Determine whether each statement is true or false. See Example 4.cot 30° < tan 40°

Verified step by step guidance

1. Step 1: Recall the definitions of cotangent and tangent. The cotangent of an angle is the reciprocal of the tangent of that angle. Therefore, \( \cot \theta = \frac{1}{\tan \theta} \).
2. Step 2: Calculate \( \cot 30^\circ \). Since \( \tan 30^\circ = \frac{1}{\sqrt{3}} \), \( \cot 30^\circ = \sqrt{3} \).
3. Step 3: Calculate \( \tan 40^\circ \). Use a calculator or trigonometric table to find \( \tan 40^\circ \).
4. Step 4: Compare the values of \( \cot 30^\circ \) and \( \tan 40^\circ \).
5. Step 5: Determine if \( \cot 30^\circ < \tan 40^\circ \) is true or false based on the comparison.

Key Concepts

Cotangent and Tangent Functions

The cotangent function, denoted as cot(θ), is the reciprocal of the tangent function, tan(θ). This means cot(θ) = 1/tan(θ). Understanding these functions is crucial for comparing their values, especially at specific angles like 30° and 40°.

Angle Measurement in Trigonometry

In trigonometry, angles can be measured in degrees or radians. The question involves angles measured in degrees, specifically 30° and 40°. Knowing the values of trigonometric functions at these angles is essential for determining the truth of the statement.

Comparison of Trigonometric Values

To determine if cot(30°) is less than tan(40°), one must calculate the actual values of these functions. This involves using trigonometric tables or calculators. Understanding how to evaluate and compare these values is key to answering the question accurately.