Problem 17 - Algebra and Trigonometry

Question

\(13-24\) . Find the degree measure of the angle with the given radian measure. $$ 3 $$

Step-by-Step Solution

Answer

The angle measures approximately 171.9 degrees.

Step 1: Understand the Relationship

First, understand that to convert radians to degrees, we use the relationship that 180 degrees is equivalent to \( \pi \) radians. This means that 1 radian equals \( \frac{180}{\pi} \) degrees.

Step 2: Set Up the Conversion Formula

The formula to convert from radians to degrees is given by: \[\text{Degrees} = \text{Radians} \times \frac{180}{\pi}\]In this problem, the radian measure is 3.

Step 3: Plug the Value into the Formula

Substitute the radian measure into the conversion formula:\[\text{Degrees} = 3 \times \frac{180}{\pi}\]

Step 4: Perform the Calculation

Calculate the degree measure:\[\text{Degrees} = \frac{3 \times 180}{\pi} = \frac{540}{\pi}\]Using a calculator, approximate the value of \( \pi \approx 3.14159 \). Then:\[\text{Degrees} \approx \frac{540}{3.14159} \approx 171.89\]

Step 5: Round the Result

Round the approximate degree measure to one decimal place: \(\text{Degrees} \approx 171.9\)

Key Concepts

Radians

Radians are a way to measure angles, just like degrees. Instead of dividing a circle into 360 parts (like degrees), radians relate directly to the circumference of the circle. One full circle is equal to approximately 6.28 radians, which is equivalent to 2π radians.

• This comes from the formula for the circumference of a circle, which is 2πr, where r is the radius.
• In a unit circle, where the radius is 1, the circumference is 2π, and hence a full circle is 2π radians.

To visualize a radian, imagine wrapping the radius of a circle along its edge: the length that stretches once around the circle equals 2π radians.

Degrees

Degrees are perhaps the most familiar way of measuring angles. In this system, a full circle is divided into 360 equal parts called degrees. This means that:

• 90 degrees represents a quarter circle, or a right angle.
• 180 degrees is a straight line, half of a circle.

One advantage of using degrees is their intuitive sense of measuring, making it easier to visualize and calculate basic geometric shapes. It's common in everyday settings to hear angles described in degrees, from navigation to architecture. By converting radians to degrees, you can apply radian measurements in everyday contexts.

Angle Measurement

When it comes to measuring angles, both radians and degrees serve a purpose, depending on the context. Degrees break down angles into 360 parts, which is practical for many everyday uses. However, radians provide a more direct mathematical relationship with the circle's geometry. In physics and engineering, radians are often preferred because they lead to simpler and more natural formulas.

• Understanding both systems is fundamental in mathematics.
• The ability to convert between radians and degrees ensures versatility in problem-solving and application.

Whether you're in a classroom or tackling a real-world problem, being comfortable with both systems broadens your understanding of how angles work.

Pi in Radians

Pi (\(\pi\)) is a fundamental constant in mathematics, approximately equal to 3.14159. It represents the ratio of a circle's circumference to its diameter. In radian measure, \(\pi\) acts as a natural unit for angles:

• Half a circle is \(\pi\) radians.
• A full circle corresponds to \(2\pi\) radians.

Pi's role in radian measure eliminates the need for more complex calculations when working with circles. Its simplicity can be particularly helpful in calculus and trigonometry, where angle measures directly impact calculations.