Problem 83 - A Graphical Approach to Precalculus with Limits

Question

Convert each radian measure to degrees. Round answers to the nearest minute. $$2$$

Step-by-Step Solution

Answer

2 radians is approximately 114 degrees, 35 minutes.

Step 1: Understanding the Relationship Between Radians and Degrees

First, understand that a full circle is divided into 360 degrees and also equal to \(2\pi\) radians. This implies that \(1\text{ radian} = \frac{180}{\pi}\text{ degrees}\).

Step 2: Converting Radians to Degrees

To convert 2 radians to degrees, multiply by \(\frac{180}{\pi}\). That is:\[2 \text{ radians} \times \frac{180}{\pi}\text{ degrees per radian} = \frac{360}{\pi}\text{ degrees}\].

Step 3: Calculating the Degree Measure

Calculate \(\frac{360}{\pi}\) using a calculator:\[\frac{360}{3.141592653589793} \approx 114.591559\].This is the degree measure before rounding to the nearest minute.

Step 4: Converting the Decimal Part to Minutes

The degree measure is 114 degrees and 0.591559 of a degree. To convert 0.591559 degrees to minutes, multiply by 60 (since 1 degree = 60 minutes):\[0.591559 \times 60 \approx 35.49354\].

Step 5: Rounding to the Nearest Minute

Round the minutes to the nearest whole number. Thus, 35.49354 minutes rounds to 35 minutes.

Step 6: Presenting the Final Answer

The radian measure of 2 converted to degrees is approximately \(114\) degrees and \(35\) minutes.

Key Concepts

Radians and Degrees Relationship

When exploring the relationship between radians and degrees, it's like comparing two different languages for measuring angles. Just as converting between languages requires a dictionary, converting between radians and degrees requires understanding their relationship.
A circle, in terms of degrees, is 360 degrees. In terms of radians, a circle is expressed as \(2\pi\) radians. From here, you can determine that 1 radian equals \(\frac{180}{\pi}\) degrees.

• This concept helps us switch from the radian language to the degree language.
• It is essential because many real-world applications use degrees, but mathematical equations often use radians.

Whether you're rotating a wheel or calculating the area of a sector, understanding this relationship is key to accurately interpreting angle measurements.

Full Circle Measurement

A full circle can be measured in both degrees and radians, and both offer a complete look at what it means to make a 360-degree rotation. A full rotation equals \(360\) degrees and \(2\pi\) radians, serving as a universal standard for measuring complete cycles. This dual measurement approach is useful across various disciplines:

• In geometry and trigonometry, it simplifies calculations and helps in mapping angles on a coordinate plane.
• In physics, understanding angular velocity and acceleration often requires fluent conversion between these units.

Knowing both measures allows you to seamlessly transition between problems or projects that use either format, making it a fundamental skill for math and science enthusiasts.

Rounding Off Minutes

After converting radians to degrees, sometimes your answer will not be in a whole number. This is where rounding off comes in, especially when converting the decimal portion of a degree into minutes.
A degree consists of 60 minutes, much like an hour consists of 60 minutes of time. To find the minutes after converting, multiply the decimal fraction of the degree by 60. For example, when you get a degree measure of 114.591559, you take the decimal \(0.591559\), multiply by 60, and arrive at approximately 35.49354 minutes.

• This rounded figure, 35 minutes, simplifies interpretation and application, often necessary for precise engineering designs or geographical navigation.

Rounding off helps ensure your angle measurements are as clear and usable as possible, serving practical purposes across various fields.