Degrees, minutes and seconds: smaller pieces of a turn

You are expressing angle minutes as parts of one degree. Write 30′ as decimal degrees.

1. 30′ = 3060$\frac{30}{60}$° = 12$\frac{1}{2}$° = 0.5°.60 minutes fill one degree, so 30 fill half of it.

You are finding the turn size left after removing the smaller DMS angle. Subtract 18°10′20″ from 42°30′50″.

1. 50″ − 20″ = 30″.Both amounts count seconds and the first is large enough.
2. 30′ − 10′ = 20′.No borrowing was required, so the minute column is unchanged.
3. 42° − 18° = 24°.Subtract the whole-degree column independently.

You are finding the positive opening still needed to complete 180°. Find the supplement of 112°40′.

1. 180°0′ = 179°60′.Exchange one degree for 60 minutes so you can subtract 40′.
2. 179°60′ − 112°40′ = 67°20′.179 − 112 = 67 and 60 − 40 = 20, with the same units in each column.

A telescope starts level, pointing at the horizon. The display on its mount shows that it has now been tilted upward through 34°19′52″. Pointing straight up, at the point directly overhead, takes a tilt of exactly 90° from level. Through what further angle must the telescope be tilted upward to point straight up? Give the angle in degrees, minutes and seconds, show each exchange you make, and check your answer by addition.

1. Set up the subtraction 90° − 34°19′52″. Write 90° as 90°00′00″ so the columns line up: 90° over 34°, 00′ over 19′, 00″ over 52″.Straight up is 90° from level, so the tilt still needed is 90° minus the tilt already made. Only like units can be subtracted. Writing 00′ and 00″ shows that the top minutes and seconds columns are empty.
2. The seconds column needs 00″ − 52″, which cannot be done. The minutes column has no minute to lend. So first borrow one degree as 60 minutes: 90°00′00″ = 89°60′00″.1° = 60′, so trading one degree for 60 minutes leaves the angle the same size. The seconds can only borrow from the minutes column, so that column must hold some minutes before it can lend one.
3. Now borrow one of those minutes as 60 seconds: 89°60′00″ = 89°59′60″.1′ = 60″, so the angle is still unchanged. Together the two exchanges give the rule 90° = 89°59′60″. Now every top entry is at least the entry below it: 89 ≥ 34, 59 ≥ 19 and 60 ≥ 52.
4. Subtract column by column. Seconds: 60″ − 52″ = 8″. Minutes: 59′ − 19′ = 40′. Degrees: 89° − 34° = 55°.After the exchanges each column holds enough of its own unit. Each column can therefore be subtracted on its own with no further borrowing.
5. Combine the three column results: 55°40′08″.The minutes (40) and the seconds (8) are each less than 60. Nothing needs to be carried, so the angle is already in standard degrees-minutes-seconds form.

A ship's navigator sets a straight course turned 27°48′35″ clockwise from due north, so the ship heads between north and east. Measured clockwise from due north, due east is a quarter turn of 90° and due south is a half turn of 180°. (a) Find the complement of the course angle: the further clockwise turn that would swing the ship from its course to due east. (b) Find the supplement of the course angle: the further clockwise turn that would swing the ship from its course to due south. Give both answers in degrees, minutes and seconds, and check each one by adding it to 27°48′35″.

1. Write each missing partner as a subtraction. Turn to due east = 90° − 27°48′35″ (the complement). Turn to due south = 180° − 27°48′35″ (the supplement).Turning clockwise from north, the course angle and the further turn together make the whole turn to the target. That is a quarter turn of 90° to reach east and a half turn of 180° to reach south. So each missing partner is the whole turn minus the part already turned.
2. Line up the columns for the complement, 90°0′0″ over 27°48′35″. Borrow 1° as 60′ to get 89°60′0″. Then borrow 1′ of those as 60″ to get 89°59′60″.The top angle has 0′ and 0″, so 48′ and 35″ cannot be taken away. Since 1° = 60′ and 1′ = 60″, the exchanges change only how 90° is written, not its size: 89°59′60″ = 89°60′ = 90°. The exchanges are in sixties, not in tens as in ordinary subtraction.
3. Subtract each column. Seconds: 60″ − 35″ = 25″. Minutes: 59′ − 48′ = 11′. Degrees: 89° − 27° = 62°. The complement is 62°11′25″.After the double borrow, each top entry is at least as large as the entry beneath it (60 ≥ 35, 59 ≥ 48, 89 ≥ 27). So each column subtracts on its own with no further exchanges.
4. Line up the columns for the supplement, 180°0′0″ over 27°48′35″. Rewrite 180° as 179°59′60″: borrow 1° as 60′ to get 179°60′0″, then borrow 1′ as 60″.The half turn also has 0′ and 0″, so it needs the same double borrow. 179°59′60″ still equals 180°.
5. Subtract each column. Seconds: 60″ − 35″ = 25″. Minutes: 59′ − 48′ = 11′. Degrees: 179° − 27° = 152°. The supplement is 152°11′25″.The seconds and minutes columns hold the same numbers as in the complement, so they give the same 11′25″. Only the degree column changes, because the top now has 179° instead of 89°.
6. Compare the two partners: 152°11′25″ − 62°11′25″ = 90°0′0″.Since 180° − θ = (90° − θ) + 90°, the supplement must be exactly 90° more than the complement, with the same minutes and seconds. The matching 11′25″ in both answers shows the smaller columns were handled the same way in both subtractions.

A metal shop is remaking a steel bracket from an old drawing. The drawing marks the angle between the bracket's two arms as 71°14′51″. The shop's laser cutter accepts angles only in decimal degrees, with up to four decimal places. Use 1° = 60′ and 1′ = 60″, so 1° = 3600″. (a) Write 71°14′51″ in decimal degrees. This is the number to type into the cutter. (b) A coworker suggests copying the digits and typing 71.1451. Would the cutter's angle then be too large or too small, and by how many degrees?

1. Split the angle into its columns: 71°14′51″ = 71° + 14′ + 51″. Keep the 71 whole degrees as they are.The degrees are already in the unit the cutter wants. Only the 14 minutes and 51 seconds are smaller pieces of a degree that still need converting.
2. Turn the minutes into degrees: 14′ = 14 ÷ 60 degrees = 1460$\frac{14}{60}$°.1° = 60′, so one minute is 160$\frac{1}{60}$ of a degree. A minute is a sixtieth of a degree, not a hundredth, so the digits 14 cannot just be placed after the decimal point.
3. Turn the seconds into degrees: 51″ = 51 ÷ 3600 degrees = 513600$\frac{51}{3600}$°.1° = 60′ = 60 × 60″ = 3600″, so one second is 13600$\frac{1}{3600}$ of a degree. Dividing by only 60 would turn the seconds into minutes, not into degrees.
4. Put the minutes over 3600 and add: 1460$\frac{14}{60}$ = 14 × 6060 × 60$\frac{14\times 60}{60\times 60}$ = 8403600$\frac{840}{3600}$, so 8403600$\frac{840}{3600}$ + 513600$\frac{51}{3600}$ = 8913600$\frac{891}{3600}$.Fractions can be added only over a common denominator, and 3600 = 60 × 60 is a multiple of 60. Exact fractions also avoid rounding 14 ÷ 60 = 0.2333… too early. Any rounding waits until the end.
5. Divide: 8913600$\frac{891}{3600}$ = 99400$\frac{99}{400}$ = 247510000$\frac{2475}{10000}$ = 0.2475. On a calculator, 891 ÷ 3600 = 0.2475.Dividing top and bottom by 9 gives 99400$\frac{99}{400}$. Multiplying top and bottom by 25 turns it into ten-thousandths. So 14′51″ is exactly 0.2475 of a degree, with four decimal places and nothing to round.
6. Add back the whole degrees: 71 + 0.2475 = 71.2475°.Decimal degrees = D + M60$\frac{M}{60}$ + S3600$\frac{S}{3600}$, here with D = 71, M = 14 and S = 51.
7. (b) Compare the copied entry with the true value: 71.2475 − 71.1451 = 0.1024.71.1451 is the smaller number, so the cutter would make the angle 0.1024° too small. Copying the digits treats 14′51″ as 145110000$\frac{1451}{10000}$ of a degree, but it is really 8913600$\frac{891}{3600}$ = 247510000$\frac{2475}{10000}$ of a degree. The error is more than 6 minutes, since 0.1024 × 60 = 6.144′.

A hiking club is printing a trail guide that lists the latitude of each landmark in degrees, minutes and seconds, rounded to the nearest second. (Latitude is an angle. It measures how far north or south of the equator a place lies.) A member's phone app gives the latitude of a waterfall on the club's Blue Loop trail as 39.4147° N, in decimal degrees. Using 1° = 60′ and 1′ = 60″, write the waterfall's latitude in degrees, minutes and seconds, to the nearest second, for the guide.

1. Split off the whole degrees: 39.4147° = 39° + 0.4147°. The 39° goes straight into the answer.The digits before the decimal point already count complete degrees. Only the leftover 0.4147° is less than one degree, so only that part needs to be rewritten in the smaller units.
2. Turn the leftover degree into minutes: 0.4147 × 60 = 24.882′.1° = 60′, so any part of a degree is worth 60 times as many minutes. Multiply only the leftover 0.4147, not 39.4147. The digits 4147 are ten-thousandths of a degree, not minutes and seconds. Reading them as 39°41′47″ would give about 39.6964°, which is a different latitude.
3. Keep the whole minutes, 24′, and carry the leftover 0.882′ to the next step.24.882′ is 24 complete minutes and most of another minute. Rounding up to 25′ here would throw the leftover away. The 0.882′ is still part of the angle, and it becomes the seconds.
4. Turn the leftover minute into seconds: 0.882 × 60 = 52.92″.1′ = 60″, so a part of a minute times 60 gives the same amount in seconds.
5. Round at the end: 52.92″ ≈ 53″ to the nearest second. Since 53 is less than 60, nothing carries into the minutes column.Seconds are the last unit, so this is the only place to round. Rounding earlier changes the answer. For example, rounding 24.882′ to 24.9′ first would give 0.9 × 60 = 54″, which is one second too many.
6. Put the columns together: 39°24′53″, keeping the N from the app.Each column now holds a whole number below its limit (24 < 60 and 53 < 60), so no column needs adjusting. The N still shows that the waterfall is north of the equator.

You are describing the same turn with a different measuring unit. Convert 12.9999° to DMS to the nearest second.

1. Keep 12°. Compute 0.9999 × 60 = 59.994′.Only the fractional degree changes into minutes.
2. Keep 59′. Compute 0.994 × 60 = 59.64″.Only the fractional minute changes into seconds.
3. 59.64″ rounds to 60″. Exchange 60″ for 1′, giving 12°60′0″.Nearest-second rounding goes upward, and 60 seconds make one minute.
4. Exchange 60′ for 1°, giving 13°0′0″.Standard DMS form uses fewer than 60 minutes and fewer than 60 seconds.