Systems of measurement of angles

There are three systems of measurement of angles
(i) Sexagesimal or English system
(ii) Centesimal or French system
(iii) Circular system
Sexagesimal or English system:

Systems of measurement of angles

(i) Sexagesimal or English system : when right angle is divided into 90 equal parts called degree then we use a sexagesimal system. Here $1^{0}$ = one degree. So, $1^{0} =\frac{1}{90}$.
$1^{0}$ is divided into 60 equal parts called minutes and is denoted as $1^{'}$. Again each minute is divided as 60 parts called seconds and is denoted as $1^{"}$.
1 right angle = $90^{0}$
$1^{0}$ = 60 minutes = $60^{'}$
$1^{'}$ = 60 seconds = $60^{"}$
Example : Convert a sexagesimal number to a decimal one. Input degrees/hours, minutes and seconds and get the decimal number:
$32^{0}$ 3 minutes 5 seconds :
$32^{0}$ will remains the same.
As we know that $1^{0}$ = 60 minutes
∴ 3 minutes = $\frac{3}{60}$ = 0.05 minutes
For 5 seconds ⇒ $\frac{5}{60}$ = 0.083 minutes
0.083 minutes to seconds = $\frac{0.083}{60}$ = 0.00138 seconds
$32^{0}$ 3 minutes 5 seconds = 32 degree + 0.05 minutes + 0.00138 seconds = 32.05138

Convert a sexagesimal number to a decimal one.

Deg/Hs
Min
Sec
Dec. number

Convert a decimal number to a sexagesimal one.

Dec. number
Deg/Hs
Min
Sec

Centesimal System : when right angle is divided into 100 equal parts called grades then we use a Centesimal System system.
1 grade is divided into 100 equal parts called minutes and is denoted as $100^{'}$. Again each minute is divided as 100 parts called seconds and is denoted as $100^{"}$.
1 right angle = 100 grades = $100^{g}$
1 grade = 100 minutes = $100^{'}$
1 minute = 100 seconds = $100^{"}$

Circular system : In this system degrees are given in radian.
1 radian is the measure of an angle subtended at the center of the circle by an arc length equal to the radius of the circle. It is denoted by $1^{c}$.

The circle with center 'O' and radius OA and OB. Thus the angle formed at the center is $\angle$AOB is central angle.
$\angle$AOB = $1^{c}$

Formula to convert radians to degree and vice versa:
$1^{c} =\frac{180}{\Pi}$

$1^{0} =\frac{\Pi}{180}$

Degrees Radian
$0^{0}$
0
$30^{0}$
$\frac{\pi}{6}$
$45^{0}$
$\frac{\pi}{4}$
$60^{0}$
$\frac{\pi}{3}$
$90^{0}$
$\frac{\pi}{2}$
$120^{0}$
$\frac{2\pi}{3}$
$150^{0}$
$\frac{5\pi}{6}$
$180^{0}$
$\pi$
$210^{0}$
$\frac{7\pi}{6}$
$240^{0}$
$\frac{4\pi}{3}$
$270^{0}$
$\frac{3\pi}{2}$
$300^{0}$
$\frac{5\pi}{3}$
$330^{0}$
$\frac{11\pi}{6}$
$360^{0}$
$2\pi$

11th grade math

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