(a) Find the radian and degree measures of the central angle θ subtended by the given arc of len

step 1 So in the given question, we are given the arc length S of a circle, arc length as 7 centimeters

(a) Find the radian and degree measures of the central angle...

(a) Find the radian and degree measures of the central angle $\boldsymbol{\theta}$ subtended by the given arc of length $s$ on a circle of radius $r$ (b) Find the area of the sector determined by $\boldsymbol{\theta}$.
$$s=7 \mathrm{cm}, \quad r=4 \mathrm{cm}$$

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Key Concepts

Central Angle and its Radian Measure

In a circle, the central angle (measured in radians) is determined using the relationship ? = s / r, where s is the arc length and r is the radius. This concept connects the linear measure of the arc to the angular measure at the center, forming the basis of circular geometry.

Conversion Between Radians and Degrees

Angles can be expressed in radians or degrees. The conversion from radians to degrees is performed by multiplying by 180/?, making it easier to interpret angles in the context of familiar degree measures, while preserving the accuracy of calculations in geometry and trigonometry.

Arc Length

Arc length is the distance along a portion of the circumference of a circle. It is directly proportional to the central angle in radians and the radius of the circle, emphasizing the fundamental link between circular motion and angular displacement.

Area of a Sector

The area of a sector, a 'slice' of a circle defined by two radii and the intercepted arc, is calculated using the formula (1/2)r²? when the angle is in radians. This formula quantifies how the area scales with both the size of the radius and the central angle.

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