RELATIONSHIP BETWEEN DEGREE AND RADIAN MEASURE

We have degree and radian units to measure angles. One measuring unit is better than another if it can be defined in a simpler and more intuitive way.

For example, in measuring temperature, Celsius unit is better than Fahrenheit as Celsius was defined using 0° and 100° for freezing and boiling points of water.

Radian measure is better for conversion and calculations. Radian measure is more convenient for analysis whereas degree measure of an angle is more convenient to communicate the concept between people.

Greek Mathematicians observed the relation of π which arises from circumference of a circle and thus, π plays a crucial role in radian measure.

In unit circle, a full rotation corresponds to 360° whereas, a full rotation is related to 2π radians, the circumference of the unit circle.

Thus, we have the following relations :

2π radians = 360°

Divide each side by 2. 

π radians = 180° 

Then, 

1 radian = (180/π)° or 1° = (π/180) radians

x radians = (180x/π)° or x° = (πx/180) radians

Observe that the scale used in radians is much smaller than the scale in degrees. The smaller scale makes the graphs of trigonometric functions more visible and usable.

The above relation gives a way to convert radians into degrees or degrees into radians.

Points to Remember

1. The ratio of the circumference of any circle to its diameter is always a constant. This constant is denoted by the irrational number π.

2. Mark a point P on a unit circle and put the unit circle on the number line so that P touches the number 0. Allow the circle to roll along the number line. The point P will touch the number 2π on the number line when the circle rolls to one complete revolution to the right.

3. If the unit of angle measure is not specified, then the angle is understood to be in radians.

4. Consider a sector of a circle with radius r. If θ is the central angle of the sector, then

Clearly, the calculation in radian measure is much easier to work with.

5. The values of π and 22/7 correct to four decimal places are 3.1416 and 3.1429 respectively. Thus, π and 22/7 are approximately equal correct upto two decimal places. Hence, π ≈ 227.

6. 

radian ≈ 57°17'45" and 1° ≈ 0.017453 radian

1' = [π/(180 ⋅ 60)] radian ≈ 0.000291 radian

1" = [π/(180 ⋅ 60 ⋅ 60)] radian ≈ 0.000005 radian

7. The radian measures and the corresponding degree measures for some known angles are given below. 

And also

2π  =  360°

3π/2  =  270°

π  =  180°

π/2  =  90°

π/3  =  60°

π/4  =  45°

π/6  =  30°

8. sin 90° = 1 but,sin 90 ≠ 1 (in radian measure).

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