Determine whether each statement is true or false. See Example 4. cos 28° < sin 28° (Hint: sin 28° = cos 62°)

Verified step by step guidance

1. Recall the complementary angle identity: \(\sin \theta = \cos (90^\circ - \theta)\). Using this, verify the hint given: \(\sin 28^\circ = \cos 62^\circ\).
2. Rewrite the inequality \(\cos 28^\circ < \sin 28^\circ\) using the hint as \(\cos 28^\circ < \cos 62^\circ\).
3. Understand the behavior of the cosine function between \(0^\circ\) and \(90^\circ\): cosine decreases as the angle increases in this interval.
4. Since \(28^\circ < 62^\circ\) and cosine is decreasing in this range, \(\cos 28^\circ\) is greater than \(\cos 62^\circ\).
5. Conclude that the original inequality \(\cos 28^\circ < \sin 28^\circ\) is false because it is equivalent to \(\cos 28^\circ < \cos 62^\circ\), which is not true.

Key Concepts

Complementary Angle Relationship

In trigonometry, the sine of an angle equals the cosine of its complement, meaning sin(θ) = cos(90° - θ). This relationship helps compare trigonometric values by converting sine functions into cosine functions or vice versa, as shown by sin 28° = cos 62°.

Properties of the Cosine Function in the First Quadrant

Cosine values decrease as the angle increases from 0° to 90°. Since 28° < 62°, cos 28° is greater than cos 62°, which helps determine the inequality between cos 28° and sin 28°.

Comparing Trigonometric Values

To compare trigonometric expressions, it is useful to rewrite them using known identities or evaluate their approximate values. Understanding how sine and cosine values change with angles allows for determining inequalities without a calculator.