Problem 44 - Algebra and Trigonometry

Question

Write the first trigonometric function in terms of the second for \(\theta\) in the given quadrant. $$ \csc \theta, \quad \cot \theta ; \quad \theta \text { in Quadrant III } $$

Step-by-Step Solution

Answer

\(\csc \theta = -\sqrt{\cot^2 \theta + 1}\)

Step 1: Understand the Given Functions

We are given \(\csc \theta\) and \(\cot \theta\), and we need to express \(\csc \theta\) in terms of \(\cot \theta\) while considering that \(\theta\) is in Quadrant III.

Step 2: Recall Definitions and Signs in Quadrant III

Recall that \(\csc \theta = \frac{1}{\sin \theta}\) and \(\cot \theta = \frac{\cos \theta}{\sin \theta}\). In Quadrant III, both \(\sin \theta\) and \(\cos \theta\) are negative, but \(\cot \theta\) is positive since it is the ratio of the two negative numbers.

Step 3: Express \(\sin \theta\) in Terms of \(\cot \theta\)

The expression for \(\cot \theta\) is \(\frac{\cos \theta}{\sin \theta} = \sqrt{\cot^2 \theta+1}\) due to the identity \(\cot^2 \theta = \csc^2 \theta - 1\). Rearranging gives \(\csc^2 \theta = \cot^2 \theta + 1\). Therefore, \(\csc \theta = \pm \sqrt{\cot^2 \theta + 1}\). Since \(\sin \theta\) is negative in Quadrant III, \(\csc \theta\) is also negative.

Step 4: Provide the Expression for \( \csc \theta \)

With these relationships and considering the sign, we have \(\csc \theta = -\sqrt{\cot^2 \theta + 1}\) in Quadrant III.

Key Concepts

Cosecant Function

The cosecant function, denoted as \( \csc \theta \), is a fundamental trigonometric function that is the reciprocal of the sine function. This means its value is the inverse of sine. You can write it as:

• \( \csc \theta = \frac{1}{\sin \theta} \)

Knowing this reciprocal relationship is crucial for solving various trigonometry problems where you are required to convert or express trigonometric functions in terms of other trigonometric forms.
In practice, understanding \( \csc \theta \) is helpful when tackling angles that are located in specific quadrants, as this determines the sign of the sine, and consequently, the sign of the cosecant function as well. Since \( \sin \theta \) can have either positive or negative values, this directly affects \( \csc \theta \). Understanding its behavior can help you predict its graph and find solutions to trigonometric equations.

Cotangent Function

Cotangent, denoted as \( \cot \theta \), is another important trigonometric function. It represents the ratio of the cosine function to the sine function. Formally, it's defined as:

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

This definition shows that when \( \cos \theta \) and \( \sin \theta \) have like signs, \( \cot \theta \) becomes positive.
The cotangent function, much like the tangent function, can also be described in terms of the reciprocal of the tangent:

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

It's important to be cautious about the sign of \( \cot \theta \), as it varies with the quadrant where \( \theta \) is located.

Quadrant III

The third quadrant of the Cartesian plane is an interesting area where both sine and cosine functions are negative. When focusing on trigonometric functions, the sign changes as you move from one quadrant to another. This leads us to the cosine, sine, and tangent functions being negative or positive in different quadrants.

• In Quadrant III, \( \sin \theta \) is negative
• \( \cos \theta \) is negative
• However, \( \tan \theta \) and consequently \( \cot \theta \) are positive since they are the quotient of two negative numbers

This quadrant is where both \( x \) and \( y \) values are negative, meaning negative angles or working with angle extensions can come into play. Understanding the behavior of functions in Quadrant III is crucial for predicting their graphs and solving equations when \( \theta \) is positioned here. The knowledge about the sign and value of trigonometric functions is integral for handling more complicated calculations involving angles specific to Quadrant III.