1.WHAT ACTUALLY "A RADIAN" IS?2.WHY DO WE NEED RADIANS FOR PERFORMING CALCULATIONS IN MATHS AND PHYSICS?3.WHY IN CALCULAS,ALL TRIG FUNCTIONS ARE IN RADIANS?

1)

A unit of angular measure equal to the angle subtended at the center of a circle by an arc equal in length to the radius of the circle, approximately 57°17'44.6"

2)

The reason people use radians is convenience. Degrees is a totally artifitial unit, which is hard to relate to anything geometrical.
If you draw a unit circle (radius=1), and consider different angles from it's center, the the length of the chord will be equal to the value of the angle in radians.

The degrees just gave us confusion about sin and the others. You see,
sin x ≠ sin x°,
but
sin x = sin (x radians)

Here the sin function without an angle unit is a circular function, but that with an angle unit is a trigonometric function. Clearly when we use radians we can easily change it into a real number, but in degrees you must convert it first to radians by multiplying pi/180, which is not convenient.

Another is that in calculus, figures such as x sin x, x + cos x, etc., are apparent. Here you cannot use a degree for x, rather, you must use a real number, and that is radians. Because radians are simply numbers; they are unitless (π radians is the same as π). since the result of sin, or cos is always a real number (not an angle), we can now add them, without bothering about the units.

3)

Radians specify an angle by measuring the length around the path of the unit circle and constitute a special argument to the sine and cosine functions

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regards

Ramesh