Question

The following angles are given in radians. Convert them to degrees: \(\pi / 6 \mathrm{rad}, 0.70 \mathrm{rad}, 1.5 \pi \mathrm{rad}, 5 \pi \mathrm{rad}\).

Short answer

\(\frac{\pi}{6}, 0.70, 1.5\pi, 5\pi\) rad are \(30^\circ, 40.05^\circ, 270^\circ, 900^\circ\) respectively.

Step-by-step solution

Understand the Conversion Factor

Radians and degrees are two ways to measure angles. To convert an angle from radians to degrees, we use the fact that \(180^\circ = \pi \text{ radians}\). So, the conversion factor is \(\frac{180^\circ}{\pi}\).

Convert Each Angle to Degrees

Each given angle in radians needs to be multiplied by \(\frac{180^\circ}{\pi}\) to convert it to degrees.For \(\frac{\pi}{6}\) rad:\[ \frac{\pi}{6} \times \frac{180^\circ}{\pi} = 30^\circ \]For \(0.70\) rad:\[ 0.70 \times \frac{180^\circ}{\pi} \approx 40.05^\circ \]For \(1.5\pi\) rad:\[ 1.5\pi \times \frac{180^\circ}{\pi} = 270^\circ \]For \(5\pi\) rad:\[ 5\pi \times \frac{180^\circ}{\pi} = 900^\circ \]

Compile the Results

The angles converted from radians to degrees are:- \(\frac{\pi}{6}\) rad = \(30^\circ\)- \(0.70\) rad \(\approx 40.05^\circ\)- \(1.5\pi\) rad = \(270^\circ\)- \(5\pi\) rad = \(900^\circ\)

Key concepts

Understanding Angle Measurement

In mathematics, an angle is a figure formed by two rays, called the sides of the angle, sharing a common endpoint, known as the vertex of the angle. Angles can be measured in degrees or radians, two distinct systems that allow us to quantify the 'openness' between the rays.
Measure in degrees is more common in everyday contexts, like describing directions or calculating latitude and longitude. One complete revolution is divided into 360 equal parts, each known as a degree (°).
Radians, however, provide a more natural measure in the realm of mathematics due to their relation with arc length. In this system, one complete revolution around a circle's circumference corresponds to an angle of 2π radians. This makes radians especially convenient in calculus and trigonometry, where growth rates and periodic phenomena are frequently analyzed.

The Role of Trigonometry in Angle Conversion

Trigonometry is the branch of mathematics that deals with the relationships between the sides and angles of triangles. Understanding this relationship is key to angle conversion.
Since trigonometry connects algebra and geometry, it encompasses various functions, like sine, cosine, and tangent, which all depend on angle measurement. This is why radian and degree conversions are fundamental in solving trigonometric equations.
The step-by-step conversion method uses the ratio \\(\frac{180^\circ}{\pi}\) because there are 180 degrees in half a circle (or π radians). For instance, when converting π/6 radians to degrees, multiplying by \\(\frac{180^\circ}{\pi}\) yields 30°. This reflects how deeply angle measurement is integrated into trigonometric calculations.

Relevance in Mathematics Education

Understanding radians and degrees is essential not only in math classes but also in many applications, such as physics and engineering.
Mathematics education emphasizes this because a solid comprehension of these concepts forms the foundation for more complex topics, like advanced trigonometry and calculus.
Practical exercises, like converting radians to degrees, help students build mathematical literacy and flexible problem-solving skills. By mastering both degree and radian measurements, students enhance their ability to visualize and solve a wide array of mathematical problems.
It's crucial for learners to practice these conversions to become comfortable and confident in using both units, as they form the basis for a variety of mathematical and scientific analyses.