Find the approximate value of s, to four decimal places, in the interval [0, π/2] that makes each statement true.


tan s = 0.2126

Verified step by step guidance

1. Identify the trigonometric function involved: \( \tan(s) = 0.2126 \).
2. Recall that the tangent function, \( \tan(s) \), is the ratio of the opposite side to the adjacent side in a right triangle.
3. To find \( s \), use the inverse tangent function: \( s = \tan^{-1}(0.2126) \).
4. Ensure that the solution is within the given interval \([0, \frac{\pi}{2}]\).
5. Use a calculator to find \( s \) to four decimal places, ensuring the calculator is set to the correct mode (radians).

Key Concepts

Tangent Function

The tangent function, denoted as tan, is a fundamental trigonometric function defined as the ratio of the opposite side to the adjacent side in a right triangle. It can also be expressed in terms of sine and cosine as tan(s) = sin(s)/cos(s). Understanding the behavior of the tangent function, especially within specific intervals, is crucial for solving equations involving tan.

Inverse Trigonometric Functions

Inverse trigonometric functions, such as arctan, are used to find angles when the value of a trigonometric function is known. For example, if tan(s) = 0.2126, then s can be found using s = arctan(0.2126). These functions are essential for determining angles in various contexts, particularly when working with specific values of trigonometric ratios.

Quadrants and Angle Ranges

Trigonometric functions have different signs in different quadrants of the unit circle. The interval [0, π/2] corresponds to the first quadrant, where both sine and cosine are positive, and thus tangent is also positive. Recognizing the appropriate quadrant is important for determining the correct angle that satisfies the given trigonometric equation.