t}{\cos t} \].

Step 4: Simplify the expression

Since in the third quadrant, \( \cos t \) is negative and \( \cot t = \frac{1}{\tan t} = \frac{1}{\frac{\sin t}{\cos t}} = \frac{\cos t}{\sin t} \), substitute back to simplify \( \csc t \). Therefore, \( \csc t = \frac{\cot t}{\cos t} = \frac{1}{\sin t} \), and \( \sin t = \frac{1}{\sqrt{1 + \cot^2 t}} \). Hence, \( \csc t = \sqrt{1 + \cot^2 t} \).

Key Concepts

Cosecant Function

The cosecant function, denoted by \( \csc t \), is one of the six fundamental trigonometric functions. It is the reciprocal of the sine function, which means it is defined as \( \csc t = \frac{1}{\sin t} \). This function is particularly useful in scenarios where the sine value might be small, making its reciprocal substantially larger.

To better understand this function, consider the unit circle, where \( \sin t \) represents the y-coordinate of the point on the circle for a given angle \( t \). When \( \sin t \) approaches zero, \( \csc t \) grows very large, demonstrating its reciprocal nature.

• Reciprocal Relationship: \( \csc t = \frac{1}{\sin t} \).
• Zero Requirement: The sine function must not be zero for \( \csc t \) to be defined.
• Graphical Behavior: It resembles a series of arches, becoming infinite as \( \sin t \) nears zero.

This understanding is crucial to solve equations involving \( \csc t \), especially when translating between different trigonometric expressions.

Cotangent Function

The cotangent function, represented by \( \cot t \), is another primary trigonometric function. It is the reciprocal of the tangent function. Therefore, its formula is given by \( \cot t = \frac{1}{\tan t} \) or alternatively, \( \cot t = \frac{\cos t}{\sin t} \). This dual representation highlights how \( \cot t \) relates both to the tangent and the cosine and sine functions.

Here's a quick guide to understanding \( \cot t \):

• Reciprocal Identity: As \( \cot t = \frac{1}{\tan t} \), it acts in parallel to the behavior of tangent.
• Ratio Definition: Being \( \cot t = \frac{\cos t}{\sin t} \) means it reflects the ratio of cosine to sine values.
• Graphical Behavior: \( \cot t \) looks like a series of repeating curves, likely shifting based on the circle's quadrant.

Understanding these aspects can help when solving trigonometric equations and understanding their graphical representation.

Trigonometric Functions in Quadrants

The behavior of trigonometric functions changes as we move through different quadrants of the unit circle. Each of the four quadrants has distinct characteristics. For functions like sine, cosine, and consequently, cosecant and cotangent, understanding their quadrant-specific behavior is essential.

In Quadrant III:

• Sine and Cosine: Both \( \sin t \) and \( \cos t \) are negative. This is because the reference angles fall into the negative section of both the x and y axes.
• Cosecant and Cotangent: Since both the sine and cosine functions are negative, \( \cot t = \frac{\cos t}{\sin t} \) becomes positive, as the ratio of two negative numbers results in a positive value. Calculating \( \csc t \) requires careful consideration because \( \csc t = \frac{1}{\sin t} \) will be negative since \( \sin t \) is negative.

The ability to navigate these quadrant-specific behaviors allows for more accurate problem-solving and is foundational in trigonometry. Practicing these concepts in various scenarios can help solidify your understanding.