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

Verified step by step guidance

1. Step 1: Recall that the cosecant function, \( \csc \theta \), is the reciprocal of the sine function, \( \sin \theta \). Therefore, \( \csc \theta = \frac{1}{\sin \theta} \).
2. Step 2: Calculate \( \sin 20^\circ \) and \( \sin 30^\circ \). Note that \( \sin 30^\circ = \frac{1}{2} \).
3. Step 3: Determine \( \csc 20^\circ \) by finding the reciprocal of \( \sin 20^\circ \), and \( \csc 30^\circ \) by finding the reciprocal of \( \sin 30^\circ \).
4. Step 4: Compare \( \csc 20^\circ \) and \( \csc 30^\circ \) by evaluating whether \( \frac{1}{\sin 20^\circ} < \frac{1}{\sin 30^\circ} \).
5. Step 5: Conclude whether the statement \( \csc 20^\circ < \csc 30^\circ \) is true or false based on the comparison in Step 4.

Key Concepts

Cosecant Function

The cosecant function, denoted as csc, is the reciprocal of the sine function. It is defined as csc(θ) = 1/sin(θ). Understanding the values of sine at specific angles is crucial for evaluating cosecant, as it directly influences the comparison of cosecant values for different angles.

Angle Comparison

When comparing trigonometric functions of different angles, it is essential to understand the behavior of these functions within the unit circle. For angles in the first quadrant (0° to 90°), sine values increase, which means cosecant values will decrease as the angle increases. This property is vital for determining the truth of statements involving inequalities between cosecant values.

Trigonometric Inequalities

Trigonometric inequalities involve comparing the values of trigonometric functions at different angles. To solve these inequalities, one must evaluate the functions at the specified angles and understand their relationships. In this case, determining whether csc(20°) is less than csc(30°) requires calculating or estimating the cosecant values and applying the properties of the sine function.