Problem 79 - A Graphical Approach to Precalculus with Limits

Question

Convert each degree measure to radians. Round to the nearest hundredth. $$139^{\circ} 10^{\prime}$$

Step-by-Step Solution

Answer

The angle \(139^{\circ} 10^{\prime}\) is approximately \(2.43\) radians.

Step 1: Understand the problem

The problem requires us to convert an angle given in degrees and minutes, specifically \(139^{\circ} 10^{\prime}\), to radians and round the result to the nearest hundredth.

Step 2: Convert minutes to degrees

Degrees are expressed with a whole number plus minutes, where \(1^{\prime} = \frac{1}{60}^{\circ}\). Convert the \(10^{\prime}\) portion of the angle to degrees: \[ 10^{\prime} = \frac{10}{60}^{\circ} = 0.16667^{\circ} \]

Step 3: Calculate the total degrees

Add the converted minutes to the whole degrees to get the total degrees: \[ 139^{\circ} + 0.16667^{\circ} = 139.16667^{\circ} \]

Step 4: Convert degrees to radians

Use the conversion factor \( \pi \text{ radians} = 180^{\circ} \). Convert the total degrees to radians: \[ 139.16667^{\circ} \times \frac{\pi}{180} \approx 2.4296 \text{ radians} \]

Step 5: Round the result

Round the result to the nearest hundredth, which gives approximately: \[ 2.43 \text{ radians} \]

Key Concepts

Angle Measurement

In mathematics, angles play a crucial role in geometry and trigonometry. An angle is a measure of rotation between two intersecting lines. Understanding how angles are measured helps interpret geometric shapes and improve calculation precision. Angles can be measured using various units. Two common units are degrees and radians.
- **Degrees**: This is the most common unit of measuring angles. A full circle is divided into 360 degrees. - **Radians**: Radians measure angles based on the radius of the circle. In a full circle, the angle is 2π radians.
The ability to convert between these units is essential in solving geometric problems, especially when dealing with fields such as physics and engineering.

Degree and Minutes

When measuring angles in degrees, sometimes we need more precision. This is where minutes and seconds come in. Similar to time, where 1 hour equals 60 minutes, in the context of angles, 1 degree is divided into 60 minutes ('). This allows for more detailed description and calculation of angles.
For example, if we have an angle of 139 degrees and 10 minutes (denoted as \(139^{\circ} 10^{\prime}\)), it implies there are an additional 10/60 degrees added to the 139 degrees.- To simplify calculations, convert minutes to a decimal by dividing by 60. This practice helps in improving the precision of angle measurements when working in fields that demand high accuracy.

Radians

Radians are another unit of measuring angles, considered more mathematically significant than degrees. This is because radians relate directly to arc length and circle radius, providing a natural measure of angles within the context of the unit circle.
- **Definition**: One radian is the angle formed when the arc length is equal to the radius of the circle. This leads to the fact that in a full circle (with circumference 2πr), there are 2π radians. - **Usage**: In calculus and advanced mathematics, radians are preferred, especially due to their properties in sinusoidal functions and other trigonometric calculations.
Understanding radians helps in linking the geometric concepts of length and angle. This connection is pivotal in more advanced mathematics, engineering, and physics computations.

Mathematics Conversion

Mathematics conversions are the backbone of problem-solving in various fields. The ability to convert units—like angles from degrees to radians—is fundamental for solving real-world problems efficiently. When converting between these units, understanding the foundational conversion relationships is key.
- **Degrees to Radians Conversion**: The conversion formula is rooted in the equivalence of 180 degrees to π radians. To convert, use: \[ ext{Radians} = ext{Degrees} \times \frac{\pi}{180} \]This formula transforms angles measured in degrees into their radian equivalent.
By mastering this conversion, calculations across various fields requiring different unit usage become more straightforward. Whether in engineering tasks needing precise mathematical models or in navigation tasks demanding accurate angular readings, understanding these conversions widens the capability to tackle a plethora of activities.