Problem 29 - Basic Technical Mathematics with Calculus

Question

Change the given angles to radian measure. $$125.2^{\circ}$$

Step-by-Step Solution

Answer

Approximately 2.184 radians.

Step 1: Understanding the Conversion Factor

To convert degrees to radians, we use the conversion factor \( \frac{\pi}{180} \). This is because 180 degrees is equivalent to \( \pi \) radians.

Step 2: Apply the Conversion Formula

To convert \( 125.2^{\circ} \) to radians, we multiply by the conversion factor:\[125.2^{\circ} \times \frac{\pi}{180} = ?\]

Step 3: Calculate the Radian Measure

Now, let's perform the multiplication: \[125.2 \times \frac{\pi}{180} = \frac{125.2\pi}{180}\]

Step 4: Simplify the Fraction

Simplify \( \frac{125.2\pi}{180} \) if possible. In this case, it simplifies to approximately 2.184 radians (depending on desired precision).

Key Concepts

Angle Conversion

Angle conversion refers to changing the measurement of angles from one unit to another. The two most common units used to measure angles are degrees and radians. Degrees are often used in everyday contexts, like describing temperatures or navigation. Radians, however, are primarily used in higher-level mathematics, especially in calculus and trigonometry.

When converting angles, it is crucial to understand the relationship between these units to avoid mistakes. A complete circle is equal to 360 degrees, which is also equal to \(2\pi\) radians. This relationship provides the foundation for converting angles between degrees and radians. Whether converting from degrees to radians or the other way around, the conversion ensures that the angle's size remains consistent regardless of measurement unit.

Understanding angle conversion is essential for solving mathematical problems that require trigonometric computations, as many functions in mathematics are defined using radians rather than degrees.

Degree to Radian

The degree to radian conversion is an essential skill for students, especially those involved in trigonometry and calculus. Degrees are the "common" unit, used often in elementary discussions of geometry. However, when you move into evaluation and analysis of trigonometric functions, radians become much more useful.

• One complete circle in degrees is 360 degrees.
• In radians, it's \(2\pi\) radians. So, half a circle - 180 degrees - equals \(\pi\) radians.

For converting degrees to radians, we utilize a straightforward formula by multiplying the degree measure by the conversion factor \(\frac{\pi}{180}\). This operation scales the degree measure properly into radians.

For example, to convert \(125.2^\circ\) to radians:

• Multiply \(125.2\) by \(\frac{\pi}{180}\).
• Compute: \[125.2 \times \frac{\pi}{180} = \frac{125.2\pi}{180}\]
• Typically, simplification follows, to yield a neat expression or an approximation, such as 2.184 radians.

Conversion Factor

The conversion factor is a key component in changing degree measurements into radian measures. The purpose of a conversion factor is to enable a seamless transition between units without altering the original value's meaning. In the case of radians and degrees, the conversion factor is \(\frac{\pi}{180}\).

• This factor comes from the fact that \(180^\circ\) equals \(\pi\) radians.
• This implies we can divide the number of degrees by 180, then multiply by \(\pi\) to achieve the equivalent radian measure.

Using the conversion factor transforms the units while keeping the angle equivalent. This standard approach allows anyone to easily switch between using radians and degrees, which can optimize problem solving in both educational and professional settings. In our worked example, using the conversion factor allowed the angle \(125.2^\circ\) to be precisely converted into \(2.184\) radians.