Initially right triangles were used to define trigonometric ratios and angles were measured in degrees. But right triangles have limitations as they involve only acute angles.

**Radian Measure**

Initially right
triangles were used to define trigonometric ratios and angles were measured in
degrees. But right triangles have limitations as they involve only acute
angles. In degrees a full rotation corresponds to 360* ^{â—¦}* where the choice of 360 dates back thousands of years to the
Babylonians. They might have chosen 360 based on the number of days in a year.
But it does have the nice property of breaking into smaller angles like 30

(i) All circles are
similar. Thus, for a given central angle in any circle, the ratio of the
intercepted arc length to the radius is always constant.

(ii) When s = r, we
have an angle of 1 radian. Thus, one radian is the angle made at the centre of
a circle by an arc with length equal to the radius of the circle.

(iii) Since the
lengths s and r have same unit, Î¸ is unitless and thus, we do not use any
notation to denote radians.

(iv) Î¸ = 1 radian
measure, if s = r

Î¸ = 2 radian measure,
if s = 2r

Thus, in general Î¸ = k
radian measure, if s = kr.

Hence, radian measure
of an angle tells us how many radius lengths, we need to sweep out along the
circle to subtend the angle Î¸.

(v) Radian angle
measurement can be related to the edge of the unit circle. In radian system, we
measure an angle by measuring the distance travelled along the edge of the unit
circle to where the terminal side of the angle intercepts the unit circle .

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* ^{â—¦}* and 100

In unit circle, a full
rotation corresponds to 360* ^{â—¦}* whereas, a full
rotation is related to 2

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.

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11th Mathematics : UNIT 3 : Trigonometry : Radian Measure | Definition, Formula, Solved Example Problems, Exercise | Trigonometry | Mathematics

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