Quadrants and Measurement of Angles
Let $ \displaystyle XO{X}’$ and $ \displaystyle YO{Y}’$ be two mutually perpendicular lines in a plane and $ \displaystyle OX$ be the initial half line. The whole plane is divided into 4 different regions $ \displaystyle XOY,YO{X}’,{X}’O{Y}’\text{ and }{Y}’OX$. These regions are called quadrants and are respectively called 1st, 2nd, 3rd and 4th quadrants. The angle is said to be in 1st, 2nd, 3rd and 4th quadrants according as the terminal side lies in the 1st, 2nd, 3rd and 4th quadrants. If the terminal side coincides with one of the axes, then angle is called the quadrant angle.
For any angle $ \displaystyle \alpha $ which is not a quadrant angle and when the number of revolutions is not more than one and the radius vector rotates in an anticlockwise direction.
$ \displaystyle {{0}^{{}^\circ }}<\alpha <{{90}^{{}^\circ }}$ if $ \displaystyle \alpha $ lies in the 1st quadrant
$ \displaystyle {{90}^{\circ }}<\alpha <{{180}^{\circ }}$ if $ \displaystyle \alpha $ lies in the 2nd quadrant
$ \displaystyle {{180}^{\circ }}<\alpha <{{270}^{\circ }}$ if $ \displaystyle \alpha $ lies in the 3rd quadrant
$ \displaystyle {{270}^{\circ }}<\alpha <{{360}^{\circ }}$ if $ \displaystyle \alpha $ lies in the 4th quadrant
when the terminal side coincides with $ \displaystyle OY$ angle formed = $ \displaystyle {{90}^{\circ }}$
when the terminal side coincides with $ \displaystyle O{X}’$ angle formed = $ \displaystyle {{180}^{\circ }}$
when the terminal side coincides with $ \displaystyle O{Y}’$ angle formed = $ \displaystyle {{270}^{\circ }}$
when the terminal side coincides with $ \displaystyle OX$ angle formed = $ \displaystyle {{360}^{\circ }}$
Units of measurement of angles
In geometry angles are measured in terms of right angle. But in order to measure smaller angle we introduce smaller units of angle.
There are three systems of unites for measurement of angles.
- Sexagesimal system
- Centesimal system
- Radian or circular measure
1. Sexagesimal or British System
In this system of measurement a right angles is divided into 90 equal parts called Degrees. Each degree is then divided into 60 equal parts called Minutes and each minute is further divided into 60 equal parts called Seconds.
A degree, a minute and a second are respectively denoted by the symbol $ \displaystyle {{1}^{\circ }},{{1}^{‘}}\text{ and }{{1}^{{”}}}$.
Thus,
$ \displaystyle \begin{array}{l}1\text{ right angle}={{90}^{{}^\circ }}\\{{1}^{{}^\circ }}={{60}^{‘}}\\{1}’={{60}^{{”}}}\end{array}$
Generally this system of measurement of angle is used but it is not a convenient system because of multipliers of 90 and 60.
2. Centesimal or French System
In this system of measurement a right angle is divided into 100 equal parts called Grades. Each grades is then divided into 100 equal parts called minutes and each minute is further divided into 100 equal parts called Seconds.
Thus,
$ \displaystyle \begin{array}{l}1\text{ right angle}={{100}^{g}}\\{{1}^{{}^\circ }}={{100}^{‘}}\\{1}’={{100}^{{”}}}\end{array}$
Angle $latex \displaystyle {{90}^{\circ }}$ is called a right angle.
Note: $ \displaystyle {{1}’}$ of centesimal system $ \displaystyle \ne {1}’$ of Sexagesimal system
$ \displaystyle {{1}^{{”}}}$ of centesimal system $ \displaystyle \ne {{1}^{{”}}}$ of Sexagesimal system
3. Radian or circular measure
The angle subtended at the center by an arc of circle whose length is equal to the radius of the circle is called radian and is denoted by $ \displaystyle {{1}^{c}}$.
We shall show that this angle is independent of the radius of the circle considered.
Let O be the center of a circle of radius r and let length of arc AB = r.
Then by the definition of radian $ \displaystyle \angle AOB=1\text{ radian}$
Meaning of pie ($ \displaystyle \pi $)
The ratio of circumference and diameter of a circle is always constant and this constant is denoted by Greek letter $ \displaystyle \pi $.
$ \displaystyle \pi =\frac{{22}}{7}\text{ (roughly)}=3.1415…$
$ \displaystyle \pi $ is an irrational number. Its more accurate value is $ \displaystyle \frac{{355}}{{113}}$ (roughly). The exact numerical value of $ \displaystyle \pi $ cannot be found. Thus, if r be the radius of a circle and c be its circumference then,
$ \displaystyle \frac{c}{{2r}}=\pi \text{ or }c=2\pi r$
