I personally recommend to avoid using Sine,Cosine and other trigonometric function to solve geometry related problems as other easier ways seem more handy and quick.Here is my solution:
<BCA is an inscribe angle and AB is the diameter,so <BCA = 90 degree.So as ABC is a right triangle so we can use Pythagorean Theorem on this triangle.Since sum of the interior angles of any triangle is equal to 180 degrees therefore
m<BCA+x+2x = 180 => 90+3x = 180 => x = 30 => Triangle ABC is a 30-60-90 triangle and we may use properties of 30-60-90 triangle to solve this question.Sides of a 30-60-90 triangle are in the ratio 1:\(sqrt3\):2.Since hypotenuse of Triangle ABC is equal to Diameter AB = 2r of the circle;where r is radius of the circle.So the sides of Triangle ABc are r,r\sqrt{3} and 2r.
=> Quantity A : Area of Triangle ABC = (1/2)*(base)*(height) = (1/2)*(BC)*(AC) = = (1/2)*(r)*(r\(sqrt3\)) = \((sqrt3/2)*r^2\)
Since 2x = 2*30 = 60 so Triangle OBD is an equilateral triangle and formula for area of an equilateral triangle is \((sqrt3/4)*s^2\),where s is the side of the equilateral triangle.Side of equilateral Triangle OBD is equal to radius r of the circle
Quantity B : Twice the Area of Triangle ABC = \(2*(sqrt3/4)*r^2\) = \((sqrt3/2)*r^2\) which is equal to Quantity => Quantity A = Quantity B.Hence answer is C.
This question has been taken from tests of Greatest Prep.Here is the link to the website of Greatest Prep :
http://www.greatestprep.com