What is the meaning of the value of tangent?

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The unit circle gif came around again and I can wrap my head around sin and cos as measures of the x and y components of the unit circle. but the tangent doesn’t seem to be a measurement of anything and I have never really understood that.

TIA

In: Mathematics
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It is the [length of the tangent](https://www.onlinemathlearning.com/unit-circle.html) to the unit circle.

That’s how it got its name, but in actuality it is just useful to have a third ratio for a right triangle which does not include the hypotenuse. Often you’ll have two of three.

`tan(x) = sin(x) / cos(x)`, but when you expand out the definitions of sin and cos with the same right angle triangle, you have `(opposite / hypotenuse) / (adjacent / hypotenuse)` or simplified, `opposite / adjacent` , which is the ratio of the lengths of the two non-diagonal lines in a right-angle triangle. Take a step back and look at that… the tangent function basically converts between angles (degrees) and slope (the mathematical graph concept, and what a derivative calculates).

Examples:

`tan(0 degrees) = 0` which is correct for a flat line

`tan(45 degrees) = 1`which is correct for a straight diagonal line moving upwards.

`tan(90 degrees)` does not exist, which is correct because a vertical line has no slope.

etc.

tan^-1 (2) = 63.4 degrees which, when drawn, should look right for a slope of 2.

The tangent (in trigonometry) is mathematically equivalent to the slope of the hypoteneuse.