NEW DIAGRAM:
On a unit circle, sin x = square root 3 over 2, with center O.
Vertical line from top to center of circle is PO.
Horizontal line from middle to right of circle is OE.
Another vertical line of AB is parallel to PO.
And radius of circle (line from O to A) is 1 unit.
From A to E is the arc.
From E to B is another arc.
) is the angle 60 degree
Fine the area of the shaded region (where the ::::: are).
Sorry it's a bad diagram, can't really draw a circle on here..
..............P
..............|.........A
..............|....... "|:
..............|......"..|::
..............|...."....|::.
..............|.." )....|:::
..----------- O --------+--- E
..............|.."......|:::
..............|...."....|::'
..............|......"..|::
..............|........"|:
..............|.........B
a 30,60 , 90 triangle has sides
1 =AO, sqrt 3 = EA and hypotenuse 2
AB = 2 sqrt 3
your angle AOB is 2*60 = 120
OE the altitude is 1
Your area = (1/2)(1)(2 sqrt 3)
= sqrt 3
sorry 1 = EO, the side along the x axis
But the question ask to find the area of AE and EB together
To find the area of the shaded region, we need to subtract the area of sector AOE from the area of triangle AEB.
Step 1: Find the area of sector AOE.
The area of a sector can be found using the formula A = (θ/360) * π * r^2, where θ is the angle in degrees and r is the radius.
In this case, the angle θ is 60 degrees and the radius r is 1 unit.
So the area of sector AOE = (60/360) * π * (1)^2 = (1/6) * π.
Step 2: Find the area of triangle AEB.
The base of triangle AEB is the distance between points A and B, which is the same as the length of the arc EB.
To find the length of the arc, we use the formula s = r * θ, where s is the length of the arc, r is the radius, and θ is the angle in radians.
In this case, the angle θ is 60 degrees which is equal to π/3 radians.
So the length of arc EB = (1) * (π/3) = π/3.
Now, the height of triangle AEB is the distance between points E and the horizontal line OE.
Since the sin of angle x is given as sqrt(3)/2, we can use the fact that sin(x) = opposite/hypotenuse.
In this triangle, the opposite side is the height we are looking for and the hypotenuse is the radius, which is 1 unit.
So the height = sin(x) * radius = (sqrt(3)/2) * 1 = sqrt(3)/2.
Therefore, the area of triangle AEB = (1/2) * base * height = (1/2) * (π/3) * (sqrt(3)/2) = (π/6) * (sqrt(3)/2) = (π * sqrt(3))/12.
Step 3: Subtract the area of sector AOE from the area of triangle AEB.
Area of shaded region = Area of triangle AEB - Area of sector AOE
= (π * sqrt(3))/12 - (1/6) * π
= (π * sqrt(3))/12 - (2π)/12
= (π * sqrt(3) - 2π)/12
= (π(sqrt(3) - 2))/12.
Therefore, the area of the shaded region is (π(sqrt(3) - 2))/12 square units.