Determine whether each statement is true or false. See Example 4. csc 20° < csc 30°

Verified step by step guidance

1. Recall the definition of the cosecant function: \(\csc \theta = \frac{1}{\sin \theta}\). This means that to compare \(\csc 20^\circ\) and \(\csc 30^\circ\), we need to compare \(\frac{1}{\sin 20^\circ}\) and \(\frac{1}{\sin 30^\circ}\).
2. Since \(\csc \theta\) is the reciprocal of \(\sin \theta\), the inequality \(\csc 20^\circ < \csc 30^\circ\) is equivalent to \(\frac{1}{\sin 20^\circ} < \frac{1}{\sin 30^\circ}\).
3. To compare these, consider the values of \(\sin 20^\circ\) and \(\sin 30^\circ\). Remember that \(\sin 30^\circ = \frac{1}{2}\), which is a known exact value.
4. Since \(\sin 20^\circ\) is less than \(\sin 30^\circ\), and both sines are positive in the first quadrant, the reciprocal relationship means \(\csc 20^\circ\) will be greater than \(\csc 30^\circ\).
5. Therefore, the statement \(\csc 20^\circ < \csc 30^\circ\) is false because \(\csc 20^\circ\) is actually greater than \(\csc 30^\circ\).

Key Concepts

Definition of the Cosecant Function

The cosecant function, csc(θ), is the reciprocal of the sine function, defined as csc(θ) = 1/sin(θ). It is important to understand this relationship to compare values of csc at different angles.

Behavior of the Sine Function in the First Quadrant

In the first quadrant (0° to 90°), the sine function increases as the angle increases. Since sine values increase, their reciprocals (cosecants) decrease, which affects the comparison between csc 20° and csc 30°.

Comparing Trigonometric Values

To determine inequalities involving trigonometric functions, it is essential to evaluate or estimate their values accurately. For example, knowing sin 20° ≈ 0.342 and sin 30° = 0.5 helps conclude that csc 20° > csc 30°.