In Exercises 7–12, find the radian measure of the central angle of a circle of radius r that intercepts an arc of length s. Radius, r: 10 inches Arc Length, s: 40 inches

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Recall the formula that relates the arc length \(s\), the radius \(r\), and the central angle \(\theta\) in radians: \[ s = r \times \theta \]

Identify the given values from the problem: radius \(r = 10\) inches and arc length \(s = 40\) inches.

Substitute the known values into the formula: \[ 40 = 10 \times \theta \]

Solve for the central angle \(\theta\) by dividing both sides of the equation by the radius \(r\): \[ \theta = \frac{40}{10} \]

Simplify the fraction to express \(\theta\) in radians, which will give the radian measure of the central angle.

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주요 개념

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Radian Measure of an Angle

A radian is a unit of angular measure based on the radius of a circle. One radian is the angle subtended at the center of a circle by an arc whose length is equal to the radius. It provides a natural way to relate angles to arc lengths without converting to degrees.

Relationship Between Arc Length, Radius, and Central Angle

The central angle θ (in radians) of a circle is related to the arc length s and radius r by the formula θ = s / r. This formula allows you to find the angle when the arc length and radius are known, making it essential for solving problems involving circular arcs.

Units and Conversion in Trigonometry

Understanding units is crucial; arc length is measured in linear units (e.g., inches), radius in the same units, and the central angle in radians (a dimensionless measure). Ensuring consistent units and interpreting the result in radians is key to correctly solving and applying trigonometric problems.