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
To find the area of the shaded region, we need to find the area of the sector (the region enclosed by the arc) and subtract the area of the triangle.
First, let's calculate the area of the sector. The formula for the area of a sector is:
Area of sector = (angle / 360 degrees) * π * r^2
In this case, the angle is 60 degrees (as given), and the radius is 1 unit (from the center O to the point A). So we have:
Area of sector = (60 / 360) * π * 1^2
= (1/6) * π
Now, let's calculate the area of the triangle. The base of the triangle is the length of the line segment AB, which is the same as the length of the arc EB. The formula to calculate the length of an arc is:
Arc length = (angle / 360 degrees) * 2 * π * r
In this case, the angle is 60 degrees and the radius is 1 unit, so we have:
Arc length = (60 / 360) * 2 * π * 1
= (1/3) * π
The height of the triangle can be calculated as the difference between the y-coordinate of the point A and the y-coordinate of the point E. The y-coordinate of A is √3/2 (as given in the question), and the y-coordinate of E is 1 (since it lies on the unit circle). So we have:
Height of triangle = √3/2 - 1
Finally, we can calculate the area of the triangle using the formula:
Area of triangle = (1/2) * base * height
Substituting the values we've calculated, we have:
Area of triangle = (1/2) * (1/3) * π * (√3/2 - 1)
Now, to find the area of the shaded region, we subtract the area of the triangle from the area of the sector:
Area of shaded region = Area of sector - Area of triangle
= (1/6) * π - (1/2) * (1/3) * π * (√3/2 - 1)