Problem 361 - Precalculus

Question

Convert \(\frac{5 \pi}{6}\) radians to degrees.

Step-by-Step Solution

Answer

\( \frac{5 \pi}{6} \) radians is equal to 150 degrees.

Step 1: Understand the Conversion Formula

Radian and degree are two different units for measuring angles. To convert an angle from radians to degrees, we use the conversion formula: \[ \text{Degrees} = \text{Radians} \times \left( \frac{180}{\pi} \right) \]

Step 2: Identify the Given Radian Measure

The problem provides the angle in radians as \( \frac{5 \pi}{6} \). This will be the value we plug into our conversion formula.

Step 3: Apply the Conversion Formula

Substitute \( \frac{5 \pi}{6} \) into the conversion formula:\[ \text{Degrees} = \frac{5 \pi}{6} \times \left( \frac{180}{\pi} \right) \]Here, we'll multiply \( \frac{5 \pi}{6} \) by \( \frac{180}{\pi} \).

Step 4: Perform Cancellation

In the expression \( \frac{5 \pi}{6} \times \frac{180}{\pi} \), note that the \( \pi \) in the numerator and denominator cancel each other out, simplifying to:\[ \text{Degrees} = \frac{5}{6} \times 180 \]

Step 5: Calculate the Result

Calculate \( \frac{5}{6} \) of 180:\[\text{Degrees} = \frac{5}{6} \times 180 = 5 \times 30 = 150\]Thus, \( \frac{5 \pi}{6} \) radians is equal to 150 degrees.

Key Concepts

Angle Measurement

In mathematics, angles are a way to describe the rotation between two rays or line segments that share a common endpoint, known as the vertex. You're likely familiar with measuring angles in degrees, such as the 90 degrees of a right angle. However, angles can also be measured in another unit called radians.
Radians are based on the concept of the radius of a circle. One radian is the angle you get when you take the radius of a circle and wrap it around the circle's edge. Since the circumference of a circle is always approximately \(2\pi\) times the radius, the full circle is \(2\pi\) radians.
Understanding how to move between degrees and radians is crucial in math, particularly in trigonometry and calculus. It allows for easier calculations when dealing with circular and periodic functions.

Conversion Formula

To convert between radians and degrees, we use a simple mathematical formula. This formula relates the two units of measure by leveraging the fact that a full rotation (complete circle) is equal to \(360\) degrees or \(2\pi\) radians. As such, the conversion formula from radians to degrees is:

• \(\text{Degrees} = \text{Radians} \times \left( \frac{180}{\pi} \right)\)

This formula multiplies the given radian value by \(\frac{180}{\pi}\) to change it into degrees. The fraction \(\frac{180}{\pi}\) comes from dividing the degrees of a circle by the radians of a circle, effectively scaling the radian to its corresponding degree measure.
This straightforward equation becomes a powerful tool for converting measurements in physics, engineering, and even navigation, where precision in angle measurement is key.

Trigonometry Concepts

Trigonometry is a branch of mathematics that deals with angles, triangles, and the relationships between them. It introduces and relies heavily on angle measurements like degrees and radians.
Trigonometric functions like sine, cosine, and tangent are periodic functions, meaning they repeat their values in regular intervals. For these functions, radians often prove more convenient than degrees because the radian measures are directly related to the properties of circles and their periodic nature.
Thus, understanding radians within trigonometry helps in exploring and solving problems involving waves, oscillations, and circles. Trigonometric identities and functions are easier to apply when angles are measured in radians, especially in calculus where functions often rely on radian measures to determine derivatives and integrals.

Degrees and Radians

Degrees and radians both serve as units for measuring angles, but they suit different contexts and purposes. A degree is a more intuitive and historic measure—many people are taught about angles within the framework of degrees first.
Degrees divide a full circle into \(360\) equal parts, making it easily relatable to everyday measures and intuitive to understand. This means a quarter of a circle, or a right angle, measures \(90\) degrees.

• A full circle = \(360\) degrees
• Half a circle = \(180\) degrees
• Quarter of a circle (right angle) = \(90\) degrees

Radians offer a more natural fit in mathematical disciplines, particularly because they relate an angle directly to the inherent geometry of a circle. An understanding of both units and the ability to convert between them is essential, as different fields might prefer one measure over the other for precision and convenience.