A piece of paper in the shape of a sector of a circle of radius 10 cm and of angle 216 degree just covers the lateral surface of a right circular cone of vertical angle 2θ .Then sinθ is
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To find sinθ, we need to use the given information about the shape of the paper and the cone.
Let's start by visualizing the problem. The paper is in the shape of a sector of a circle, meaning it is a portion of the entire circle. The radius of this circle is given as 10 cm.
We are told that this piece of paper covers the lateral surface of a right circular cone. The cone has a vertical angle of 2θ. The vertical angle of a cone is the angle formed between the height (vertical line) and the slant height (diagonal line from the apex to any point on the lateral surface). Let's call the slant height of our cone 'r', which we'll need later.
Now, let's analyze the cone and the sector of the paper. The sector of the paper is curved and wraps around the lateral surface of the cone. The circumference of the circular base of the cone is equal to the arc length of the sector of the paper. The arc length of a sector of a circle with radius 'r' and angle 'θ' is given by the formula:
Arc length = 2πr * (θ/360)
In our case, the arc length is equal to the circumference of the base of the cone. The formula for the circumference of a circle with radius 'r' is:
Circumference = 2πr
So, we can set these two expressions equal to each other:
2πr = 2πr * (θ/360)
Dividing both sides by 2πr, we get:
1 = θ/360
Simplifying further:
θ = 360
Now, we know that the vertical angle of the cone is 2θ. Plugging in the value we found:
Vertical angle = 2(360) = 720 degrees
However, we are asked to find sinθ, where θ is half of the vertical angle. So, we need to calculate sin(720/2):
sin(360) = sinθ
Since the sine function is periodic with a period of 2π, or 360 degrees, sinθ at 360 degrees is the same as sinθ at 0 degrees. Therefore:
sinθ = sin(0) = 0
So, sinθ is equal to 0.