Problem 15 - Algebra 2 and Trigonometry

Question

In \(13-22\) , find the degree measure of each angle whose radian measure is given. \(\frac{\pi}{10}\)

Step-by-Step Solution

Answer

The angle is 18 degrees.

Step 1: Understand the Conversion Formula

To convert an angle from radians to degrees, use the formula: \[ Degrees = Radians \times \frac{180}{\pi} \]This formula comes from the fact that \( \pi \) radians equals 180 degrees.

Step 2: Substitute Given Radian Value

We substitute the given radian measure, \( \frac{\pi}{10} \), into the conversion formula:\[ Degrees = \frac{\pi}{10} \times \frac{180}{\pi} \]

Step 3: Simplify the Expression

Simplify the expression by cancelling out \( \pi \) from the numerator and the denominator:\[ Degrees = \frac{1}{10} \times 180 \]

Step 4: Calculate the Result

Now, calculate \( \frac{1}{10} \times 180 \) to find the degree measure:\[ Degrees = 18 \]

Key Concepts

Angle Measurement

Angle measurement is a concept used to describe the amount of rotation between two rays or lines that share a common endpoint, known as the vertex of the angle. Understanding angle measurements is essential for fields like geometry, trigonometry, and navigation.

• Angles are commonly measured in degrees or radians.
• One whole turn around a circle is equivalent to 360 degrees or \(2\pi\) radians.
• The measurement system you choose can affect the ease of calculation in different mathematical contexts.

Degrees and radians provide two different scales to describe the same geometric concept. By understanding both, we can more easily switch between different problem-solving approaches. Each system is beneficial for various applications: degrees are typically easier to visualize, while radians can simplify calculus-based problems.

Radian Measure

The radian measure is a way to express angles using the radius of a circle. Unlike degrees, which divide a circle into 360 equal parts, radians relate the angle to the circle's radius, offering a more natural mathematical form.

• One radian is the angle created when the arc length is equal to the radius of the circle.
• A full circle is \(2\pi\) radians, which equates to 360 degrees.
• Common radian measures are \(\pi/6\), \(\pi/4\), \(\pi/3\), \(\pi/2\), and their multiples.

Radians are particularly advantageous in calculus and trigonometry. Formulas such as the derivatives of trigonometric functions often yield simpler results when the angles are in radians. This characteristic makes radians integral in advanced mathematics.

Degree Measure

The degree measure is a widely used system for quantifying angles, particularly in everyday scenarios and elementary education. It divides the circumference of a circle into 360 equal parts, making it an intuitive way to think about rotations and turns.

• 360 degrees is the equivalent of a full circle.
• Degrees are sub-divided into minutes and seconds for even more precision (e.g., 53° 35' 30").
• Common angle measurements in degrees include 30°, 45°, 90°, and 180°.

One key advantage of using degrees is their simplicity in measurement and understanding, making them accessible and practical in teaching and everyday use. Furthermore, converting between degrees and radians enables deeper understanding and flexibility in mathematical problems and applications.