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


sec s = 1.0806

Verified step by step guidance

1. Recognize that \( \sec(s) = \frac{1}{\cos(s)} \). Therefore, \( \sec(s) = 1.0806 \) implies \( \frac{1}{\cos(s)} = 1.0806 \).
2. Rearrange the equation to solve for \( \cos(s) \): \( \cos(s) = \frac{1}{1.0806} \).
3. Calculate \( \cos(s) \) using the value from the previous step.
4. Use the inverse cosine function to find \( s \): \( s = \cos^{-1}(\cos(s)) \).
5. Ensure that the value of \( s \) is within the interval \([0, \frac{\pi}{2}]\).

Key Concepts

Secant Function

The secant function, denoted as sec(s), is the reciprocal of the cosine function. It is defined as sec(s) = 1/cos(s). Understanding this relationship is crucial for solving equations involving secant, as it allows us to express sec(s) in terms of cosine, which can then be manipulated to find the angle s.

Inverse Trigonometric Functions

Inverse trigonometric functions, such as arccosine, are used to find angles when the value of a trigonometric function is known. For example, if sec(s) = 1.0806, we can first find cos(s) and then use the arccos function to determine the angle s. This concept is essential for solving for angles in trigonometric equations.

Interval Restrictions

The interval [0, π/2] indicates that we are only considering angles in the first quadrant, where both sine and cosine are positive. This restriction is important because it ensures that the values of the trigonometric functions behave predictably, allowing us to find a unique solution for s within this range.