Question

Express the following sexagesimal measure as radian measure and centesimal system:
A. 135${^\circ}$
B. 270${^\circ}$
C. 72${^\circ}$

Answer 1

Hint: In order to solve the above question we must know sexagesimal, centesimal and radian measure. Sexagesimal system is the measure which has base 60. Radian measure radian is the measure of a central angle subtended by an arc that is equal in length to the radius of the circle.
Centesimal system When the right angle is divided into 100 equal parts called grades.

Complete step-by-step answer:
Let's discuss sexagesimal, centesimal and radian measure in more detail.
Sexagesimal: It was an ancient system of counting, calculation and numerical notation that used the power of 60 much as the decimal system uses powers of 10.
Radian measure: Radian is the SI unit for measuring angles. The length of an arc of a unit circle is numerically equal to the measurement in radians of the angle that it subtends.
Centesimal system: In this system, a right angle is divided into 100 centesimal degrees each centesimal degree into 100 centesimal minutes and each centesimal minutes into centesimal seconds.
Let's convert all the given values into radian measure first.
For converting degree value into radians we will multiply the given value by $\dfrac{\pi }{{180}}$
For 135 degree:
$ \Rightarrow {135^0} \times \dfrac{\pi }{{180}}$ (on cancelling the common factors of denominator and numerator)
$ \Rightarrow \dfrac{{3\pi }}{4}$ (radian value)
For 270 degree:
$ \Rightarrow {270^0} \times \dfrac{\pi }{{{{180}^0}}}$ (on cancelling the common factors of denominator and numerator)
$ \Rightarrow \dfrac{{3\pi }}{2}$
For 72 degree:
$ \Rightarrow 72 \times \dfrac{\pi }{{{{180}^0}}}$ (on cancelling the common factors of denominator and numerator)
$ \Rightarrow \dfrac{{2\pi }}{5}$
Now conversion in centesimal system
For Centesimal system;
900 = 100 grades
In 135 degree”
$ \Rightarrow {1^0} = \dfrac{{100}}{{90}}$ (on multiplying RHS and LHS by 15)
$ \Rightarrow {135^0} \times {1^0} = \dfrac{{100}}{{90}} \times 135 $
$\Rightarrow {135^0} = 150 $ (we have multiplied by 135 on RHS and LHS)
In 270 degree:
$ \Rightarrow {90^0} = 100$ (we will multiply on RHS and LHS by 3)
$ \Rightarrow {270^0} = 300$
In 72 degree:
$ \Rightarrow {1^0} = \dfrac{{100}}{{90}}$
$ \Rightarrow {72^0} \times {1^0} = \dfrac{{100}}{{90}} \times 72$ (we have multiplied by 72 on RHS and LHS)

$ \Rightarrow {72^0} = 80$

Note:
As the types of systems we have discussed in the above question are degree measuring systems, in the similar manner we have various number system conversions such as, decimal number system with base 10, binary number system with base 2, octal number system with base 8 etc.