Problem 41 - A Graphical Approach to Precalculus with Limits

Question

Convert each angle measure to degrees, minutes, and seconds. Use a calculator as necessary. Round to the nearest second. $$89.9004^{\circ}$$

Step-by-Step Solution

Answer

\( 89^\circ\, 54'\, 1'' \).

Step 1: Separate degrees

\( 89.9004^\circ \): whole degrees = \( 89^\circ \), decimal part = \( 0.9004 \).

Step 2: Convert to minutes

\( 0.9004 \times 60 = 54.024 \) minutes. Whole minutes = 54'.

Step 3: Convert to seconds

\( 0.024 \times 60 = 1.44 \approx 1'' \) (rounded to nearest second).

Step 4: Final answer

\( 89.9004^\circ = 89^\circ\, 54'\, 1'' \).

Key Concepts

Degrees, Minutes, Seconds

Degrees, Minutes, and Seconds are units used to measure angles and geographical coordinates. Understanding these components is crucial for tasks such as precise navigation or astronomy.

• Degrees (") are the primary measure of angles. Each complete rotation is 360 degrees.
• Minutes (') are smaller divisions of a degree. Each degree has 60 minutes.
• Seconds ('') are even more precise, with each minute divided into 60 seconds.

Think about them like you would time: 1 hour = 60 minutes = 3600 seconds. This system allows for very precise measurements of angles. Using these units can seem complex, but it's straightforward once you grasp the decimal-to-fraction conversion.
For example, converting a decimal degree to this format involves applying the following steps:

• The whole number represents the degrees.
• Multiply the decimal by 60 to get minutes.
• Take the remaining decimal and multiply by 60 again to find the seconds.

Understanding this breakdown ensures clarity, especially when precise angle measurements are necessary.

Radians to Degrees

Radians and degrees are two different ways to measure angles, yet they can be converted to one another.
This concept is essential in higher-level mathematics, physics, and engineering. A circle’s circumference is directly related to its radius, which is where radians come into play.

• One full circle is 360 degrees or 2π radians.
• To convert from radians to degrees, use the formula: \[ ext{Degrees} = ext{Radians} \times \left( \frac{180}{\pi} \right) \]
• Conversely, going from degrees to radians involves: \[ ext{Radians} = ext{Degrees} \times \left( \frac{\pi}{180} \right) \]

To illustrate, let's convert \(1\) radian to degrees. Apply the formula: \[ 1 \times \frac{180}{\pi} \approx 57.2958^{\circ} \] Remember, knowing these formulas lets you switch between systems effortlessly, helping in various applications from unit circle trigonometry to solving complex trigonometric functions.

Angle Measurement

Angle measurement is fundamental in understanding shapes, navigation, and even time.
When we discuss angle measurement, we are referring to the amount of turn between two lines that join at a common point. Various units can express this: degrees, radians, and sometimes even gradians. The choice of units may depend on the particular field or application.

• Degrees are often used in geometry and everyday situations like telling time, navigating, or in general education.
• Radians are typically used in calculus and physics, offering a natural connection to other mathematical constants and functions.
• Gradians, less commonly used, where a right angle is 100 grad, offer an alternate measure mostly in specific engineering fields.

To measure an angle, one can use a protractor for degrees or rely on mathematical calculations for radians.
Knowing the context for each measurement helps you select the right tool or method for any situation, enhancing accuracy and understanding.