Problem 362 - Precalculus

Question

Convert \(-620^{\circ}\) to radians.

Step-by-Step Solution

Answer

The radian value is \(-\frac{31\pi}{9}\).

Step 1: Understand the Conversion Formula

To convert degrees to radians, we use the formula: \( \text{radians} = \text{degrees} \times \frac{\pi}{180} \). This formula is derived from the fact that \(180^{\circ} = \pi\) radians.

Step 2: Substitute the Degree Value

Substitute \(-620^{\circ}\) into the conversion formula: \[ \text{radians} = -620 \times \frac{\pi}{180} \].

Step 3: Simplify the Fraction

Calculate \(-620 \times \frac{\pi}{180}\). Start by simplifying the fraction \(\frac{-620}{180}\).

Step 4: Reduce the Fraction

First, find the greatest common divisor (GCD) of 620 and 180, which is 20. Simplifying gives \(\frac{-620}{180} = \frac{-31}{9}\).

Step 5: Calculate the Final Radian Value

Substitute back the simplified fraction into the formula: \(\text{radians} = \frac{-31}{9}\pi\). Therefore, \(-620^{\circ}\) in radians is \(-\frac{31\pi}{9}\).

Key Concepts

Radian Measure

When we talk about the radian measure in mathematics, we are referring to a way of describing angles. Unlike degrees, which are more familiar in everyday contexts, radians offer a more natural mathematical unit.

A radian is based on the radius of a circle. Imagine wrapping the radius of a circle along its circumference: the angle that it creates on the circle is exactly one radian. There are two key points to remember about radians:

• One full circle is equal to 2π radians.
• A straight line or a half circle measures π radians.

Recognizing these basic facts makes it easier to picture angles in terms of radians, which is crucial for performing various calculations in geometry and trigonometry.

Degree Measure

The degree measure is the more familiar way for most people to measure angles. This system divides a full circle into 360 equal parts, which we call degrees.

Here are some key insights:

• 1 degree is 1/360 of a complete circle.
• Basic geometric shapes often use degree angles — for example, a right angle is 90 degrees, a straight line is 180 degrees.

Degrees are widely used because they offer simple, whole-number angles in many daily applications — like maps, clocks, and building plans.

However, when performing complex mathematics, and especially in calculus, radians are more efficient and simpler to use.

Angle Conversion

Angle conversion is an essential concept that allows you to switch between degree and radian measures. These conversions rely on the relationship between the two measures:

The core formula for converting degrees to radians is
\[ ext{radians} = ext{degrees} imes \frac{\pi}{180} \]

Conversely, if you need to convert from radians to degrees, you'll use:
\[ ext{degrees} = ext{radians} \times \frac{180}{\pi} \]

These formulas arise because a full circle is both 360 degrees and 2π radians. By understanding this basic relationship, you can easily transition between radians and degrees.

This skill is especially useful when solving trigonometric problems and working with calculus, where angles might be given in one system but you need them in another.

Math Formulas

In mathematics, formulas serve as tools that simplify and solve problems. The degree to radian conversion relies heavily on the core formula:

\[ \text{radians} = \text{degrees} \times \frac{\pi}{180} \]

This formula is derived from the equivalence of 180 degrees to π radians, a key fact when examining circular motion and various periodic phenomenon. The efficiency of mathematical formulas lies in their ability to transform complex data into understandable results.

For example, when handling negative angles, like in our original problem of converting \(-620^\circ\) to radians, applying the formula provides a standard method of finding equivalent radian measures as
\[-\frac{31\pi}{9}\].

Mastering formulas and their proper application is essential for anyone looking to deeply understand mathematics, as they provide simplified paths through otherwise complicated calculations.