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The definition of radian system is equal to the angle of the center of an arc with a long radius, which is called 1 radian angle, which is expressed by the symbol rad and read as radian. A system that measures angles in radians is called a radian system.
With the vertex of the known angle A as the center and any value R as the radius, the ratio of the arc length of the angle A to R is a constant value (independent of R), and we call the positive angle of L=R an angle of 1 radian. The angle is measured in units of 1 radian angle. This measuring system is called radian system to show the difference with another angle measuring system-angle system.
Edit the characteristics of the arc system in this section. The angle formed by the counterclockwise rotation of the ray corresponding to one side of any angle is called the positive angle; The angle formed by clockwise rotation is called negative angle; The light stays in its original position without any rotation, so we also regard it as an angle, which is called a zero-degree angle.
No matter whether the angle system or the arc system is adopted, the set of angles and the set of real numbers R are in one-to-one correspondence: each angle corresponds to a unique real number.
The radian value of positive angle is positive (positive real number), the radian value of negative angle is negative (negative real number), and the radian value of zero angle is zero.
The basic idea of editing arc system in this paragraph is to make the radius and circumference of a circle have the same measurement unit, and then measure the angle with the ratio of the corresponding arc length to the radius of the circle. The prototype of this idea originated in India. The famous Indian mathematician Aliyepito (476)? -550? The circumference of a circle is 2 1600 minutes, and the radius of a circle is 3438 minutes (that is, pi is 3. 142), but Aliyepito did not explicitly put forward the concept of radian system. The strict concept of radian was put forward by the Swiss mathematician Euler (1707- 1783) in 1748. Euler is different from Aliyepito in that the radius is 1 unit, so the arc length of the semicircle is π, and the sine value at this time is 0, so it is recorded as sinπ= 0. Similarly, the arc length of the circumference of 1/4 is π/2, and the sine at this time is 1, so it is recorded as sin (π/2) = 6544. Thus, the central angles of semicircle and 1/4 arc expressed by π and π/2 respectively are established. Other angles can also be analogized.
The essence of editing this arc system is to unify the units for measuring arcs and radii, thus greatly simplifying related formulas and operations, especially in advanced mathematics.
Edit the size of 1 radian in this paragraph. Angle of one radian: the central angle of an arc with a longer radius is called 1 radian.
|a|=l/r
1 radian is approximately equal to 57.3.
It is about 57 17' 45 ".
But it is exactly equal to 180/π.
180 =π radian
The formula of sector area S= 1/2LR is proved by radian system, where l is the arc length of sector and r is the radius of circle. If the central angle a of a circle with radius r is opposite to the arc length l, the sign of | A | = L/R(A) is determined by the rotation direction. )