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I think it would be simpler if trigonometric ratios are defined as the ratios of sides on any general traingle. Let's take any traingle two of whose angles are $x$ and $y$ so that the third angle automatically gets fixed. The side containing $x$ and $y$ will act as base. The side along the other arm of angle $x$ touching the base will act as hypotenuse and the remaining side, i.e. the other arm of angle $y$ will be the opposite side. Now that we have tgree well-defined sides, we can define $sin$ of angle $x$ with respect to reference angle $y$ as $sin_yx$= $\frac{opposite side}{hypotenuse}$. Similar is the case with other trigonometric ratios. I've also got expressions of $sin_yx$ in terms of $siny$ and $sinx$, i.e in terms of $90^•$ reference angle. $$sin_yx=\frac{sinx}{siny}$$ Similarly, $$cos_yx=cosx+cotysinx$$

The identities turn out to be:

$$sin^2_yx+cos^2_yx-2sin_yxcos_yxcosy=1$$

So, why isn't trigonometry like this? It would have made things a bit simpler if we could work with general traingles and wouldn't have to draw perpendiculars.

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    Functions of one variable are simpler than functions of two. That's why you learn to count before you learn to add, and why you learn one-variable Calculus before multivariable. – 2017-02-03
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    So, the trigonometry that I'm talking about, is it in use and is it studied in multivariable calculus? – 2017-02-03
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    Not that I know of, no. – 2017-02-03
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    It might be a high price to pay to avoid drawing perpendiculars. Your proposal doubles the number of trigonometric functions, making it more than twice as difficult to recognize when you can apply your new *identities* in simplification. – 2017-02-03
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    arbitrary triangles are covered by the [Law of Sines](https://en.wikipedia.org/wiki/Law_of_sines) and the [Law of Cosines](https://en.wikipedia.org/wiki/Law_of_cosines). You haven't to draw any perpendiculars. – 2017-02-09
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    you should write $\sin x$ (\sin x) instead of $sinx$ (sinx) – 2017-02-09

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Trigonometric functions defined on general triangles would be a terrible thing.

They would depend on two angles, and one would have to distinguish between the left- and right-side functions. The trigonometric identities, such as the addition formulas, would become very unhandy, and numeric tables weigh tons.

You couldn't reason on the trigonometric circle anymore but on an ugly surface (possibly an ellipsoid).

That would just have killed trigonometry (or made it accessible to specialized professionals only), and slowed down some scientific progress by centuries.

You can get a taste of that by looking at the elliptical funtions of Jacobi (https://en.wikipedia.org/wiki/Jacobi_elliptic_functions#Minor_functions).

On the opposite, mathematicians found it very convenient that the general triangles can be solved by reduction to problems on right angled triangles.

Even better, spherical triangles, one level of complexity higher (https://en.wikipedia.org/wiki/Spherical_trigonometry), can also be addressed this way.

Mathematics are about generalizing by simplifying, not the converse.

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    Yeah, but suppose we've two sides and all angles of any traingle given. Then, wouldn't it be easier to be able to directly use trig ratios of one angle with respect to other instead of drawing perpendiculars? – 2017-02-03
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    @dove: my answer is unambiguous: no. – 2017-02-03