Determine whether each statement is true or false. See Example 4. tan 28° ≤ tan 40°

Key Concepts

Tangent Function

The tangent function, denoted as tan(θ), is a fundamental trigonometric function defined as the ratio of the opposite side to the adjacent side in a right triangle. It is also expressed as tan(θ) = sin(θ) / cos(θ). The tangent function is periodic and increases from negative infinity to positive infinity as the angle approaches 90 degrees.

Monotonicity of the Tangent Function

The tangent function is monotonically increasing in the interval (0°, 90°). This means that as the angle increases within this range, the value of the tangent function also increases. Therefore, if θ1 < θ2, then tan(θ1) < tan(θ2) for angles in this interval, which is crucial for comparing tangent values.

Comparison of Angles

When comparing angles in trigonometry, it is essential to understand their relative sizes. If one angle is less than another, and both angles are within the same range where the tangent function is increasing, the tangent of the smaller angle will also be less than the tangent of the larger angle. This principle allows us to determine the truth of statements involving inequalities of tangent values.