Problem 10 - Precalculus

Question

Find the measure (in radians) of a central angle \(\theta\) that intercepts an are of length \(s\) on a circle with radius \(r\). \(r=1 \mathrm{m}, s=2 \mathrm{cm}\)

Step-by-Step Solution

Answer

The central angle \( \theta \) is 0.02 radians.

Step 1: Convert Units if Necessary

Before proceeding, ensure that all measurements are in the same unit. Here, the radius is given as 1 meter and the arc length as 2 centimeters. Convert the arc length into meters: 2 cm = 0.02 m. Now both the radius and arc length are in meters.

Step 2: Use the Formula for Central Angle

The formula to find the central angle \( \theta \) in radians is \[ \theta = \frac{s}{r} \] where \( s \) is the arc length and \( r \) is the radius of the circle.

Step 3: Substitute the Values into the Formula

Substitute the values for \( s \) and \( r \) into the formula: \[ \theta = \frac{0.02}{1} = 0.02 \]

Step 4: Interpret the Result

The central angle \( \theta \) that intercepts the arc with the given length on the circle is 0.02 radians.

Key Concepts

Radians

Radians are a way of measuring angles. Unlike degrees, which divide a circle into 360 parts, radians are based on the relationship between a circle's arc length and its radius.
A full circle in radians is equivalent to the ratio of the circle's circumference to its radius. This is approximately 6.28318 radians, but is more commonly known as 2π radians. By using radians, calculations in mathematics, especially those involving trigonometric functions, become much simpler and more convenient.

• A quarter circle is \(\frac{\pi}{2}\) radians.
• Half a circle is \(\pi\) radians.
• And a full circle makes 2\(\pi\) radians.

Understanding radians as a measurement allows us to connect the geometric world with mathematical functions efficiently. They are essential for solving problems involving arcs and circles.

Arc Length

Arc length is a portion of the circumference of a circle. When dealing with circles, knowing arc length helps us understand various properties related to circles and their sectors.
The formula to find the arc length \(s\), in terms of a central angle \(\theta\) in radians and the circle's radius \(r\), is given by:\[ s = \theta \times r \]So, if you know the circle's radius and the central angle in radians, you can easily determine the length of an arc. This relationship is particularly useful in real-world applications, such as designing gears or determining distances along a circular path.

• The arc length represents a direct relationship between the angle and the radius.
• An increase in the central angle or the circle's radius will lead to a longer arc.
• Arc lengths are crucial in understanding the geometry of circles and trigonometry.

Understanding arc length and its calculation ensure effective solving of problems related to circular motion and design.

Circle Radius

The radius of a circle is the distance from the center of the circle to any point on its circumference. The radius is one of the most critical aspects of a circle, defining its size and influencing every property associated with it.

• The radius is half of the diameter.
• Any changes in the radius result in proportional changes in circumference and area.
• The formula for the area of a circle is \(\pi r^2\), and the circumference is \(2\pi r\).

In the context of central angles and arc lengths, the radius forms the basis for calculating how large an angle or how long an arc is relative to the circle as a whole. It links the linear dimensions to angular measures efficiently.
Having the radius allows you to determine other essential properties of the circle and better understand the mathematical concept of circular geometry. Studying how the radius affects other properties is crucial in fields like physics, engineering, and more.