Problem 38 - Algebra 2 and Trigonometry

Question

In \(33-38,\) find the exact radian measure \(\theta\) of an angle with the smallest absolute value that satisfies the equation. $$ \cot \theta=\sqrt{3} $$

Step-by-Step Solution

Answer

The exact radian measure \(\theta\) is \(\frac{\pi}{6}\).

Step 1: Understand the Cotangent Function

The cotangent function, \(\cot \theta\), is defined as the reciprocal of the tangent function, i.e., \(\cot \theta = \frac{1}{\tan \theta}\). This means that \(\tan \theta = \frac{1}{\sqrt{3}}\). We want to find an angle where \(\cot \theta = \sqrt{3}\) and \(\tan \theta = \frac{1}{\sqrt{3}}\).

Step 2: Recall Common Angle Values

The tangent of \(\frac{\pi}{6}\) radians, which is 30 degrees, is \(\frac{1}{\sqrt{3}}\). This makes \(\cot \frac{\pi}{6} = \sqrt{3}\). Thus, \(\theta = \frac{\pi}{6}\) satisfies our equation.

Step 3: Ensure Smallest Absolute Value Requirement

The angle \(\theta = \frac{\pi}{6}\) already satisfies \(\cot \theta = \sqrt{3}\). Since \(\frac{\pi}{6}\) is the smallest positive angle for this cotangent value, it also has the smallest absolute value.

Key Concepts

Understanding the Cotangent Function

The cotangent function is an essential concept in trigonometry, representing the reciprocal of the tangent function. This is denoted as \( \cot \theta \). Think of it as the flip side of tangent. Where tangent measures the ratio of the opposite to the adjacent side in a right triangle, cotangent measures the reverse. Hence, the equation \( \cot \theta = \frac{1}{\tan \theta} \) holds true.

In practical terms, if you know the tangent of an angle, you can easily find the cotangent by taking its reciprocal. For instance, if \( \tan \theta = \frac{1}{\sqrt{3}} \), then \( \cot \theta = \sqrt{3} \). This relationships is particularly useful in solving problems that require finding angles based on their trigonometric identities.

Exploring the Tangent Function

The tangent function is one of the primary trigonometric functions, often denoted as \( \tan \theta \). It represents the ratio of the opposite side to the adjacent side of an angle in a right triangle. Tan is a useful function not just in geometry but also in calculus and other areas of mathematics.

When it comes to specific angles, the tangent function has well-known values. For example, \( \tan \frac{\pi}{6} = \frac{1}{\sqrt{3}} \), which directly helps in solving problems where the cotangent is \( \sqrt{3} \). Understanding the tangent function and its values at common angles provides a powerful tool for solving trigonometric equations.

Common Angle Values in Trigonometry

Certain angles in trigonometry are referred to as "common angle values" because their trigonometric functions take on simple values. These angles usually include \( 30^{\circ}, 45^{\circ}, \) and \( 60^{\circ} \), or their radian equivalents \( \frac{\pi}{6}, \frac{\pi}{4}, \) and \( \frac{\pi}{3} \) respectively.

These angles are especially useful because they often simplify calculations in trigonometry. For example:

• \( \tan \frac{\pi}{6} = \frac{1}{\sqrt{3}} \)
• \( \tan \frac{\pi}{4} = 1 \)
• \( \tan \frac{\pi}{3} = \sqrt{3} \)

By knowing these values by heart, solving equations like \( \cot \theta = \sqrt{3} \) becomes straightforward as \( \theta \) then can be \( \frac{\pi}{6} \).

Reciprocal Trigonometric Functions

Reciprocal trigonometric functions are derived from the primary trigonometric functions and are used to make certain calculations more convenient or solve specific equations. Reciprocals include:

• Secant (\( \sec \theta \)) as the reciprocal of cosine \( \cos \theta \)
• Cosecant (\( \csc \theta \)) as the reciprocal of sine \( \sin \theta \)
• Cotangent (\( \cot \theta \)) as the reciprocal of tangent \( \tan \theta \)

Each of these functions are valuable in situations where the primary trigonometric functions represent inconvenient ratios. In our example where \( \cot \theta = \sqrt{3} \), determining the cotangent required a specific knowledge of these properties. Understanding and using reciprocal trigonometric functions can streamline solving many mathematical and engineering problems.