Problem 23 - Algebra and Trigonometry

Question

Find the degree measure of the angle with the given radian measure. $$ \frac{\pi}{10} $$

Step-by-Step Solution

Answer

18 degrees

Step 1: Understand the problem

The task is to convert an angle given in radians, \( \frac{\pi}{10} \), to degrees.

Step 2: Recall the conversion formula

To convert from radians to degrees, use the formula: \( \text{Degrees} = \text{Radians} \times \frac{180}{\pi} \).

Step 3: Substitute the given value

Substitute \( \frac{\pi}{10} \) into the conversion formula: \[ \text{Degrees} = \frac{\pi}{10} \times \frac{180}{\pi} \].

Step 4: Perform the calculation

The \( \pi \) in the numerator and denominator cancel out, so calculate: \( \frac{180}{10} \).

Step 5: Simplify the result

Divide 180 by 10 to get the degree measure: 18 degrees.

Key Concepts

angle measurement

Angle measurement is about determining the size of the angle, typically using either degrees or radians. Angles are a way to describe the rotation from one ray to another around their common endpoint. Understanding how to measure angles is important in geometry, trigonometry, and everyday applications like navigation and engineering.

There are two main units used for measuring angles:

• **Degrees** - The most common unit, where a full circle is divided into 360 parts, making each part one degree.
• **Radians** - The standard unit in mathematics, where a full circle is divided into 2π, making one full rotation equivalent to 2π radians.

Understanding the relationship between these units is crucial for converting from one to another. Knowing how to measure angles gives you the foundation to tackle more complex mathematical problems.

degree measure

A degree is one of the units commonly used to measure angles. It is represented by the symbol °. When thinking about degrees, imagine cutting a pizza into 360 slices. Each slice represents one degree of the full circle, which is useful for both geometry and practical applications like telling time.

A measure in degrees makes it very intuitive to understand and visualize angles:

• A right angle is 90°
• A straight angle is 180°
• A full rotation or circle is 360°

When working with degrees, calculations and comparisons can become easier for those used to this measurement. It connects neatly with various aspects of math and is often more tangible in everyday contexts than radians.

conversion formula

Converting between radians and degrees is essential to switch from mathematics-focused calculations to more practical applications. The conversion formula helps facilitate this transition by providing a straightforward method: \[\text{Degrees} = \text{Radians} \times \frac{180}{\pi}\]This formula arises because a full circle is 360° and 2π radians. When converting, you scale the radian measure by this ratio to get the equivalent in degrees.

Key steps in using the conversion formula:

• Identify the angle in radians.
• Apply the formula by multiplying the radians by \(\frac{180}{\pi}\).
• Simplify the expression to find the degree measure.

By mastering this conversion, you can seamlessly translate between the radians used in mathematical equations and the degrees commonly used in real-world scenarios.