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

Verified step by step guidance

1. Recall the definitions of cotangent and tangent: \(\cot \theta = \frac{1}{\tan \theta}\) and \(\tan \theta = \frac{\sin \theta}{\cos \theta}\).
2. Express \(\cot 30^\circ\) in terms of tangent: \(\cot 30^\circ = \frac{1}{\tan 30^\circ}\).
3. Use known exact values or approximate values for \(\tan 30^\circ\) and \(\tan 40^\circ\) to compare them. For example, \(\tan 30^\circ = \frac{1}{\sqrt{3}}\).
4. Calculate \(\cot 30^\circ\) by taking the reciprocal of \(\tan 30^\circ\), which gives \(\cot 30^\circ = \sqrt{3}\).
5. Compare \(\cot 30^\circ\) and \(\tan 40^\circ\) by evaluating or approximating \(\tan 40^\circ\) and then determine if \(\cot 30^\circ < \tan 40^\circ\) is true or false.

Key Concepts

Definition of Cotangent and Tangent Functions

Cotangent and tangent are trigonometric functions related to angles in a right triangle. Tangent of an angle is the ratio of the opposite side to the adjacent side, while cotangent is the reciprocal of tangent, or adjacent over opposite. Understanding these definitions helps compare their values for given angles.

Evaluating Trigonometric Functions at Specific Angles

To compare cot 30° and tan 40°, one must know or calculate their approximate values. Using known exact values or a calculator, cot 30° equals √3 (~1.732), and tan 40° is approximately 0.839. This evaluation is essential to determine the truth of the inequality.

Inequality Comparison of Trigonometric Values

Comparing trigonometric values involves understanding inequalities and numerical approximations. After evaluating cot 30° and tan 40°, comparing their magnitudes determines if the statement cot 30° < tan 40° is true or false. This concept ensures logical reasoning in trigonometric comparisons.