Problem 111 - Precalculus

Question

If \(\csc \theta=-\frac{a}{b},\) where \(a\) and \(b\) are positive, and if \(\theta\) lies in quadrant IV, find cot \(\theta\)

Step-by-Step Solution

Answer

\( \cot \theta = -\frac{\sqrt{a^2 - b^2}}{b} \).

Step 1: Understand the cosecant function

The cosecant function is the reciprocal of the sine function, so \ \( \csc \theta = \frac{1}{\sin \theta} \). Given \( \csc \theta = -\frac{a}{b} \) implies \( \sin \theta = -\frac{b}{a} \).

Step 2: Use Pythagorean Identity

The Pythagorean identity states \( \sin^2 \theta + \cos^2 \theta = 1 \). Substitute \( \sin \theta = -\frac{b}{a} \) into the identity to find \( \cos \theta \): \ \( \left(-\frac{b}{a}\right)^2 + \cos^2 \theta = 1 \, \frac{b^2}{a^2} + \cos^2 \theta = 1 \, \cos^2 \theta = 1 - \frac{b^2}{a^2} = \frac{a^2 - b^2}{a^2} \).

Step 3: Determine cosine sign in Quadrant IV

In Quadrant IV, the cosine function is positive. Therefore, \( \cos \theta = \frac{\sqrt{a^2 - b^2}}{a} \).

Step 4: Calculate cotangent

The cotangent function is the ratio of cosine to sine: \( \cot \theta = \frac{\cos \theta}{\sin \theta} \). Substitute the values found: \ \( \cot \theta = \frac{\frac{\sqrt{a^2 - b^2}}{a}}{-\frac{b}{a}} = -\frac{\sqrt{a^2 - b^2}}{b} \).

Key Concepts

Cosecant Function

The cosecant function, expressed as \( \csc \theta \), is a crucial part of trigonometry. It is defined as the reciprocal of the sine function. This means \( \csc \theta = \frac{1}{\sin \theta} \). If you know the cosecant value, you can easily find the sine by taking its reciprocal. For example, if \( \csc \theta = -\frac{a}{b} \), then \( \sin \theta = -\frac{b}{a} \).

The negative sign indicates that the sine is negative for the given angle \( \theta \). In trigonometry, the signs of the trigonometric functions vary depending on the quadrant in which the angle lies. In this specific exercise, because \( \theta \) is in Quadrant IV, the sine is negative, perfectly aligning with our given relationship. Knowing these reciprocal identities helps in converting between trigonometric functions, simplifying complex equations, and solving various trigonometric problems with ease.

Pythagorean Identity

The Pythagorean identity is one of the most fundamental tools in trigonometry. It states that for any angle \( \theta \), the identity \( \sin^2 \theta + \cos^2 \theta = 1 \) holds true. This identity forms the foundation of many trigonometric transformations and calculations. It relates sine and cosine through the sum of their squares.

In the context of our exercise, you substitute the value of \( \sin \theta \) to find \( \cos \theta \) using this identity. Given \( \sin \theta = -\frac{b}{a} \), you plug it into the Pythagorean identity:

• \( \left(-\frac{b}{a}\right)^2 + \cos^2 \theta = 1 \) simplifies to \( \frac{b^2}{a^2} + \cos^2 \theta = 1 \).
• Rearranging gives \( \cos^2 \theta = 1 - \frac{b^2}{a^2} \).
• This can be rewritten as \( \cos^2 \theta = \frac{a^2 - b^2}{a^2} \).

This demonstrates how the identity helps to solve for cosine when sine is known. In Quadrant IV, where cosine values are positive, knowing \( \cos \theta \) as \( \frac{\sqrt{a^2 - b^2}}{a} \) becomes crucial in further calculations like determining the cotangent.

Cotangent Function

The cotangent function, denoted as \( \cot \theta \), is another important trigonometric function. It represents the ratio of the cosine function to the sine function, mathematically described as \( \cot \theta = \frac{\cos \theta}{\sin \theta} \). Understanding this relationship helps in solving trigonometric equations and analyzing periodic functions.

In the exercise, you need to find \( \cot \theta \) given that \( \csc \theta = -\frac{a}{b} \). We found earlier that \( \sin \theta = -\frac{b}{a} \) and \( \cos \theta = \frac{\sqrt{a^2 - b^2}}{a} \). Using these, you calculate:

• Substitute the known values into the formula, \( \cot \theta = \frac{\frac{\sqrt{a^2 - b^2}}{a}}{-\frac{b}{a}} \).
• This simplifies to \( -\frac{\sqrt{a^2 - b^2}}{b} \).

This solution shows how the cotangent function combines both sine and cosine, emphasizing its dependency on these two major trigonometric entities. Understanding and applying the cotangent function is key to grasping broader trigonometric concepts.