Problem 34 - Algebra 2 and Trigonometry

Question

In \(28-37, \theta\) is the radian measure of a central angle that intercepts an arc of length \(s\) in a circle with a radius of length \(r .\) If \(r=4.2\) and \(s=21,\) find \(\theta\)

Step-by-Step Solution

Answer

\(\theta = 5\) radians.

Step 1: Understand the Formula

Realize that the relationship between arc length \(s\), radius \(r\), and angle \(\theta\) in radians is given by the formula: \(s = r\theta\). We are going to use this formula to find \(\theta\).

Step 2: Plug Values into the Formula

Insert the given values of \(s\) and \(r\) into the formula: \(21 = 4.2\theta\).

Step 3: Solve for \(\theta\)

Rearrange the equation to solve for \(\theta\) by dividing both sides of the equation by 4.2: \(\theta = \frac{21}{4.2}\).

Step 4: Perform the Division

Calculate the value of \(\theta\) by performing the division: \(\theta = 5\).

Key Concepts

Central Angle

A central angle in a circle is an angle whose apex (vertex) is located at the center of the circle, and whose sides (arms) extend to the circumference, creating what we call an arc on the circle's edge. The measure of this central angle plays a crucial part in calculating both the arc length and the relation to the circle's radius.

• Central angles are measured in radians, unlike traditional degrees.
• A complete rotation around the circle amounts to an angle of \(2\pi\) radians.
• Understanding central angles as part of this system allows us to apply important formulas connecting angles to arc length.

The radian measure simplifies calculations in circular geometry, particularly when finding lengths and areas associated with the circle. Knowing the measure of a central angle is essential for determining how much of the circle's perimeter (or arc length) it intercepts.

Arc Length

Arc length refers to the distance measured along the curved line forming the arc, which is a segment of the circle's circumference.

Calculating the arc length involves the formula: \[ s = r\theta \] where:

• \(s\) is the arc length.
• \(r\) is the radius of the circle.
• \(\theta\) is the central angle in radians.

This formula is derived from the proportion of the circle's circumference that is represented by the arc. If you understand this concept, determining arc length becomes straightforward, significantly benefiting from using radians. For example, when given the radius and arc length, it is easy to determine the central angle, or vice versa, using this foundational relationship.

Radius

The radius of a circle is a line from its center to any point on its circumference. It is crucial for many calculations involving circles, including those for arc length and central angles.

• The radius is constant for a given circle, making it a key component in the circle's formulas.
• It doubles to become the diameter.
• Working with the radius simplifies the expression of relationships within the circle through the radian measure.

When solving problems concerning circles, the radius helps create ratios and relationships between angles and distances, especially when working with radian measures. This allows for direct calculations of arc lengths and ultimately transforms the abstract notion of radian angles into concrete measures of distance on a circle.

Angle Conversion

Angle conversion refers to the process of changing angle measurements from degrees to radians, or vice versa. This conversion is essential because many formulas in geometry, especially those involving circles, are more naturally expressed in radians.

To convert:

• From degrees to radians, multiply by \( \frac{\pi}{180} \).
• From radians to degrees, multiply by \( \frac{180}{\pi} \).

Understanding and performing these conversions allow you to apply formulas in their most natural context, ensuring accuracy and compatibility with mathematical constants like \( \pi \). This is particularly useful in problems involving circles, where radian measure directly ties the angle to the circle's geometry, making these conversions essential for solving many geometry problems.