Problem 9 - Precalculus Mathematics for Calculus

Question

Find the radian measure of the angle with the given degree measure. $$1080^{\circ}$$

Step-by-Step Solution

Answer

The radian measure of \(1080^{\circ}\) is \(6\pi\).

Step 1: Understanding the Problem

We need to convert the angle from degrees to radians. The given angle is \(1080^{\circ}\).

Step 2: Conversion Factor

Recall that \(180^{\circ}\) is equivalent to \(\pi\) radians. Therefore, the conversion factor from degrees to radians is \(\frac{\pi}{180}\).

Step 3: Apply the Conversion

Multiply the degree measure by the conversion factor: \[ 1080^{\circ} \times \frac{\pi}{180} \].

Step 4: Simplify the Expression

Calculate the multiplication: \[ 1080 \times \frac{\pi}{180} = 6\pi \].

Step 5: Verify the Result

Check the calculation: \(1080 \div 180 = 6\), so the radian measure is \(6\pi\). The result makes sense since one full rotation is \(2\pi\) radians, and \(1080^{\circ}\) is three full rotations.

Key Concepts

Radian Measure

Understanding radian measure is essential for converting angles. Unlike degrees, which divide a circle into 360 equal parts, radians offer a more natural way of measuring angles using the concept of arc length. The radian is based on the radius of the circle, making it a unit that directly relates the angle to the proportion of the circular arc it subtends.

In a full circle, there are exactly \(2\pi\) radians. Therefore, if you measure an angle in radians, you are essentially asking how many radii would wrap around the circle's arc if laid end to end.

It's helpful to remember that:

• \(\pi\) radians are equivalent to a half-circle or \(180^\circ\).
• \(2\pi\) radians make up a full circle or \(360^\circ\).

Radian measure greatly simplifies many mathematics equations and calculations, especially those involving trigonometric functions.

Degree Measure

Degree measure is perhaps the most familiar unit for angles, often used in everyday situations. A circle is divided into 360 degrees, making it much easier to visualize smaller segments of rotations or turns.

This unit may have roots in ancient calendars, which approximated the year as 360 days. It is an arbitrary but very practical number that allows for straightforward divisions into halves, thirds, quarters, and more.

Here are some key points about degree measure:

• A complete circle is \(360^\circ\).
• Half a circle is \(180^\circ\).
• Common right angles are \(90^\circ\).

Understanding degree measure can provide immediate relatability for tasks involving turning, such as navigation or geometry.

Conversion Factor

Conversion is key when switching between radians and degrees. The pivotal conversion factor is \(\frac{\pi}{180}\), which allows the transformation of degrees into radians. This factor emerges from the relationship between a half-circle in both systems: \(180^\circ = \pi\) radians.

The process of converting an angle from degrees to radians can be summarized as follows:

• Multiply the degree value by \(\frac{\pi}{180}\).
• This calculation scales down the degree value to its corresponding radian measurement.

This relationship simplifies electrical engineering, physics, and other fields that rely heavily on angle calculations.

Mathematical Calculations

Mathematical calculations involving radians and degrees do not have to be intimidating. Once you understand the conversion factor, problems like the one in our exercise become straightforward.

Consider our example with an angle of \(1080^\circ\). The calculation employs the conversion formula:
\[1080^{\circ} \times \frac{\pi}{180}\]To solve it:

• Divide \(1080\) by \(180\) to simplify, giving a result of \(6\).
• Thus, the angle is \(6\pi\) radians.

This type of calculation is fundamental in various applications like rotational motion or wave analysis. Being able to interchange between these measures fluently can expand your problem-solving toolbox in both academic and real-world contexts.