Problem 27 - Algebra 2

Question

Rewrite each degree measure in radians and each radian measure in degrees. \(-\frac{\pi}{3}\)

Step-by-Step Solution

Answer

The radian measure \(-\frac{\pi}{3}\) is equivalent to -60 degrees.

Step 1: Understand the Conversion Formula

To convert radians to degrees, we use the formula: Degrees = Radians × \(\frac{180}{\pi}\). This is because \(\pi\) radians is equivalent to 180 degrees.

Step 2: Apply the Formula

Apply the conversion formula to the given radian measure, \(-\frac{\pi}{3}\). Multiply \(-\frac{\pi}{3}\) by \(\frac{180}{\pi}\) to convert it to degrees.

Step 3: Simplify the Expression

Perform the multiplication: \(-\frac{\pi}{3} \times \frac{180}{\pi} = -\frac{180}{3}\). Simplify \(-\frac{180}{3}\) to get -60 degrees.

Key Concepts

Degree Measure

In our everyday life, angles are often expressed in degrees. It is a way to understand and measure the rotation or turn between two different directions. A full circle is divided into 360 equal parts, each part being a degree. This helps with interpretations and calculations in various fields such as geography, astronomy, and even simple tasks like drawing angles in mathematics. Degrees provide a straightforward way to describe the rotation from a starting line.

• 360 degrees in a full circle
• 90 degrees for a quarter of a turn, commonly known as a right angle
• 180 degrees as a straight line

All measurements in degrees are inherently positive. However, moving clockwise from a starting point creates negative degree measures.

Radian Measure

Unlike degrees, radians measure angles based on the arc length relative to the radius of a circle. This is a fundamental concept in mathematics, particularly calculus and trigonometry, because it provides a natural way of linking linear and angular measurements. So what is a radian? One radian is the angle made when the arc length is equal to the radius. In a complete circle, the total angle is \(2\pi\) radians. This is why radians are often connected to \(\pi\), an irrational number approximately equal to 3.14159.

• One full circle equals \(2\pi\) radians.
• Half a circle is \(\pi\) radians.
• A quarter circle (right angle) is \(\frac{\pi}{2}\) radians.

Working with radians can seem tricky at first due to their algebraic form, but they simplify many mathematical formulas, particularly involving trigonometric functions.

Conversion Formula

When dealing with angle measures, especially in trigonometry and calculus classes, it's crucial to switch between radians and degrees. The conversion formula provides a bridge between these two systems. To translate an angle from radians to degrees, use the formula: Degrees = Radians × \(\frac{180}{\pi}\).This formula is derived from the fact that \(\pi\) radians is equal to 180 degrees. Simplifying angles using this relationship allows for effortless conversions and deeper understanding in mathematical computations.For instance, if an angle is given in radians, you would:

• Multiply the radian value by 180.
• Divide the result by \(\pi\).

This quick operation turns radian measures into degrees without much hassle, making it a handy tool for students to have at their disposal.

Simplification

Once you apply the conversion formula, you might end up with a fraction that needs simplifying. This is an essential step to fully translate radian measures into a clear degree measure, making them practical and easy to understand.Let’s say we have an angle given as \(-\frac{\pi}{3}\) radians. After applying the conversion formula:

• Calculate: \(-\frac{\pi}{3} \times \frac{180}{\pi} = -\frac{180}{3}\)
• Simplify the fraction to obtain: \(-60\) degrees.

The simplification makes the conversion neat and straightforward. It helps verify that the conversion was applied correctly. Always remember to double-check your multiplication and division for errors. Keeping operations clear prevents mistakes and solidifies understanding of the conversion process.