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

Verified step by step guidance

1. insert step 1: Understand the problem by identifying that you need to compare the tangent values of two angles, 28° and 40°.
2. insert step 2: Recall that the tangent function, \( \tan(\theta) \), is an increasing function in the interval \( 0° < \theta < 90° \).
3. insert step 3: Since both 28° and 40° are within this interval, the tangent of a larger angle will be greater than the tangent of a smaller angle.
4. insert step 4: Compare the angles: 28° is less than 40°, so \( \tan(28°) \) should be less than \( \tan(40°) \).
5. insert step 5: Conclude that the statement \( \tan(28°) \leq \tan(40°) \) is true based on the properties of the tangent function.

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.