Problem 37 - Precalculus with Calculus Previews

Question

In Problems \(33-40,\) convert the given angle from radians to degrees. $$ 5 \pi $$

Step-by-Step Solution

Answer

The angle \(5\pi\) radians is equal to 900 degrees.

Step 1: Understand the Problem

We need to convert an angle from radians to degrees. The given angle is \(5\pi\) radians.

Step 2: Recall the Conversion Formula

To convert radians to degrees, use the formula: \( \text{Degrees} = \text{Radians} \times \frac{180}{\pi} \).

Step 3: Plug in the Radian Measure

Substitute \(5\pi\) for the radian measure in the conversion formula: \( \text{Degrees} = 5\pi \times \frac{180}{\pi} \).

Step 4: Simplify the Expression

Cancel \(\pi\) in the numerator and the denominator: \( \text{Degrees} = 5 \times 180 \).

Step 5: Calculate the Result

Multiply \(5\) by \(180\) to get the degree measure: \( \text{Degrees} = 900 \).

Key Concepts

Radians to Degrees

The angle measurement can be quite tricky when you consider different systems, like radians and degrees. Each system has its own way of expressing angles, and one vital task is converting between them.

Radian measures are often used in mathematics due to their natural connection with the radius of a circle. Meanwhile, degrees are more common in everyday contexts.

• There are approximately 57.3 degrees in one radian.
• Picturing this visually, a radian is the angle created when one radius length is laid along the circumference of a circle.
• Degrees divide a circle into 360 parts, making it a more intuitive system for many people.

Understanding how to make these conversions allows seamless calculations and a better grasp of mathematical problems.

Conversion Formula

To convert an angle from radians to degrees, you use a specific formula that revolves around the relationship between these two units.

The formula is:\[\text{Degrees} = \text{Radians} \times \frac{180}{\pi}\]This equation works because:

• A complete circle is 360 degrees, which corresponds to \(2\pi\) radians.
• Therefore, there are \(\frac{180}{\pi}\) degrees per radian.

By plugging the radian value into this formula, it seamlessly converts the angle measurement into degrees. This conversion is foundational and simplifies many complex mathematical problems, especially those involving trigonometry.

Multiplication in Conversion

Once you have the conversion formula ready, it's time to perform the actual calculation using multiplication.

For instance, if you have a value like \(5\pi\) radians and wish to convert it into degrees, here's what you do:

• Insert the radian value into the formula: \(5\pi \times \frac{180}{\pi}\).
• Notice that \(\pi\) in the numerator and denominator cancel out.
• This leaves you with \(5 \times 180\) degrees.
• Finally, perform the multiplication: \(5 \times 180 = 900\) degrees.

Conducting this process involves straightforward arithmetic, but it's crucial to carefully maintain accuracy during each step. Multiplication in conversion ensures you arrive at the precise degree measurement that corresponds to the original radian value.