Problem 127 - A Graphical Approach to Precalculus with Limits

Question

Use fundamental identities to find each expression. Write \(\sin \theta\) in terms of \(\cot \theta\) if \(\theta\) is in quadrant III.

Step-by-Step Solution

Answer

\(\sin \theta = -\sqrt{\frac{1}{1 + \cot^2 \theta}}\) in quadrant III.

Step 1: Understand Cotangent Identity

Start with the definition of cotangent, which is given by the identity \(\cot \theta = \frac{1}{\tan \theta}\). Since \(\tan \theta = \frac{\sin \theta}{\cos \theta}\), we can rewrite \(\cot \theta\) as \(\cot \theta = \frac{\cos \theta}{\sin \theta}\).

Step 2: Rearrange for Sine

From the identity \(\cot \theta = \frac{\cos \theta}{\sin \theta}\), express \(\sin \theta\) in terms of \(\cos \theta\) and \(\cot \theta\). This gives us \(\sin \theta = \frac{\cos \theta}{\cot \theta}\).

Step 3: Use Pythagorean Identity

Apply the Pythagorean identity \(\sin^2 \theta + \cos^2 \theta = 1\). To express \(\sin \theta\) using \(\cot \theta\), we first need \(\cos \theta\) in terms of \(\sin \theta\). Rearrange to get \(\cos \theta = \sqrt{1 - \sin^2 \theta}\).

Step 4: Substitute and Solve for Sine

Substitute \(\cos \theta = \sqrt{1 - \sin^2 \theta}\) in the expression \(\sin \theta = \frac{\cos \theta}{\cot \theta}\) to find \(\sin \theta = \frac{\sqrt{1 - \sin^2 \theta}}{\cot \theta}\). Since \(\theta\) is in quadrant III, \(\sin \theta\) and \(\cos \theta\) are both negative. Thus, solve \(\sin^2 \theta + \left(\frac{\cot \theta \cdot \sin \theta}{1}\right)^2 = 1\) to get \(\sin^2 \theta(1 + \cot^2 \theta) = 1\).

Step 5: Find the Exact Expression

Rearrange to solve for \(\sin^2 \theta\), giving \(\sin^2 \theta = \frac{1}{1 + \cot^2 \theta}\). Therefore, \(\sin \theta = -\sqrt{\frac{1}{1 + \cot^2 \theta}}\). The negative sign is due to \(\theta\) being in the third quadrant where sine is negative.

Key Concepts

Cotangent

The cotangent of an angle, often symbolized as \(\cot \theta\), is a fundamental trigonometric function. Unlike the more commonly discussed sine and cosine, cotangent relates closely to the tangent function. Specifically, \(\cot \theta\) is the reciprocal of the tangent, defined by the identity:

• \(\cot \theta = \frac{1}{\tan \theta}\)
• Since \(\tan \theta = \frac{\sin \theta}{\cos \theta}\), we can further express \(\cot \theta\) as \(\frac{\cos \theta}{\sin \theta}\).

This identity is particularly useful when you need to convert between different trigonometric functions. For example, in the exercise, knowing \(\cot \theta\) helps us express \(\sin \theta\) in terms of \(\cos \theta\) and \(\cot \theta\), as \(\sin \theta = \frac{\cos \theta}{\cot \theta}\). Understanding this relationship is key to transforming expressions involving trigonometric functions.

Pythagorean Identity

The Pythagorean Identity is a crucial equation in trigonometry, establishing a fundamental relation between sine and cosine. Its most common form is:

• \(\sin^2 \theta + \cos^2 \theta = 1\)

This identity stems from the Pythagorean Theorem in geometry, and it ensures that for any angle \(\theta\), the squared sum of its sine and cosine will always equal one. This property becomes handy when you need to find one function in terms of the other.
In the exercise above, the identity is crucial for expressing \(\cos \theta\) in terms of \(\sin \theta\), allowing you to deduce \(\sin \theta = \frac{\cos \theta}{\cot \theta}\) while also knowing the sign change because of trigonometric nature in different quadrants.

Quadrant Analysis

Analyzing the quadrant in which an angle lies is essential for determining the signs of its trigonometric functions. The coordinate plane is divided into four quadrants, each with different characteristics for the signs of sine, cosine, and tangent functions:

• Quadrant I: Both sine and cosine are positive.
• Quadrant II: Sine is positive, and cosine is negative.
• Quadrant III: Both sine and cosine are negative.
• Quadrant IV: Sine is negative, and cosine is positive.

In our problem, since \(\theta\) is in Quadrant III, both \(\sin \theta\) and \(\cos \theta\) are negative. This sign change is fundamental to correctly solving the exercise. When finding \(\sin \theta\) in terms of \(\cot \theta\), recognizing that \(\sin \theta = -\sqrt{\frac{1}{1 + \cot^2 \theta}}\) involves acknowledging the negative sign due to \(\theta\) being in the third quadrant, a critical observation in trigonometric analysis.