Problem 105 - Algebra and Trigonometry Real Mathematics, Real People

Question

Convert the angle measure from radians to degrees. Round your answer to three decimal places. $$8.5$$

Step-by-Step Solution

Answer

To convert 8.5 radians into degrees gives a result of approximately 487.183 degrees when rounded to three decimal places.

Step 1: Identifying the Given Value

Firstly, identify the given value in radians. In this case, that is 8.5 radians.

Step 2: Applying the Conversion Factor

As the conversion factor for radians to degrees is '180/\(\pi\)', the task is now to multiply the given value with this. This will give: 8.5 * 180/\(\pi\).

Step 3: Calculating and Rounding the Result

Perform the calculation and round the result to three decimal places.

Key Concepts

Trigonometry

Trigonometry is a branch of mathematics that studies relationships between the sides and angles of triangles. The unit circle, where the radius is one, plays a pivotal role in trigonometry and is where the concept of radians comes into play. Radians measure angles based on the distance traveled along the circumference of the unit circle. One radian is the angle created when the length of the arc equals the radius of the circle. Understanding radians is essential for calculating various functions like sine, cosine, and tangent, which are fundamental in trigonometry.

In the context of our exercise, we're starting with an angle in radians, which is a common scenario in trigonometry. Students often start with the radian measure because trigonometric functions on calculators often use radians for their calculations.

Unit Conversion

Unit conversion is a critical skill in mathematics, science, and various technical fields. It entails changing the measurement of a quantity from one unit to another while keeping the value of that quantity the same. When converting from radians to degrees, you are changing the way you express the measurement of an angle, not the size of the angle itself. The formula for converting radians to degrees is to multiply the number of radians by \frac{180}{\(pi\)} since 180 degrees is equal to \(pi\) radians.

This conversion is necessary because some contexts require degrees, such as navigation and aviation, while others, like theoretical mathematics and physics, often use radians. For our exercise, multiplying 8.5 by the conversion factor allows us to express the angle in a unit that might be more familiar or practical for certain applications.

Angle Measurement

The measurement of angles is a fundamental aspect of geometry and trigonometry. There are two primary units for measuring angles: degrees and radians. A degree is 1/360th of a complete rotation around a circle, and it’s often used for practical applications like geography and everyday measurements.

On the other hand, a radian is a more natural choice in pure mathematics and physics as it arises from the properties of the circle itself. To put it simply, if you were to unwrap the edge of the circle and lay it flat, the angle that corresponds to the arc length equal to the radius of the circle would be one radian. For this reason, radians are incredibly useful in calculations involving the circular motion and waves.

Converting radians to degrees, as in our exercise, is often done to make the results more interpretable in contexts where degrees are the conventional unit of measure. The given angle of 8.5 radians has to be converted using the conversion factor to yield a result in degrees, which is often required for various real-world applications like in understanding bearings in navigation.