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The definitions have existed for a long time and basically the reason we write $\tan(x) =\frac{\sin(x)}{\cos(x)}$ or $\sec(x) = \frac{1}{\cos(x)}$ etc. is because in those days people looked up trig values from a table, not using calculators. So it is easier to look up say $\sec(x)$ values than calculate $\frac{1}{\cos(x)}$ in order to get the same answer. With time and usage these terms stuck and have been inducted as part of the family.

I'll leave some links to videos which explain it better, one is from one of my favorite channels 3Blue1Brown ([Tattoos On Math][1]) and the other is from an amazing guy called Simon Clark. ([Why $\sin$ and $\cos$ don't mean anything][2]).

Edit: Forgot to mention, to be honest $\sin(x)$ and $\cos(x)$ are the only trigonometric values we need, the rest can be derived. But the world is sometimes a real scary place without $\tan(x)$, $\cot(x)$, $\sec(x)$ and $\operatorname{cosec}(x)$.
[1]: https://www.youtube.com/watch?v=IxNb1WG_Ido [2]: https://www.youtube.com/watch?v=AzVL432lEWA

The definitions have existed for a long time and basically the reason we write $\tan(x) =\frac{\sin(x)}{\cos(x)}$ or $\sec(x) = \frac{1}{\cos(x)}$ etc. is because in those days people looked up trig values from a table, not using calculators. So it is easier to look up say $\sec(x)$ values than calculate $\frac{1}{\cos(x)}$ in order to get the same answer. With time and usage these terms stuck and have been inducted as part of the family.

I'll leave some links to videos which explain it better, one is from one of my favorite channels 3Blue1Brown (Tattoos On Math) and the other is from an amazing guy called Simon Clark. (Why $\sin$ and $\cos$ don't mean anything).

Edit: Forgot to mention, to be honest $\sin(x)$ and $\cos(x)$ are the only trigonometric values we need, the rest can be derived. But the world is sometimes a real scary place without $\tan(x)$, $\cot(x)$, $\sec(x)$ and $\operatorname{cosec}(x)$.

The definitions have existed for a long time and basically the reason we write $\tan(x) =\frac{\sin(x)}{\cos(x)}$ or $\sec(x) = \frac{1}{\cos(x)} \,\,etc.$ is because in those days people looked up trig values from a table, not using calculators. So it is easier to look up say sec(x) values than calculate $\frac{1}{cos(x)}$ in order to get the same answer. With time and usage these terms stuck and have been inducted as part of the family.

I'll leave some links to videos which explain it better, one is from one of my favorite channels 3Blue1Brown ([Tattoos On Math][1]) and the other is from an amazing guy called Simon Clark. ([Why sin and cos don't mean anything][2]).

Edit: Forgot to mention, to be honest $\sin(x)$ and $\cos(x)$ are the only trigonometric values we need, the rest cant be derived. But the world is sometimes a real scary place without tan(x), cot(x), sec(x) and cosec(x).
[1]: https://www.youtube.com/watch?v=IxNb1WG_Ido [2]: https://www.youtube.com/watch?v=AzVL432lEWA

The definitions have existed for a long time and basically the reason we write $\tan(x) =\frac{\sin(x)}{\cos(x)}$ or $\sec(x) = \frac{1}{\cos(x)}$ etc. is because in those days people looked up trig values from a table, not using calculators. So it is easier to look up say $\sec(x)$ values than calculate $\frac{1}{\cos(x)}$ in order to get the same answer. With time and usage these terms stuck and have been inducted as part of the family.

I'll leave some links to videos which explain it better, one is from one of my favorite channels 3Blue1Brown ([Tattoos On Math][1]) and the other is from an amazing guy called Simon Clark. ([Why $\sin$ and $\cos$ don't mean anything][2]).

Edit: Forgot to mention, to be honest $\sin(x)$ and $\cos(x)$ are the only trigonometric values we need, the rest can be derived. But the world is sometimes a real scary place without $\tan(x)$, $\cot(x)$, $\sec(x)$ and $\operatorname{cosec}(x)$.
[1]: https://www.youtube.com/watch?v=IxNb1WG_Ido [2]: https://www.youtube.com/watch?v=AzVL432lEWA

The definitions have existed for a long time and basically the reason we write $\tan(x) =\frac{\sin(x)}{\cos(x)}$ or $\sec(x) = \frac{1}{\cos(x)} \,\,etc..$ is because in those days people looked up trig values from a table  ,not using calculators  . So it is easier to look up say sec(x) values than calculate $\frac{1}{cos(x)}$ in order to get the same answer. With time and usage these terms stuck and have been inducted as part of the family.

I'll leave some links to videos which explain it better  ,one is from one of my favorite channels 3Blue1Brown([Tattoos On Math][1]) and the other is from an amazing guy called Simon Clark.([Why sin and cos don't mean anything][2]  )

Edit  : Forgot to mention  , to be honest $\sin(x)$ and $\cos(x)$ are the only trigonometric values we need  ,the rest cant be derived  .But the world is sometimes a real scary place without tan(x),cot(x), sec(x) and cosec(x).
[1]: https://www.youtube.com/watch?v=IxNb1WG_Ido [2]: https://www.youtube.com/watch?v=AzVL432lEWA

The definitions have existed for a long time and basically the reason we write $\tan(x) =\frac{\sin(x)}{\cos(x)}$ or $\sec(x) = \frac{1}{\cos(x)} \,\,etc.$ is because in those days people looked up trig values from a table, not using calculators. So it is easier to look up say sec(x) values than calculate $\frac{1}{cos(x)}$ in order to get the same answer. With time and usage these terms stuck and have been inducted as part of the family.

I'll leave some links to videos which explain it better, one is from one of my favorite channels 3Blue1Brown ([Tattoos On Math][1]) and the other is from an amazing guy called Simon Clark.  ([Why sin and cos don't mean anything][2]).

Edit: Forgot to mention, to be honest $\sin(x)$ and $\cos(x)$ are the only trigonometric values we need, the rest cant be derived. But the world is sometimes a real scary place without tan(x), cot(x), sec(x) and cosec(x).
[1]: https://www.youtube.com/watch?v=IxNb1WG_Ido [2]: https://www.youtube.com/watch?v=AzVL432lEWA