What Are the Inner and Outer Radii of y=arccos(x)?

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In summary, the outer radius is the distance from the center of a circle or sphere to the outer edge, while the inner radius is the distance from the center to the inner edge. They are directly related, with the outer radius increasing as the inner radius decreases. They can also be the same, resulting in a solid shape with no empty space inside. The formula for calculating them depends on the shape, and the unit of measurement can vary.
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cmab
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What would the inner and outter radius of y=arccos(x)
 
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cmab said:
What would the inner and outter radius of y=arccos(x)

You need to state the whole problem. I assume this function is one part of specifying a volume of rotation. How are the boundaries defined and where is the axis of rotation?
 
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be

The inner and outer radius of y=arccos(x) would depend on the specific context in which the equation is being used. If we are considering the graph of this equation in the Cartesian coordinate system, then the inner radius would be 0 and the outer radius would be 1. This is because the range of the arccosine function is limited to the interval [0, π], and the graph of this function forms a semi-circle with a radius of 1 centered at the origin.

However, if we are looking at this equation in the context of polar coordinates, then the inner and outer radius would be different. In polar coordinates, the equation y=arccos(x) can be written as r=arccos(θ), where r is the distance from the origin and θ is the angle measured from the positive x-axis. In this case, the inner radius would be 0, as the point (0,0) is the origin. The outer radius would depend on the value of θ. As θ approaches π/2, the outer radius would approach 1, and as θ approaches 0, the outer radius would approach infinity.

Overall, the inner and outer radius of y=arccos(x) can vary depending on the context in which it is being used, but in general, the inner radius would be 0 and the outer radius would be 1.
 

Related to What Are the Inner and Outer Radii of y=arccos(x)?

What is the difference between outer and inner radius?

The outer radius is the distance from the center of a circle or sphere to the outer edge, while the inner radius is the distance from the center to the inner edge. Essentially, the outer radius is the total radius, while the inner radius is the radius of the empty space inside.

How are outer and inner radius related?

The outer and inner radius are directly related to each other. As the outer radius increases, the inner radius decreases, and vice versa. This is because the total radius remains constant.

Can the outer and inner radius be the same?

Yes, the outer and inner radius can be the same. In this case, the shape would be a solid sphere or cylinder with no empty space inside.

What is the formula for calculating outer and inner radius?

The formula for calculating outer and inner radius depends on the shape. For a circle, the outer radius is the same as the radius, while the inner radius is 0. For a cylinder, the outer radius is the radius of the circular base, while the inner radius is the distance from the center of the cylinder to its inner edge. For a sphere, the outer radius is the radius of the sphere, while the inner radius is 0.

What is the unit of measurement for outer and inner radius?

The unit of measurement for outer and inner radius can vary depending on the context. For example, the radius of a circle may be measured in inches, while the radius of a planet may be measured in kilometers.

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