Definition of arc system
The central angle of an arc with a long radius is called the angle of 1 radian, and the system for measuring the angle with radians is called the radian system. Taking the vertex of the known angle A as the center, taking any value R as the radius as the arc, and the ratio of the arc length of the angle A to R is a constant [independent of R], we call the positive angle of L=R an angle of 1 radian. Taking 1 radian angle as the unit of measuring angle, this measuring system is called radian system to show the difference from another angle measuring system-angle system.
Edit the basic idea of arc system in this section.
The basic idea of arc system is to make the radius and circumference of a circle have the same measurement unit, and then measure the angle by 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 [i.e. π 3. 142], but Arie Pito did not explicitly put forward the concept of arc 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.
Edit the essence of this arc system
The essence of arc system is to unify the units for measuring arc and radius, which greatly simplifies the related formulas and operations, especially in advanced mathematics, and its advantages are particularly obvious.
Edit the size of 1 radian in this paragraph.
A radian angle: the angle of the center of an arc with a long radius is called 1 radian angle. 1 radian is about 57.3, which is about 57 17' 45 ",but it is exactly equal to 180/π 180 = π rad. Prove the sector area formula S= 1/2LR by arc system, where l is the arc length of the sector. )
Edit the conversion formula of arc system and angle system in this paragraph.
360 = 2πrad——→ 180 =πrad——→ 1 =π/ 180 rad≈0.0 1745 rad——→ 1 rad = 180/π≈57.30 = 57 18
After understanding, you can understand the multiplication and division of angles! I hope I can help you.