length of sine curve?

Question

Does the phrase "length of the sine curve from 0 to 90 degrees" have any meaning? It is a curved line, so has length, but the axes have no length reference.

Answer 1

Ordinarily the length would be measured in units and the angle would be specified in radians (90° = π/2 radians). The length is given by

.. ∫√(1+cos(x)^2)dx from 0 to π/2



This is an elliptic integral, which does not have a closed-form solution. Numerically, it evaluates to approximately 1.9100988945138560090.

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If you want the x-axis to be degrees, and you want to ignore the units mismatch and integrate from 0 to 90, now you have the length very close to 90. It is approximately 90.006853500529589008. (It is only a little bit longer than the hypotenuse of a right triangle with sides 1 and 90.) The integrand in this case is apparently √(1+(π*cos(x*π/180)/180)^2)dx.





The answer to your question is that it really has no meaning due to the mismatch of degrees on one axis and unitless numbers on the other axis. The meaning would be defined by the context in which the question arises. That context would have to resolve the units issue.



Length as a physical measure has units associated with it. The relationship of the sine curve to the measurement scale would need to be given as part of the problem statement.

Answer 2

Length Of Sine Curve

Answer 3

But, if you are building a bridge, the axes indicate a 2D plane. A sine curve has only degrees (or radians). I figured that you would use calculus. Using a string, you would get that the string is X units in length, but what are the units?

Answer 4

Yes. Imagine placing a piece of string along the curve. Take the string and place it on one axis. How long is it? What you do is break the curve up into a very very large number of very short straight lines along the curve and add them up as the lines become infinitely small. This is a standard calculus problem. It has meaning in real life. Suppose you are building a bridge supported by a suspended cable. Given the shape of the cable and the span of the bridge, how much steel is required for the cables?

Answer 5

In my opinion 'the length of sine curve from 0 to 90 degrees' refers to the length of the curve from the point on the curve where the x-axis(that is in degrees) is zero to the point on the curve where the x-axis is 90 degrees.