In triangle ABC, the side length of the inner angle ABC is ABC respectively. It is known that C = 2 and C = 60 degrees. If SINB = 2sina, if the area of triangle ABC is equal to root 3

In triangle ABC, the side length of the inner angle ABC is ABC respectively. It is known that C = 2 and C = 60 degrees. If SINB = 2sina, if the area of triangle ABC is equal to root 3

(1) If SINB = 2sina, a / Sina = B / SINB in triangle ABC, SINB / Sina = 2 = B / a because SINB = 2sina, that is, B = 2acosc = (a ^ 2 + B ^ 2-C ^ 2) / (2Ab) C = 2, COSC = 60 °, B = 2A is brought into the above formula, 1 / 2 = (5a ^ 2-4) / (2 * 2A ^ 2) the solution is a = 2 √ 3 / 3, B = 4 √ 3 / 3 (2) if the area of triangle ABC is equal to √ 3S

As shown in the figure, P1 is a semicircular cardboard with a radius of 1. Cut it at the left lower end of P1 ·····························································, As shown in the figure, P1 is a semicircular cardboard with a radius of 1. Cut a semicircle with a radius of 1 / 2 at the left lower end of P1 to obtain the figure P2, and then cut a smaller semicircle in turn (its diameter is the radius of the previous semicircle cut) to obtain the figures P3 and P4 ···· PN. Note that the area product of the cardboard PN is Sn, and try to find S2 = (); S3 = (); and guess sn-sn-1 = () (n ≥ 2)

S2=3π/8
S3=11π/32
Sn-sn-1 = (- π / 2) (1 / 4) to the (n-1) power

As shown in the figure, P1 is a semicircular cardboard with a radius of 1. Cut a cardboard with a radius of 1 at the left lower end of P1 Figure P2 is obtained after the semicircle of 2, and then a smaller semicircle (its diameter is the radius of the previous semicircle) is cut in turn to obtain figures P3, P4,..., PN,..., and the area of paperboard PN is SN. Try to calculate s3-s2 = ___; And guess sn-sn-1=______ （n≥2）．

S 3 is one radius less than S 2, which is (1
4) So s3-s2 = - 1
2（1
4）2 π=-π
32；
Sn-Sn-1=−π
2(1
4)n−1．
So the answer is: - π
32 and − π
2(1
4)n−1．

A square cardboard with a side length of 4cm takes its opposite side as the diameter. After cutting off two semicircles, how many centimeters is the circumference of the remaining figure?

Formula: 3.14 × 4＝12.56（cm)
12.56+4 × 2=20.56（cm)
Answer: the perimeter is 20.56cm

As shown in the figure, P1 is a semicircular cardboard with a radius of 1. Cut a cardboard with a radius of 1 at the left lower end of P1 Figure P2 is obtained after the semicircle of 2, and then a smaller semicircle (its diameter is the radius of the previous semicircle) is cut in turn to obtain figures P3, P4,..., PN,..., and the area of paperboard PN is SN. Try to calculate s3-s2 = ___; And guess sn-sn-1=______ （n≥2）．

S 3 is one radius less than S 2, which is (1
4) So s3-s2 = - 1
2（1
4）2 π=-π
32；
Sn-Sn-1=−π
2(1
4)n−1．
So the answer is: - π
32 and − π
2(1
4)n−1．

As shown in the figure, P1 is a semicircular cardboard with a radius of 1. Cut a semicircular cardboard with a radius of 1 at the left lower end of P1 Figure P2 is obtained after the semicircle of 2, and then a smaller semicircle (its diameter is the radius of the previous semicircle cut off) is cut in turn to obtain circles P3, P4,..., PN... And the area of cardboard PN is Sn, then Lim n→∞Sn=______．

The semicircular area cut each time forms a circle with π
8 first, followed by 1
4 is an equal ratio sequence of common ratios,
Then Lim
n→∞a1+a2+…+an=π
eight
1−1
4=π
six
Therefore: Lim
n→∞Sn=π
2−π
6=π
three
So the answer is: π
three