Problem 28 - Precalculus

Question

Convert from degrees to radians. Leave the answers in terms of \(\pi\). $$1800^{\circ}$$

Step-by-Step Solution

Answer

The conversion of \(1800^{\circ}\) to radians is \(10\pi\).

Step 1: Understand the Conversion Formula

To convert degrees to radians, we use the conversion formula: \[ \theta_{radians} = \theta_{degrees} \times \frac{\pi}{180} \] This formula relies on the fact that \(180^{\circ} = \pi\) radians. Therefore, multiplying by \(\frac{\pi}{180}\) converts degrees to radians.

Step 2: Insert the Degree Value

Insert \(1800^{\circ}\) into the conversion formula: \[ 1800 \times \frac{\pi}{180} \] This replaces \(\theta_{degrees}\) in the formula with the given degree measure.

Step 3: Simplify the Fraction

Calculate the fraction: \[ 1800 \times \frac{\pi}{180} = \frac{1800}{180} \times \pi \] Simplify \(\frac{1800}{180}\), which results in \(10\). Therefore, the expression simplifies to \(10\pi\).

Step 4: Final Answer

The conversion of \(1800^{\circ}\) to radians is \[ 10\pi \] Thus, in terms of \(\pi\), the radians measurement is \(10\pi\).

Key Concepts

Radian Measure

Understanding the concept of radian measure is crucial in the study of angles and their conversions. A radian is a way to measure angles, very much like degrees but based on the circle itself. Imagine wrapping the radius of a circle around its circumference. The angle formed when the arc length is equal to the radius is defined as one radian.
Now, knowing that the circumference of a circle is \(2\pi\) times the radius, a full circle contains \(2\pi\) radians. This is equivalent to \(360^\circ\). Hence, \(180^\circ\) is equivalent to \(\pi\) radians.
This relationship gives us a fundamental understanding that helps in conversion between degrees and radians. In mathematical terms, multiplying an angle measured in degrees by \(\frac{\pi}{180}\) converts it into radians directly. This is essential knowledge for various mathematical fields, be it geometry, trigonometry, or calculus.

Angle Conversion Formula

The angle conversion formula is a key tool that allows you to transition between the degree and radian measures. Degrees are often easier for humans to comprehend, while radians offer a more natural measurement from a mathematical perspective.
The conversion formula is given by:

• \( \theta_{radians} = \theta_{degrees} \times \frac{\pi}{180} \)

This formula uses the fact that \(180^\circ = \pi\) radians. It efficiently translates degree measurements into radians by applying this simple multiplication.
For instance, when given \(1800^\circ\), using the formula:

• \(1800 \times \frac{\pi}{180}\)

transforms the measure into radians. The computation begins with substituting the degree value into the formula, leading naturally to the next step of simplifying fractions, ensuring clarity and accuracy in your calculations.

Simplifying Fractions

Simplifying fractions is a key mathematical skill, ensuring that expressions are reduced to their most basic form. This is particularly relevant when converting degrees to radians using fractional terms with \(\pi\).
In the example of \(1800 \times \frac{\pi}{180}\), you need to simplify the fraction \(\frac{1800}{180}\). This process involves dividing both the numerator and the denominator by their greatest common divisor, which in this case is 180. Therefore, \(\frac{1800}{180}\) simplifies to 10.
After the fractional component is simplified, the expression now reads as \(10\pi\). Keeping fractions simplified helps maintain neatness and clarity in your mathematical expressions, allowing for easier interpretation and use in subsequent calculations.