Problem 21 - Algebra and Trigonometry

Question

\(13-24\) . Find the degree measure of the angle with the given radian measure. $$ \frac{\pi}{10} $$

Step-by-Step Solution

Answer

The degree measure is 18 degrees.

Step 1: Understanding the Problem

We need to convert the given angle from radians to degrees. The angle provided is \( \frac{\pi}{10} \) radians.

Step 2: Radian to Degree Conversion Formula

To convert an angle from radians to degrees, we use the conversion formula: \[\text{Degrees} = \text{Radians} \times \frac{180}{\pi}\] This formula helps us find the degree equivalent of a given radian measure.

Step 3: Applying the Conversion Formula

Substitute \( \frac{\pi}{10} \) for radians in the conversion formula:\[\text{Degrees} = \frac{\pi}{10} \times \frac{180}{\pi}\]This will allow us to convert the radian measure to degrees.

Step 4: Simplifying the Expression

Simplify the expression by canceling out \(\pi\):\[\text{Degrees} = \frac{1}{10} \times 180\]Multiply the remaining values to find the degree measure.

Step 5: Calculating the Final Result

Multiply \( \frac{1}{10} \) by 180 to get the final degree measure:\[\text{Degrees} = 18\]The angle \( \frac{\pi}{10} \) radians is equivalent to 18 degrees.

Key Concepts

Angle Conversion

When working with angles, you might encounter the need to switch between different measurement systems. This is known as angle conversion. There are two primary units to measure angles: degrees and radians. Understanding how to convert between these two is crucial in various fields like mathematics, physics, and engineering. To convert an angle from radians to degrees, you use a specific formula:

• The formula is: \[\text{Degrees} = \text{Radians} \times \frac{180}{\pi}\]
• This conversion is based on the relationship that \(\pi\) radians is equivalent to 180 degrees.

Whenever you need to convert radians to degrees, simply plug in the value of your angle in radians into the formula and perform the multiplication.

Degree Measure

The degree measure is one of the most common ways to express angles. A full circle is 360 degrees, making it easy to divide and understand. This system is especially beneficial in everyday applications, like navigation and construction.

• Degrees can be divided into smaller units: minutes (\(60\) minutes in a degree) and seconds (\(60\) seconds in a minute).
• Degrees are symbolized by the "°" sign. For example: 45° means 45 degrees.

Degrees provide a straightforward method to express both whole and fractional angles, which is why they are widely used in various calculations.

Radian Measure

The radian measure offers another way to look at angles, primarily used in mathematical contexts. A radian is based on the radius of a circle. When the arc length is the same as the circle's radius, that angle is 1 radian.

• There are \(2\pi\) radians in a full circle.
• This makes radians a very natural unit for calculus and analysis because of their relationship with the circle's circumference.

Radians might seem less intuitive than degrees, but they offer elegance in mathematical equations, especially in trigonometry and calculus. For this reason, understanding radians is crucial for higher-level math.

Trigonometry Basics

Trigonometry is the study of angles and their relationships with functions like sine, cosine, and tangent. These concepts often use both degree and radian measurements.

• Trigonometric functions are periodic and often naturally expressed in radians.
• Understanding both degrees and radians allows for easy conversion and application in solving real-world problems.

Grasping trigonometry basics lays the foundation for solving more complex mathematical problems involving waves, oscillations, and circular motion, among others. Mastering angle conversions opens doors to a deeper understanding of these trigonometric functions.