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: 6 yardsArc Length, s: 8 yards

Verified step by step guidance

1. Identify the formula for the radian measure of a central angle: \( \theta = \frac{s}{r} \), where \( s \) is the arc length and \( r \) is the radius.
2. Substitute the given values into the formula: \( s = 8 \) yards and \( r = 6 \) yards.
3. Calculate the radian measure by dividing the arc length by the radius: \( \theta = \frac{8}{6} \).
4. Simplify the fraction to find the radian measure of the central angle.
5. Express the simplified fraction as a radian measure.

Key Concepts

Radian Measure

Radian measure is a way of measuring angles based on the radius of a circle. One radian is the angle formed when the arc length is equal to the radius of the circle. This unit is essential in trigonometry as it relates the angle directly to the circle's dimensions, making calculations involving circular motion and periodic functions more intuitive.

Arc Length Formula

The arc length of a circle can be calculated using the formula s = rθ, where s is the arc length, r is the radius, and θ is the angle in radians. This relationship allows us to find the angle when the arc length and radius are known. Understanding this formula is crucial for solving problems involving circular motion and geometry.

Central Angle

The central angle of a circle is the angle whose vertex is at the center of the circle and whose sides extend to the circumference. It is directly related to the arc length it intercepts. Knowing how to calculate the central angle using the radius and arc length is fundamental in trigonometry, especially in problems involving circles and circular sectors.