Question

Convert each angle in radians to degrees. \(\frac{9 \pi}{2}\)

Short answer

810 degrees

Step-by-step solution

Understand the Conversion Factor

To convert radians to degrees, use the conversion factor: \[ 1 \text{ radian} = \frac{180}{\text{π}} \text{ degrees}\]

Substitute the Angle in Radians

Substitute the given angle in radians into the conversion factor. \[ \frac{9\text{π}}{2} \text{ radians} \]

Apply the Conversion Factor

Multiply the given angle by the conversion factor. \[ \frac{9\text{π}}{2} \times \frac{180}{\text{π}} \text{ degrees} \]

Simplify the Expression

Simplify the expression by canceling out \(\text{π}\) and performing the multiplication. \[ \frac{9 \times 180}{2} \text{ degrees} = \frac{1620}{2} \text{ degrees} = 810 \text{ degrees} \]

Key concepts

radian measure

A radian is a unit of angular measure used in mathematics. Essentially, one radian is the angle created when the length of the arc of a circle is equal to the radius of that circle. This unit is very natural for mathematicians to utilize because it simplifies many equations. Here are some key points about radians:

• One complete revolution around a circle is equal to \(2\text{π}\) radians.
• Half of a circle (or a half revolution) is \(\text{π}\) radians.
• This unit eliminates the need to frequently use extra conversion factors in calculus and trigonometry.

These points illustrate why radians are very useful when dealing with circular motion, oscillations, and many other areas in advanced mathematics.
In essence, understanding radians is crucial for effectively navigating through many mathematical problems.

degree measure

Degrees are another unit for measuring angles, and they might be more familiar to most students. One degree is defined as \(1/360^{th}\) of a full circle. This means a full circle has \(360\) degrees in total. Here's why degrees are commonly used:

• They are simpler and more intuitive for many basic applications, especially in geometry and everyday situations.
• Common angles like \(90^{\circ}\) (right angle) and \(180^{\circ}\) (straight line) are easy to visualize and understand.

Moreover, degrees are often used in navigation, astronomy, and various engineering fields. While degrees might be more immediately accessible, understanding both radians and degrees, and how to convert between them, is essential for broader applications in mathematics and science.

angle conversion

Converting between radians and degrees is a fundamental skill in both basic and advanced mathematics. To convert an angle from radians to degrees, you use the conversion factor \(1 \text{ radian} = \frac{180}{\text{π}} \text{ degrees}\). This means you multiply the radian measure by \(\frac{180}{\text{π}}\). Let's revisit the example in the original exercise:
Given angle in radians: \(\frac{9\text{π}}{2}\)

• Step 1: Apply the conversion factor: \(\frac{9\text{π}}{2} \times \frac{180}{\text{π}} \text{ degrees}\).
• Step 2: Simplify by canceling out \(\text{π}\) and performing multiplication. Here, \(\text{π}\) cancels out, leaving \(\frac{9 \times 180}{2} \text{ degrees}\).
• Step 3: Calculate the final result, which is \(\frac{1620}{2} = 810 \text{ degrees}\).

This method shows the clarity and simplicity of converting between these two units of angular measure.
Mastering these conversions is useful because it allows for seamless transitions between problems requiring different units.