Why is tan not adj/opp?

Question

What makes tan opp/adj? Is it a simple choice?

I don't see why it cannot be the other way. Thanks in advance for your help.

Answer 1

That is how the $tangent$ function is defined.

There are functions that define ratios between different sides.

The $cotangent$ function is the ratio between $adj$ and $opp$

There are 4 more functions namely $sine$, $cosine$, $secant$ and $cosecant$ which define the rest of the possible ratios

Answer 2

One reason is that if you fix $\theta$ (appropriately) as the angle off the $x$-axis for some line $y=mx+b$, the tangent of $\theta$ is precisely the slope of the line. I find this a convincing reason to care about tangent.

Otherwise, the original function one sought to define, was the length of a segment tangent to the unit circle circle. see here

Answer 3

Back in the day before calculators, if you had to do lots of trigonometry you could buy trigonometry tables. These were little booklets with precomputed values for the trigonometric functions. The six most common were $$ \sin = \frac{\text{opposite}}{\text{hypotenuse}}\\ \cos = \frac{\text{adjacent}}{\text{hypotenuse}}\\ \tan = \frac{\text{opposite}}{\text{adjacent}}\\ \sec = \frac{\text{hypotenuse}}{\text{adjacent}}\\ \csc = \frac{\text{hypotenuse}}{\text{opposite}}\\ \cot = \frac{\text{adjacent}}{\text{opposite}} $$ You can clearly see that there are relationships between these, but they were often all given in the tables for convenience. This way, you only ever needed to find the correct entry in the table, then multiply. Multiplication is often easier than division, so it was indeed very convenient that all these values were given. Do you have the adjacent and the angle and want the opposite? Multiply by $\tan$. Do you have the opposite and the angle and want the adjacent? Multiply by $\cot$. Simple.

Since then we have gotten calculators capable of calculating the trigonometric functions quickly, and also dividing and multiplying equally quickly (at least for a human pressing buttons). Therefore not all of these were needed, and some of these functions faded into relative disuse. That it happened to be $\sin, \cos, \tan$ which survived and not the other way around is, as far as I know, just random chance.

Answer 4

Think of it using a unit circle.

Tangent comes from the Latin word tangens, the present participle of the verb tangere, which means "to touch." A line $EF$ touching the unit circle once at point $A$ makes a right angle with the radius $OA$. This gives a natural definition of tangent. $$\tan \theta = \frac{EA}{OA} = EA$$ Observe that $EA$ and $OA $ are the opposite and adjacent sides of $\triangle OAE$. This explains the choice of the ratio $$\tan = \frac{\text{opposite}}{\text{adjacent}}.$$