Problem 120 - Calculus Volume 1

Question

For the following exercises, convert each angle in radians to degrees. $$\frac{11 \pi}{2} \mathrm{rad}$$

Step-by-Step Solution

Answer

\(\frac{11\pi}{2}\) radians is 990 degrees.

Step 1: Recall the Conversion Formula

The formula to convert an angle from radians to degrees is to multiply the radians by \(\frac{180}{\pi}\). This is because \(\pi\) radians is equal to 180 degrees.

Step 2: Apply the Conversion Formula

To convert \(\frac{11\pi}{2}\) radians to degrees, multiply by \(\frac{180}{\pi}\):\[\frac{11\pi}{2} \times \frac{180}{\pi}\] This will cancel out the \(\pi\) in the numerator and denominator.

Step 3: Simplify the Expression

After applying the conversion formula, simplify the expression:\[\frac{11 \times 180}{2}\]This simplifies to \(\frac{1980}{2}\).

Step 4: Divide to Find the Result

Divide 1980 by 2 to convert radians to degrees:\[990\text{ degrees}\] Thus, \(\frac{11\pi}{2}\) radians is equal to 990 degrees.

Key Concepts

Radians to Degrees

Understanding how to convert angles from radians to degrees is an essential skill in mathematics. Radians and degrees are units used to measure angles in geometrical contexts.
- **Radians** measure based on the radius of a circle. One full circle is equal to \(2\pi\) radians. - **Degrees** divide a circle into 360 parts, with a full circle being 360 degrees.
Why convert from radians to degrees? Often, angles are measured in radians for calculation simplicity, especially in trigonometry and calculus. Engaging with problems in degrees, however, can sometimes be more intuitive.
When you switch between these two, you facilitate a vast array of mathematical applications, ensuring numbers make sense regardless of the field of study.

Conversion Formula

To change angles from radians to degrees, we rely on a fundamental conversion formula. This formula stems from the equivalence of \(\pi\) radians to 180 degrees, a key ratio that bridges the two measurement systems.
The conversion formula is as follows: multiply the angle in radians by \(\frac{180}{\pi}\). This can be broken down into these simple steps:
- Take the angle in radians, for example, \(\frac{11\pi}{2}\). - Multiply by \(\frac{180}{\pi}\), where \(\pi\) in the numerator and denominator will cancel out.
Through this conversion formula, any angle can be easily expressed in degrees, providing a clear and widely understood value.

Simplification Steps

Once you have applied the conversion formula, simplifying the expression is crucial for arriving at the final result.
Let's explore this using the example \(\frac{11\pi}{2}\) radians:

• After multiplying \(\frac{11\pi}{2} \times \frac{180}{\pi}\), you are left with the numerator \(11 \times 180\).
• This computation results in 1980.
• Next, divide 1980 by 2 to simplify further.
• The final expression calculates to 990 degrees.

Therefore, the process is streamlined by methodical multiplication and division.
Always remember these steps allow you to convert complex radian measures into degrees with ease and precision.