Perform each calculation. See Example 3. 90° ― 36° 18' 47"

Verified step by step guidance

1. Understand that the problem requires subtracting two angles given in degrees, minutes, and seconds format: \(90^\circ\) and \(36^\circ 18' 47''\).
2. Recall that 1 degree (\(1^\circ\)) equals 60 minutes (\$60'\(), and 1 minute equals 60 seconds (\)60''$). This means you may need to borrow from degrees or minutes if the subtraction in any column (degrees, minutes, seconds) is not straightforward.
3. Set up the subtraction by aligning degrees, minutes, and seconds: \(90^\circ 0' 0'' - 36^\circ 18' 47''\).
4. Since you cannot subtract 47 seconds from 0 seconds, borrow 1 minute (which is 60 seconds) from the 0 minutes. But since minutes are also 0, you will need to borrow 1 degree (which is 60 minutes) from the 90 degrees, converting it stepwise to minutes and seconds.
5. After borrowing, perform the subtraction separately for seconds, minutes, and degrees, then combine the results to express the final angle in degrees, minutes, and seconds.

Key Concepts

Angle Measurement in Degrees, Minutes, and Seconds

Angles can be expressed in degrees (°), minutes ('), and seconds ("), where 1 degree equals 60 minutes and 1 minute equals 60 seconds. This system allows for precise angle measurements, commonly used in fields like navigation and astronomy.

Subtraction of Angles in DMS Format

Subtracting angles in degrees, minutes, and seconds requires careful borrowing when the minuend's minutes or seconds are smaller than the subtrahend's. This process is similar to subtracting time, ensuring each unit is properly aligned before performing the subtraction.

Conversion Between Units for Simplification

To simplify calculations, angles in DMS can be converted entirely into seconds or decimal degrees. This conversion helps avoid errors during arithmetic operations and can be converted back to DMS after calculation for final results.