Question

AC是圆O的直径，PA,PB是圆O的切线。A,B为切点，AB=6,PA=5.求圆O的半径

求：（1）圆O的半径
（2）sin∠BAC的值
（答案：第一题是4分之15，第二题是5分之3，，，我要过程！！！）

Answer 1

AB = 6, PA =5
let OB =OA = r and ∠AOB = 2x
=> ∠AOP = ∠POB = x
consider △AOB
sin (90°-x)/r = sin2x/AB
cosx/r = sin2x/6
r = 6cosx/sin2x
= 3/sinx (1)
consider △OAP
sinx = AP/OP
= AP/√(AP^2+OA^2)
= 5/√(25+r^2) (2)
sub (2) into (1)
r(5/√(25+r^2) ) = 3
5r = 3√(25+r^2)
25r^2 = 9(25 +r^2)
r^2 = 9(25)/16
r = 15/4

(2)
BC^2 + AB^2 = OC^2
BC^2+ 36 = 225/4
BC = 9/2
sin∠BAC = BC/AC
= BC/2r
= 2BC/15
= 3/5