Problem 73 - A Graphical Approach to Precalculus with Limits

Question

Convert each radian measure to degrees. $$\frac{7 \pi}{4}$$

Step-by-Step Solution

Answer

\(\frac{7\pi}{4}\) radians is 315 degrees.

Step 1: Understand the Problem

We need to convert the given radian measure \( \frac{7\pi}{4} \) to degrees. To do this, we will use the fact that \( \pi \) radians is equal to 180 degrees. Thus, we will replace \( \pi \) with 180 to find the equivalent degree measure.

Step 2: Set Up the Conversion Formula

To convert radians to degrees, use the formula: \[ \text{Degrees} = \text{Radians} \times \frac{180}{\pi} \]Here, our radian measure is \( \frac{7\pi}{4} \). Substitute this into the formula.

Step 3: Substitute the Radian Measure

Substitute \( \frac{7\pi}{4} \) into the conversion formula:\[ \text{Degrees} = \left( \frac{7\pi}{4} \right) \times \frac{180}{\pi} \]This setup will help us cancel the \( \pi \) and simplify the expression.

Step 4: Simplify the Expression

The \( \pi \) in the numerator and denominator cancel each other:\[ \text{Degrees} = \frac{7 \times 180}{4} \]Now, simplify \( \frac{7 \times 180}{4} \) by computing the arithmetic.

Step 5: Calculate the Result

Perform the multiplication and division:\[ 7 \times 180 = 1260 \]\[ \frac{1260}{4} = 315 \]Therefore, \( \frac{7\pi}{4} \) radians is equal to 315 degrees.

Key Concepts

Radians

Radians are a way to measure angles based on the properties of circles. Unlike degrees, which divide a circle into 360 parts, radians consider the arc length relative to the radius of the circle.
The full circumference of a circle is known as the complete rotation around the circle, which is 360 degrees. However, in terms of radians, this is represented as \(2\pi\) radians since the circumference \( (C) \) of a circle is calculated using \( C = 2\pi r \), where \( r \) is the radius. Therefore, a half-circle (which is a vital component in conversion to degrees) is exactly \(\pi\) radians.
For trigonometric purposes, radians are often preferred because they allow for cleaner and more straightforward calculus operations. Recognizing that \(\pi \) radians correspond to 180 degrees is the first step in converting between these two units of angle measurement.

Degrees

Degrees are the most common unit for measuring angles, particularly in everyday use and basic geometry. The complete angle, which represents a full circle, is quantified as 360 degrees.
This division comes from ancient Babylonian mathematics and has remained in use due to its convenience for dividing segments of a circle. Every degree can further be subdivided into 60 minutes, and each minute into 60 seconds, just like time.
To convert an angle from radians to degrees, we use the relationship \(\pi \) radians = 180 degrees. This relationship allows us to apply a conversion formula: \( \text{Degrees} = \text{Radians} \times \frac{180}{\pi} \). Thus, multiplying the radian measure by \( \frac{180}{\pi} \) provides its equivalent in degrees, which is instrumental in understanding angles within different mathematical contexts.

Trigonometry

Trigonometry is a branch of mathematics that studies the relationships between side lengths and angles in triangles. It's hugely applicable in physics, engineering, and architecture, to name a few. Core to trigonometry are the concepts of radians and degrees, which provide different but equivalent methods to express angles.
Trigonometric functions like sine, cosine, and tangent often use radians because they fit more naturally with calculus and most mathematical modeling scenarios. However, in calculating angles in real-world problems or classroom geometry, degrees are frequently used due to familiarity.
Understanding how to convert between radians and degrees enhances solving trigonometric equations and is crucial when transitioning between different methods of angle measurement in various problems. This conversion lays the groundwork for deeper exploration into trigonometric properties, identities, and real-world applications.