Determine whether each statement is true or false. If false, tell why. tan 60° ≥ cot 40°

Verified step by step guidance

1. Recall the definitions of the tangent and cotangent functions: \(\tan \theta = \frac{\sin \theta}{\cos \theta}\) and \(\cot \theta = \frac{1}{\tan \theta} = \frac{\cos \theta}{\sin \theta}\).
2. Rewrite the inequality \(\tan 60^\circ \geq \cot 40^\circ\) using the cotangent identity: \(\cot 40^\circ = \tan (90^\circ - 40^\circ) = \tan 50^\circ\).
3. Now the inequality becomes \(\tan 60^\circ \geq \tan 50^\circ\). Since tangent is increasing in the interval \(0^\circ\) to \(90^\circ\), compare the angles 60° and 50° directly.
4. Because \(60^\circ > 50^\circ\) and tangent is increasing in this range, it follows that \(\tan 60^\circ > \tan 50^\circ\).
5. Therefore, the original inequality \(\tan 60^\circ \geq \cot 40^\circ\) is true.

Key Concepts

Definition of Tangent and Cotangent Functions

Tangent of an angle in a right triangle 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 values of tan 60° and cot 40° accurately.

Relationship Between Cotangent and Tangent

Cotangent of an angle is equal to the tangent of its complementary angle: cot θ = tan (90° - θ). This identity allows rewriting cot 40° as tan 50°, simplifying the comparison between tan 60° and cot 40°.

Evaluating and Comparing Trigonometric Values

To determine the truth of inequalities involving trig functions, calculate or estimate their values using known exact values or approximations. For example, tan 60° = √3 ≈ 1.732, and tan 50° ≈ 1.191, so comparing these helps decide if tan 60° ≥ cot 40° is true.