Problem 19 - Algebra and Trigonometry

Question

Find the degree measure of the angle with the given radian measure. $$ 3 $$

Step-by-Step Solution

Answer

The degree measure is \(\frac{540}{\pi}\).

Step 1: Understand the Relationship between Radians and Degrees

The relationship between radians and degrees is defined through the fact that a full circle is 360 degrees or \(2\pi\) radians. This means that \(1\text{ radian} = \frac{180}{\pi}\text{ degrees}\). We will use this conversion factor to change radians into degrees.

Step 2: Convert Radian Measure to Degrees

Given the radian measure \(3\), we will convert this into degrees using the conversion formula:\[\text{Degrees} = 3 \times \frac{180}{\pi}\]Simplifying this expression will give us the degree measure.

Step 3: Simplify the Expression

Let's simplify the expression \(3 \times \frac{180}{\pi}\):1. Calculate the multiplication: \(3 \times 180 = 540\).2. Divide the result by \(\pi\): \(\frac{540}{\pi}\). This gives us the degree measure of the angle.

Key Concepts

Angle Measurement

Angles help us describe the size or rotation of a shape. They are especially useful in geometry, physics, and engineering. When measuring angles, we usually use either degrees or radians as units. Each type of measurement has its own advantages depending on the context.

• Degrees use a base of 360 to represent a full circle. This division makes it easy for people to understand and visualize angles.
• Radians, on the other hand, are often more natural in mathematics because they align with the properties of circles in trigonometric and calculus calculations.

Knowing how to convert between degrees and radians allows us to work easily across different fields and applications.

Degrees and Radians

Degrees and radians are two ways to express the measure of an angle. Understanding both can enhance our ability to solve problems in mathematics.

• Degrees: A full circle is divided into 360 equal parts, each part being one degree. This system is intuitive because it divides the circle into segments which are multiples of 10, a format familiar in daily life.
• Radians: A radian measures the angle created when the arc length is equal to the radius of the circle. A full circle is therefore \(2\pi\) radians. This unit is crucial for complex calculations in mathematics because it incorporates the circle's geometry directly.

Both systems are important, and depending on the context, one might be more convenient than the other.

Conversion Factor for Angles

Converting between degrees and radians is straightforward if you remember the conversion factor. A circle with 360 degrees corresponds to \(2\pi\) radians. Therefore, we can determine that:

• \(1\text{ radian} = \frac{180}{\pi}\text{ degrees}\).
• \(1\text{ degree} = \frac{\pi}{180}\text{ radians}\).

To convert radians to degrees, multiply the radian measure by \(\frac{180}{\pi}\). Similarly, convert from degrees to radians by multiplying the degree measure by \(\frac{\pi}{180}\).
Understanding and remembering this conversion factor allows us to easily transition between the two measurement systems, making calculations and comprehension in mathematics much smoother.