Problem 2 - Thomas Calculus

Question

A central angle in a circle of radius 8 is subtended by an are of length 10\(\pi .\) Find the angle's radian and degree measures.

Step-by-Step Solution

Answer

The angle is \(\frac{5\pi}{4}\) radians or 225 degrees.

Step 1: Understand the Relationship

The length of an arc (\(L\)) in a circle is directly related to the central angle (\(\theta\)) subtending it, and the radius (\(r\)) of the circle. The formula is given by \(L = r\theta\), where \(\theta\) is in radians.

Step 2: Substitute and Solve for Theta in Radians

We are given \(L = 10\pi\) and \(r = 8\). Substitute these values into the formula \(L = r\theta\):\[ 10\pi = 8\theta \]Solve for \(\theta\) by dividing both sides by 8:\[ \theta = \frac{10\pi}{8} = \frac{5\pi}{4} \]

Step 3: Convert Radians to Degrees

To convert from radians to degrees, use the conversion factor \(180^\circ = \pi\) radians. Multiply \(\theta\) by \(\frac{180}{\pi}\):\[ \theta = \frac{5\pi}{4} \times \frac{180}{\pi} = \frac{5 \times 180}{4} = 225^\circ \]

Key Concepts

Arc Length

The concept of arc length is essential when studying circles. The arc length is the distance along the curved line making up the arc of a circle. It is a portion of the circle's circumference which is influenced by two main factors: the central angle and the radius of the circle.
The formula to calculate arc length is given by \(L = r\theta\), where \(L\) is the arc length, \(r\) is the radius, and \(\theta\) is the central angle in radians. This relationship highlights that the larger the central angle or the radius, the longer the arc length will be.

• When the central angle is non-zero, the arc length is greater than zero.
• An arc length would be the circle’s circumference if the central angle is \(360^\circ\) or \(2\pi\) radians.

Understanding this relationship allows you to solve various geometric problems involving circles, like in the exercise where the radius is 8 and the arc length given is \(10\pi\). By using the formula \(L = r\theta\), you can find the central angle when the arc length and radius are known.

Radian Measure

Radian measure is a way of expressing angles, different from degrees, that is based on the radius of the circle. A radian is defined as the angle created when the arc length is equal to the radius of the circle.
One full revolution around a circle (360 degrees) equals \(2\pi\) radians. This relationship between degrees and radians is pivotal for converting from one to the other.

• The formula for finding an angle in radians given the arc length \(L\) and radius \(r\) is \(\theta = \frac{L}{r}\).
• For instance, to find the central angle in radians from our exercise, given \(L = 10\pi\) and \(r = 8\), one would solve \(\theta = \frac{10\pi}{8} = \frac{5\pi}{4}\) radians.

Understanding radians helps visualize angles in terms of the circle's geometry, where each radian translates directly to a portion of the circle's circumference.

Degree Measure

Degree measure is the traditional way of measuring angles, usually used in day-to-day applications and educational settings. Degrees divide one full circle into 360 equal parts, making it intuitive and convenient for practical scenarios.
Converting between radian and degree measures is a common task, important for various applications.

• The conversion factor from radians to degrees is \(180^\circ = \pi\) radians.
• To convert from radians to degrees, multiply the radian value by \(\frac{180}{\pi}\).
• In the exercise solution, the angle \(\theta\) of \(\frac{5\pi}{4}\) radians is converted into degrees: \(\theta = \frac{5\pi}{4} \times \frac{180}{\pi} = 225^\circ\).

Having a grasp of converting between these units allows you to operate confidently in both scenarios, whether you're designing machinery or drawing geometric figures.