Problem 22 - Algebra 2

Question

Given \(\cos \theta=-\frac{4}{5}\) and \(90^{\circ}<\theta<180^{\circ},\) find the exact value of each expression. $$ \cot \frac{\theta}{2} $$

Step-by-Step Solution

Answer

The value of \( \cot \frac{\theta}{2} \) is 3.

Step 1: Determine \( \sin\theta \)

First, determine \( \sin\theta \). We know that \(\cos \theta=-\frac{4}{5}\). Using the Pythagorean Identity \(\sin^2 \theta + \cos^2 \theta = 1 \), we can solve for \( \sin\theta \):\( \sin^2 \theta = 1 - \cos^2 \theta \)\( \sin^2 \theta = 1 - (-\frac{4}{5})^2 \)\( \sin^2 \theta = 1 - \frac{16}{25} \)\( \sin^2 \theta = \frac{9}{25} \)Thus, \( \sin\theta = \sqrt{\frac{9}{25}} \). Since we know that \(90^{\circ}< \theta < 180^{\circ}\), \(\theta\) lies in the second quadrant where \(\sin \theta > 0\), we can determine that \( \sin\theta = \frac{3}{5}\).

Step 2: Apply the Half-Angle Formula

Secondly, with the value of \(\sin \theta\) and \(\cos \theta\), we apply the half-angle formula for \(\cot \frac{\theta}{2}\) which is \(\frac{1-\cos x}{\sin x}\):\(\cot \frac{\theta}{2} = \frac{1-\cos \theta}{\sin \theta}\)Substituting the values we found for \( \cos \theta \) and \( \sin \theta \):\(\cot \frac{\theta}{2} = \frac{1-(-\frac{4}{5})}{\frac{3}{5}}\)\(\cot \frac{\theta}{2} = \frac{\frac{9}{5}}{\frac{3}{5}}\)Which simplifies to \( \cot \frac{\theta}{2} = 3 \)

Key Concepts

Pythagorean Identity

The Pythagorean Identity is a fundamental relationship in trigonometry that connects the squares of sine and cosine for any angle \( \theta \). It states: \[ \sin^2 \theta + \cos^2 \theta = 1 \] This identity is derived from the Pythagorean Theorem and serves as a powerful tool in solving trigonometric problems. In the given problem, with \( \cos \theta = -\frac{4}{5} \), we need to find \( \sin \theta \). To do this:

• Use the identity \( \sin^2 \theta = 1 - \cos^2 \theta \).
• Substitute \( \cos \theta = -\frac{4}{5} \), giving \( \cos^2 \theta = \frac{16}{25} \).
• Calculate \( \sin^2 \theta = 1 - \frac{16}{25} = \frac{9}{25} \).
• Thus, \( \sin \theta = \sqrt{\frac{9}{25}} = \frac{3}{5} \) or \( -\frac{3}{5} \).

Since \( 90^\circ < \theta < 180^\circ \) places \( \theta \) in the second quadrant where sine is positive, the positive value is selected: \( \sin \theta = \frac{3}{5} \). Understanding the Pythagorean Identity helps in verifying the fundamental properties of trigonometric functions and leveraging these principles to solve various problems like the one presented.

Half-Angle Formulas

Half-angle formulas are essential tools in trigonometry used to simplify the calculations for expressions involving half angles. These formulas help in finding relationships for angles that are half of another angle. In this exercise, we apply the half-angle formula for the cotangent function. The half-angle formula for \( \cot \frac{\theta}{2} \) is given by: \[ \cot \frac{\theta}{2} = \frac{1 - \cos \theta}{\sin \theta} \] To solve the problem, we use the values:

• \( \cos \theta = -\frac{4}{5} \)
• \( \sin \theta = \frac{3}{5} \)

Inserting these into the formula yields: \[ \cot \frac{\theta}{2} = \frac{1 - (-\frac{4}{5})}{\frac{3}{5}} = \frac{\frac{9}{5}}{\frac{3}{5}} \] This simplifies to 3, which is the exact value of \( \cot \frac{\theta}{2} \). By applying the half-angle formulas, complex trigonometric expressions are simplified, aiding in easier and quicker problem solving.

Trigonometric Functions

Trigonometric functions are foundational to understanding angles and their relationships in mathematics. They connect angles to ratios of sides in right triangles and have numerous applications. The primary trigonometric functions are sine (\( \sin \)), cosine (\( \cos \)), and tangent (\( \tan \)), along with their reciprocals—cosecant (\( \csc \)), secant (\( \sec \)), and cotangent (\( \cot \)). Here is a quick overview of these functions:

• \( \sin \theta = \) Opposite side / Hypotenuse
• \( \cos \theta = \) Adjacent side / Hypotenuse
• \( \tan \theta = \) Opposite side / Adjacent side
• \( \cot \theta = 1/\tan \theta \)
• \( \sec \theta = 1/\cos \theta \)
• \( \csc \theta = 1/\sin \theta \)

Understanding these functions allows one to solve a wide array of problems involving angles and triangles. In the problem given, the function \( \cot \frac{\theta}{2} \) involves using both sine and cosine to determine the cotangent of a half angle. These trigonometric functions are interconnected, and mastering them is crucial for advancing in the study of mathematics.