Problem 42 - Precalculus

Question

Convert from radians to degrees. $$-\frac{8 \pi}{9}$$

Step-by-Step Solution

Answer

The angle \(-\frac{8\pi}{9}\) radians is equivalent to \(-160\) degrees.

Step 1: Recall the Radian to Degree Conversion Formula

To convert from radians to degrees, multiply the radian measure by \( \frac{180}{\pi} \). This formula is based on the fact that 180 degrees is equivalent to \( \pi \) radians.

Step 2: Set Up the Conversion

Substitute the given radian value into the conversion formula: \(-\frac{8\pi}{9} \times \frac{180}{\pi} \).

Step 3: Simplify the Expression

Cancel the \( \pi \) from both the numerator and denominator to simplify the calculation: \(-\frac{8}{9} \times 180\).

Step 4: Perform the Multiplication

Multiply \(-\frac{8}{9}\) by 180: \(-\frac{8 \times 180}{9} = -\frac{1440}{9}\).

Step 5: Finalize the Division

Divide 1440 by 9 to find the degree measure: \(-160\) degrees.

Key Concepts

Radian

When you think about radians, imagine a circle and its radius. A radian is a unit of angular measurement based on the radius of the circle. More specifically, a radian is the angle that is formed when the length of the circle's arc is equal to the circle's radius. This might sound complex at first, but think of it as simply a way to measure angles based on the circle's geometry instead of degrees like we usually do. Radians are especially useful in mathematics and physics due to their natural relation with the properties of the circle.

• 1 radian is approximately 57.3 degrees.
• There are 2π radians in a complete circle, equivalent to 360 degrees.

Because it's based on the radius, the radian gives a more straightforward way to understand and calculate properties in trigonometry and calculus.

Degree

Degrees are a more familiar unit of measurement when it comes to angles. The full circle is divided into 360 equal parts, and each part is one degree. This division comes from historical reasons but is widely used in everyday scenarios, from navigation to telling time on a clock.Degrees are intuitive because they map onto a full rotation or cycle perfectly. Think of a clock's face: a quarter of a turn (90 degrees) is pretty easy to visualize, as is a complete round trip back to where you started (360 degrees).

• 1 degree is equal to \(\frac{\pi}{180}\) radians.
• Visual and practical for measuring average daily angles like in geometry and navigation.

Ultimately, degrees provide a user-friendly way to express and visualize angles for many everyday applications.

Conversion Formula

Understanding the conversion formula between radians and degrees is crucial for switching between these two measurement systems. The basic idea is simple: to convert from radians to degrees, you multiply the radian value by \(\frac{180}{\pi}\). This formula uses the fact that π radians equal 180 degrees. It's particularly useful when working in disciplines like geometry or engineering where both measurements might be needed.Here's how you use it:

• Start with a radian measurement.
• Multiply it by \(\frac{180}{\pi}\) to convert to degrees.

For example, if you are given \(-\frac{8\pi}{9}\) radians, apply the formula:

• \( -\frac{8\pi}{9} \times \frac{180}{\pi} \) simplifies down to \(-160\) degrees.

This formula is incredibly helpful for making sense of problems that involve trigonometric functions or when angles need to be compatible with degrees.

Angular Measurement

Angular measurement is the method used to describe the size of an angle, typically within a circle. It's one of the basic types of measurements that underpins geometry and is vital in many sciences and engineering fields. Let's consider two main units of angular measurement: radians and degrees. Each has its pros and cons depending on the scenario it is being used in.

• Degrees: Excellent for everyday use and easily understandable to people in non-technical fields.
• Radians: Preferred in mathematical equations and when thinking in terms of the circle's geometry and trigonometry.

When you measure angles, think about which unit provides the clearest understanding or easiest calculations for your situation. Often, with complex calculations or advanced mathematics, radians are preferred due to their relations with π and natural circles.