Problem 20 - Algebra and Trigonometry with Analytic Geometry

Question

Express \(\theta\) in terms of degrees, minutes, and seconds, to the nearest second. $$\theta=4$$

Step-by-Step Solution

Answer

\( \theta = 4^\circ 0' 0'' \).

Step 1: Understand the Conversion

The goal is to convert the value of \( \theta = 4 \) from degrees to degrees, minutes, and seconds (DMS). In this system, 1 degree is equal to 60 minutes and 1 minute is equal to 60 seconds.

Step 2: Determine Whole Degrees

Since \( \theta = 4 \) is already in degrees, we already have the whole number of degrees: 4 degrees.

Step 3: Calculate Remaining Minutes and Seconds

Since there are no decimal parts, we don't have any extra parts to convert into minutes or seconds. The value is a whole number, so there are 0 minutes and 0 seconds because it is exactly 4 degrees.

Key Concepts

Angle Measurement

Angle measurement is a fundamental concept in geometry and trigonometry. It helps us to determine the size of an angle. Angles can be measured in various units, but the most common are degrees, radians, and gradians.

When using degrees, the complete circle is divided into 360 parts, each called a degree. This system makes it easy to express angles in sections of a complete rotation. Often, degrees are further divided into smaller units such as minutes and seconds for more precise measurements.

Understanding angle measurement is crucial in fields like navigation, astronomy, and even construction. It provides precision and accuracy in describing the rotation or orientation differences between lines or planes.

Degrees to Minutes and Seconds

Degrees, minutes, and seconds (DMS) is a system used to express angles in a more detailed format. This system splits the angle measurement into three components: degrees, minutes, and seconds.

• Degrees: The largest unit in the DMS system. 1 full rotation is 360 degrees.
• Minutes: Subdivisions of degrees, where 1 degree equals 60 minutes.
• Seconds: Subdivisions of minutes, where 1 minute equals 60 seconds.

For example, an angle could be expressed as 4° 30' 45", meaning 4 degrees, 30 minutes, and 45 seconds. This system is mainly used in situations where precision is vital, such as astronomy and cartography. However, in cases such as the original exercise where the angle is a whole number, the conversion to minutes and seconds simply results in 0 for both subdivisions, as seen in the step-by-step solution.

Conversion of Angles

Converting angles between different measurement systems, such as degrees, minutes, and seconds (DMS), involves some basic multiplication and division, but no complex mathematics. This process is often necessary in practical applications.

Here's how the conversion works for the DMS system:

• Degrees to Minutes: To convert degrees to minutes, multiply the fractional part of the degree by 60.
• Minutes to Seconds: To convert any remaining minutes to seconds, again multiply the fractional part by 60.

In the exercise where \(\theta = 4\), there is no fractional part after the whole number of degrees. Thus, no further conversion to minutes or seconds is needed, resulting in 4° 0' 0".

Remember, when converting, it is essential to maintain precision and accurately express the measurement to the nearest desired unit. This ensures consistency, especially in technical and scientific endeavors.