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Why should we use six trigonometric ratios when all the three sides of a right angled triangle (opposite, adjacent and hypotenuse) are connected with the three rations sine, cosine and tangent? what is the significance of inverses of these ratios(cosec, sec and cot)?

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  • $\begingroup$ The "co" functions correspond to the trig function related to the coangle (if $\theta$, $\theta^{\prime}$ are the two (non-right) angles of the right triangle, $\theta^{\prime}$ is the co-angle of $\theta$). $\endgroup$ Nov 17 '20 at 7:27
  • $\begingroup$ Even if you don't use the $\tan(x/2)$ trick in the answer below you can relate all three sides with just two ratios. The third ratio can just be expressed in terms of the other two, for example $\tan(\theta) = \sin(\theta)/\cos(\theta).$ But what makes any three (or two) of the six possible ratios better than all the other three (or four)? $\endgroup$
    – David K
    Aug 21 '21 at 3:14
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Short answer: there are three sides (adjacent $A$, hypotenuse $H$ and opposite $O$) we can choose in three unordered pairs or six ordered pairs, giving ratios such as $\sec x=\frac{H}{A}$ etc.

Long answer: You could argue only one trigonometric function is needed, $t:=\tan\frac{x}{2}$ so$$\sin x=\frac{2t}{1+t^2},\,\cos x=\frac{1-t^2}{1+t^2},\,\tan x=\frac{2t}{1-t^2}.$$Or you could argue it's convenient to name not only the above functions and their reciprocals, but also other functions such as $\operatorname{versin}x$ etc. (That link shows geometric significances of all these functions, albeit from the perspective of a circle in which we embed a right-angled triangle of hypotenuse $1$.) History has settled on a compromise where a few functions are widely memorised while others aren't (e.g. a mathematician might not know off the top of their head that $\operatorname{exsec}x:=\sec x-1=\frac{2t^2}{1-t^2}$). But history turned out that way because of how often each function is useful.

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It is not particularly helpful to have $6$ trigonometric functions when you only consider them in the context of right-angled triangles. If you simply want to compute the lengths of the sides of a triangle, and the angles in between them, then $\sin$, $\cos$, and $\tan$ are all you need. However, the reciprocal functions have nice geometric interpretations, one of which is given here:

Geometric interpretations of trigonometric functions

The 'co-functions' have another interpretation. $\cos \theta$ is sine of the co-angle of $\theta$. So, for example, in a right-angled triangle with angles $\{60,30,90\}$, $\cos 30 = \sin 60$, since $60$ is the co-angle of $30$ (the two angles have to add up to 90).

(Incidentally, this is how the function tangent got its name. It is the length of the line segment that is tangent to a point $P$, with the end points of the line segment being the $x$-axis and $P$.)

That being said, we could do just fine without the reciprocal functions. 3Blue1Brown calls them tattoos on math.

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  • $\begingroup$ Nice graphic, +1 $\endgroup$
    – rogerl
    Aug 21 '21 at 1:26
  • $\begingroup$ @rogerl: Thank you! $\endgroup$
    – Joe
    Aug 21 '21 at 18:17
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Sine, cosine, and tangent are used most often. But sometimes the other three are more convenient or more appropriate. They also show up as solutions to differential equations, physics problems, statistics problems, astronomy problems, economics problems, etc.

Bye the way. There used to be more than 6 trig functions.

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    $\begingroup$ I would argue that there still are more than six trig functions. Some are just better known than others. People still refer to the haversine formula, for example. $\endgroup$
    – David K
    Aug 21 '21 at 3:11

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