Problem 5 - Algebra 2

Question

Rewrite each degree measure in radians and each radian measure in degrees. \(130^{\circ}\)

Step-by-Step Solution

Answer

\(130^{\circ} = \frac{13\pi}{18}\) radians.

Step 1: Understand the Problem

We need to convert the angle given in degrees into radians. The angle provided is \(130^{\circ}\).

Step 2: Recall Conversion Formula

The formula to convert from degrees to radians is: \[ \text{radians} = \text{degrees} \times \frac{\pi}{180^{\circ}} \]

Step 3: Apply Conversion Formula

Substitute \(130^{\circ}\) into the conversion formula: \[ \text{radians} = 130^{\circ} \times \frac{\pi}{180^{\circ}} \]

Step 4: Simplify the Expression

Calculate the conversion: \[ \text{radians} = \frac{130\pi}{180} \] Simplify the fraction: \[ \frac{130}{180} = \frac{13}{18} \] so the expression becomes \(\frac{13\pi}{18}\).

Step 5: Write the Converted Measure

The angle \(130^{\circ}\) is equivalent to \(\frac{13\pi}{18}\) radians.

Key Concepts

Degree to Radian Conversion

Angle conversion is an important skill in mathematics, especially when dealing with trigonometry and calculus. To convert angles from degrees to radians, we use a specific formula. Degrees and radians measure angles, but they use different units. This conversion allows us to switch between these two measures effectively. The process to convert degrees into radians involves multiplying the degree value by a constant factor, which is derived from the mathematical relationship between a circle's degrees and radians. This is useful because many mathematical formulas use radians, particularly in higher-level math. So, understanding this conversion is essential to adapt angle measures to fit various calculations.

Angle Measurement

Angles can be measured in two primary units: degrees and radians. A full circle is 360 degrees or \(2\pi\) radians. While degrees are often used in everyday situations and are easier to understand initially, radians are preferred in calculus and higher mathematics because they simplify many calculations involving circular motion. Measuring angles in radians can make trigonometric equations more intuitive and easier to manipulate. Remember, one radian is the angle created when the radius of a circle is wrapped along the circle's edge, which corresponds to about 57.3 degrees. Practicing these conversions can enhance your understanding and application of math concepts.

Radian Measure

Radian measure is a way to describe an angle in terms of the radius of a circle. In simpler terms, it's how many times the radius fits along the circumference of a circle. Unlike degrees, which are a more arbitrary measure developed for convenience, radians are based on the properties of circles themselves. This makes radian measure a natural fit for many scientific and engineering applications. To visualize one radian, imagine taking the radius of a circle and laying it along the circle's edge. Understanding radian measure can be crucial in subjects like physics where angular measurements are frequently used.

Conversion Formula

To switch between angles measured in degrees to those measured in radians, we use a conversion formula. The formula is simple yet powerful: \[\text{radians} = \text{degrees} \times \frac{\pi}{180^{\circ}}\].

• This formula comes from the fact that a full circle is \(360^{\circ}\) or \(2\pi\) radians.
• By setting these proportionally equal, we derive the constant \(\frac{\pi}{180}\).

To use the formula, you simply multiply the angle in degrees by \(\frac{\pi}{180}\). For example, when converting \(130^{\circ}\), you calculate: \(130^{\circ} \times \frac{\pi}{180}\), resulting in \(\frac{13\pi}{18}\) radians. This conversion is straightforward when you apply the formula and reinforces the relationship between these two units of angle measurement.