Find the approximate value of s, to four decimal places, in the interval [0, π/2] that makes each statement true.
cos s = 0.9250

Verified step by step guidance

Identify the problem: We need to find the angle \( s \) in the interval \([0, \frac{\pi}{2}]\) such that \( \cos s = 0.9250 \).

Recall that the cosine function is the ratio of the adjacent side to the hypotenuse in a right triangle, and it is also an even function, meaning it is symmetric about the y-axis.

Use the inverse cosine function, \( \cos^{-1} \), to find the angle \( s \). This function will help us determine the angle whose cosine is 0.9250.

Calculate \( s = \cos^{-1}(0.9250) \) using a calculator or a trigonometric table to find the angle in radians.

Ensure that the calculated angle \( s \) is within the specified interval \([0, \frac{\pi}{2}]\).

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Cosine Function

The cosine function is a fundamental trigonometric function that relates the angle of a right triangle to the ratio of the length of the adjacent side to the hypotenuse. It is defined for all real numbers and is periodic with a period of 2π. In the context of the unit circle, the cosine of an angle corresponds to the x-coordinate of the point on the circle at that angle.

Inverse Trigonometric Functions

Inverse trigonometric functions, such as arccosine, are used to find the angle that corresponds to a given trigonometric ratio. For example, if cos(s) = 0.9250, then s can be found using s = arccos(0.9250). These functions are essential for solving equations involving trigonometric ratios and are typically restricted to specific intervals to ensure they return a unique angle.

Quadrants and Angle Ranges

Understanding the quadrants of the unit circle is crucial for determining the values of trigonometric functions. The interval [0, π/2] corresponds to the first quadrant, where both sine and cosine values are positive. This knowledge helps in identifying the appropriate angle that satisfies the given cosine value, ensuring that the solution lies within the specified range.

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