A unit to compare spins

Dividing a turn into 360 equal parts defines the degree. A full turn measures $360°$; half-turn, $180°$; a room, $90°$. Angles between $0°$ and $90°$ are acute; between $90°$ and $180°$, obtuse.

On a circle, the measure of an arc in degrees is the measure of its corresponding central angle. To distinguish between two arcs with the same endpoints, indicate which one is being measured.

Minor parts of the degree

A degree has 60 arc minutes and each minute has 60 arc seconds:

$$1°=60',\qquad1'=60'',\qquad1°=3600''.$$

These units measure angles. Names do not make an angle a measurement of time.

To transform $27°18'$ into decimal degrees:

$$27°18'=\left(27+\frac{18}{60}\right)°=27{,}3°.$$

In the opposite direction, in $42{,}25°$, the part $0{,}25°$ corresponds to $0{,}25\cdot60=15'$. Therefore, $42{,}25°=42°15'$.

Demo: Why do we need to regroup?

Adding units of the same type gives:

$$18°45'+7°30'=25°75'.$$

Like $75'=60'+15'=1°15'$, the form with minutes less than 60 is:

$$25°75'=26°15'.$$

The transformation changes the unit of a part of the sum, preserving the total measurement. When subtracting, we can do the opposite: $30°=29°60'$.

Complementary and supplementary

Two angles are complementary when they add up to $90°$ and supplementary when they add up to $180°$. The complement of $34°$ is $56°$; the supplement is $146°$.

These relationships reappear when we study the two acute angles of a right-angled triangle and the symmetries of the trigonometric cycle.

In everyday life

  • Coordinates. Latitudes and longitudes are given in degrees, minutes and seconds; São Paulo is close to $23^\circ 33'$ S.
  • Nautical mile. One minute of latitude measures about 1852 m, the nautical mile used by ships and planes; one second, about 31 m.
  • Cell phone GPS. The applications also show the coordinates in decimal degrees: $23{,}55^\circ = 23^\circ 33'$.
Latitudes in degrees, minutes and seconds, and the distance to the equator counted in nautical miles.

Exercises with resolution

1. Calculate $40°10\prime-12°35\prime$.

Check resolution 1

$40°10\prime=39°70\prime$. Subtracting, $39°70\prime-12°35\prime=27°35\prime$.

2. Convert $12°24\prime36\prime\prime$ to decimal degrees.

Check resolution 2

$12+24/60+36/3600=12{,}41$. The measurement is $12{,}41°$.

In the interactive book

The same subjects, with all the demonstrations and the complete list of exercises, are in the interactive Trigonometry book:

This class is a shorter introduction to the topic; In the book, the theory is accompanied by solved and proposed exercises from the chapter.

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