Use trigonometric substitution and trigonometric identity to determine the circumference of a circle of radius
Solution:
Here,
Radius of the circle (r) = 3
Since, we know, equation of the circle is represented by,
x2 + y2 = r2
Now, Substituting the values in the above equation,
x2 + y2 = 32
y =
Further differentiating the above equation with respect to x,
Now,
The equation of the circumference of the circle is,
Thus, substituting the values
Further,
Substitute
x = 3sinθ
dx = 3cosθdθ
Thus at x = 3
Integrating the above equation we will get
Therefore after interacting we get the circumference of the circle with a radius of 3,
C = 6Ï€
Use Trigonometric Substitution and Trigonometric Identity to determine the Circumference of a Circle of Radius 3.
When any two-dimensional or three-dimensional plane is being intersected and cut then there are curves obtained, which are known as conic sections. In the case of a cone, there are two identical conical shapes, which are also known as capes. The shape of these conical shapes in the cone depends upon the angle that is opting in as a result of the cut between the plane and the cone. Various shapes can be obtained depending upon the difference in the curve sizes which are circle, hyperbola, ellipse, and parabola we can study these shapes in detail, wherein
- The ellipse is one of the conic sections, which is formed as a result of the intersection of a plane with the cone at an angle.
- The circle is considered to be a conic section where the intersection happens parallel to the cone base and it can be considered to be a special type of ellipse.
- Also, hyperbola formation occurs when the plane is parallel to the cone axis and the intersection happens with both the names of the double cone.
- Lastly, we obtain a parabola when the intersecting plane cuts the surface of the cone at an angle.
The following diagram represents the various types of conic sections that are available depending on the difference in the cuts between the intersecting surface and the cone and its necks.
In this article, we will focus on the conical section of the circle. First, the circle has the following properties,