@Zergling_man

Only for numbers in the range that happens to be used for thermal expansion calculations for some materials near room temperature when the temperature change is small enough

I do not think it works for large numbers

The sine of theta in radians is also close to theta for angles smaller than a certain angle but not for angles larger than a certain angle

@shortstories That means you're looking at a coincidence, not a relationship.
Volume and length are actually related.

@Zergling_man

I would say that for small angles in radians the sine of theta being approximately equal to theta is because of a relationship

When an angle is small enough

Where the angle measures both a section of a circle and also is an angle within a triangle

When the largest side of a triangle is the same as the radius of the circle

the perimeter of a section of a circle having an angle of theta

is close to the length of a side in a right angle triangle having an angle of theta

@shortstories >I would say that for small angles in radians the sine of theta being approximately equal to theta is because of a relationship
Do you know what the sine function actually does?

@shortstories Or, better question, do you know what the cosine function does?

@Zergling_man

Small angle approximation formulas for trigononetry

https://en.wikipedia.org/wiki/Small-angle_approximation

@Zergling_man

If you have a ramp on the ground at an angle measured relative to the ground where the angle is zero if the ramp is parellel to the ground

The tangent converts the angle to a slope

The sign gives you the height of the ramp divided by it's diagnol path

The cosine gives you the horizontal distance divided by it's diagnol path

You can also do ratios of side lengths given dimensions of triangles using the names opposite, adjacent and hypoteneous but this is harder to visualize

@Zergling_man

The volume is not three times the length

the change in volume is approximately three times the change in length

If the change in length is less than 10% the original length

@shortstories Somewhere around 1.75 the result of multiplying by three and cubing pass each other. I guess that doesn't hold when you're starting from non-zero but... ... If anything, it shrinks, and should shrink towards 0? So I don't think that's true for all lengths.

@Zergling_man

If the increase in length is less than 10% and greater than 0% it approximately works

It should not work for a decrease in length

@Zergling_man

I would not know how to use the method to calculate cubes for numbers above 4/3 because 3*4/3=2

And I am looking at the digits after 1 when you cube a number above 1 written as a decimal

@shortstories >3*4/3=2
How?

@Zergling_man

1.3333333333333333^3

is approxinstely equal to

1.9999999999999999^3

according to the drecribed estimation technique

(4/3)^3 is approximately equal to 2

3"((4/3)-1)+1 = 2

You are multiplying the digits after 1 by 3 and keeping the ones digit the same for numbers greater than or equal to 1 and less than or equal to 4/3

1 of 2

@Zergling_man

You can do the same thing for bigger numbers by sutracting 1 then mu!tiplying the number by 3 then adding 1 in that order

(1.75-1)*3+1 =3.25

1.75^3=5.359 rounded

2 of 2