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## Circular System (Radian Measure) In this system, the unit of measurement of an angle is a radian. Let *r* be the radius of a circle with centre *O*. Let *A* and *B* be two points on the circle such that the length of arc *AB* is *r*. Then the measure of the central angle *AOB* is defined as 1 ra...

## Circular System (Radian Measure) In this system, the unit of measurement of an angle is a radian. Let *r* be the radius of a circle with centre *O*. Let *A* and *B* be two points on the circle such that the length of arc *AB* is *r*. Then the measure of the central angle *AOB* is defined as 1 radian. It is denoted by 1c. ### Theorem: The radian so defined is independent of the radius of the circle used and 1c = π/180° **Proof:** Let us consider a circle with centre at *O* and radius *r*. Let *AB* be an arc length *r*. Join *OA* and *OB*. Then ∠*AOB* Produce *AO* to meet the circle at *C*. Clearly, ∠*AOC* = a straight angle = 2 right angles Since measures of the angles at the centre of a circle are proportional to the lengths of the corresponding arcs subtending them: m∠*AOB*/(arc*AB*) = m∠*AOC*/(arc*ABC*) m∠*AOB* = 1/(2π) * m∠*AOC* m∠*AOB* = 1/π * m∠*AOC* 1c = m∠AOB = ( 2 right angles) / π a constant independent of *r*. Hence one radian is well defined. Also, πc = 2 right angles = 180°. Hence, a radian is a constant angle and two right angles = 180° = πc ## Activity 2: Verify the above result for circles having different radii. Let an angle have its measure θ in degrees. Then its proportion to a straight angle is the same in either case: θ/180° = rθx We use this relation to convert measure into degree and vice-versa. ### Notes: 1. To convert degree measure into radian measure, multiply degree measure by π/180°. 2. To convert radian measure into degree measure multiply radian measure by 180°/π. 3. Taking π = 3.14, we have 1°= 180°/π = 57.3248°. Here fractional degree is converted into fraction. It can be converted into minutes and seconds as follows: 0.3248° = (0.3248 x 60)'= 19.488' 19.488' = 19' + (0.488 x 60)"= 19' 29'' Thus, 1° = 57° 19' 29" 4. In the table given below degree measures are expressed in radians: | Degree | Radian | |---|---| | 15 | π/12 | | 30 | π/6 | | 45 | π/4 | | 60 | π/3 |