Measurement of Angles

The measure of an angle is the amount of rotations required to get to the terminal side from the initial side. A common measure of an angle is derived by placing its vertex at the center of a circle of some fixed radius.

There are two commonly used measurements for angles: degrees and radians

Sexagesimal System (Degree, Minute and Second)

If the initial ray \overrightarrow {OA} rotates in an anti-clockwise direction in such a way that it coincides with itself, the angle then formed is said to be of 360 degrees (360^\circ ).

One rotation (anti-clockwise) = 360^\circ

\frac{1}{2} rotation (anti-clockwise) = 180^\circ is called a straight angle
\frac{1}{4} rotation (anti-clockwise) = 90^\circ is called a right angle.


measurement-angle

1 degree (1^\circ ) is divided into 60 minutes (60') and 1 minute (1') is divided into 60 seconds (60''). As this system of measurement of angles originates from English and because90, 60 and multiplies of 6 and 10, it is known as the English System or Sexagesimal System.

Thus 1 rotation (anti-clockwise)               =          360^\circ
One degree (1^\circ )                                =          60'
One minute (1')                                        =          60''

Conversion from D^\circ M'S'' to a decimal form and vice versa.

(i)         16^\circ \,30' =  16.5^\circ {\text{As}}\,\left( {30' = \frac{{1^\circ }}{2} = 0.5^\circ } \right)
(ii)        42.25^\circ =   45^\circ \,15' \left( {0.25^\circ = \frac{{25^\circ }}{{100}} = \frac{{1^\circ }}{4} = 15'} \right)

Example:

Convert 18^\circ \,\,6'\,\,21'' to decimal form.

Solution:

1' = {\left( {\frac{1}{{60}}} \right)^\circ } and 1'' = {\left( {\frac{1}{{60}}} \right)^\prime } = {\left( {\frac{1}{{60 \times 60}}} \right)^\circ }

\begin{gathered} \therefore 18^\circ \,\,6'\,\,21'' = {\left[ {18 + 6\left( {\frac{1}{{60}}} \right) + 21\left( {\frac{1}{{60 \times 60}}} \right)} \right]^\circ } \\ \therefore 18^\circ \,\,6'\,\,21'' = \left( {18 + 0.1 + 0.005833} \right)^\circ = 18.105833^\circ \\ \end{gathered}


Example:

Convert 21.256^\circ to the D^\circ M'S'' form

Solution:

\begin{gathered} 0.256^\circ = \left( {0.256} \right)\left( {1^\circ } \right) \\ 0.256^\circ = \left( {0.256} \right)\left( {60'} \right) = 15.36' \\ \end{gathered}

and

\begin{gathered} 0.36' = \left( {0.36} \right)\left( {1'} \right) \\ 0.36' = \left( {0.36} \right)\left( {60''} \right) = 21.6'' \\ \end{gathered}

Therefore,

\begin{gathered} 21.256^\circ = 21^\circ + 0.256^\circ \\ \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = 21^\circ + 15.36' \\ \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = 21^\circ + 15' + 0.36' \\ \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = 21^\circ + 15' + 21.6'' = 21^\circ \,\,15'\,\,21''\,rounded{\text{ }}off{\text{ }}to{\text{ }}nearest{\text{ }}second \\ \end{gathered}

Thus \angle AOB = 1{\text{ radian}}