Trigonometric ratios of a triangle

Question

Why is $\sin x = \frac ph$? What are $\cos$, $\sin$ and $\csc$? Is there a derivation? I don't get trigonometric ratios. Isn't there any derivation?

Answer 1

There are three sides to a triangle. If we call then $a,b$, and $c$, then we can form $6$ ratios using those three numbers. (I suggest that you verify this by listing all possible such ratios.)

That is why there are $6$ trigonometric ratios.

The names sine, secant, and tangent come from pictures of certain geometric figures.

The sine of the complement of an angle $A$ is an important ratio. I am told that at one time it was called "sine complement A" which got shortened to "sin co A" and later the "co" migrated to the beginning of the name to become "cosine A". (I am not a specialist in history, so I may be passing along mis-information that I was taught by non-specialists.)

One usefulness of trigonometric ratios arises because corresponding sides of similar triangles have proportional lengths.

So if I know that in a right triangle, the ratio of the side opposite an angle that measures $30^{\circ}$ to the hypotenuse is $\frac{1}{2}$, then for a similar right triangle with hypotenuse $20 cm$ we can find the length of the side opposite the $30^{\circ}$ angle.

Answer 2

The short answer:

We use these functions because after many years of trying to do trigonometry in different ways, these are the definitions that people found convenient.

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The long answer:

For most of the history of (Western) trigonometry, people did not use sine, cosine, tangent, or the other functions that we learn today. Instead, they had tables of chords of angles. The chord of an angle $\alpha$ is the straight-line distance between the two endpoints of an arc of angle $\alpha$. That is, $$ \mathrm{crd\ } \alpha = 2r \sin\frac\alpha2, $$ where $r$ is the radius of your reference circle.

This was the way trigonometry apparently was done for many hundreds of years. In the 2nd century CE, the famous reference work, the Almagest by Ptolemy, had a table of chords of angles from $\frac12$ degree to $180$ degrees in $\frac12$-degree increments. But no tables of sines, cosines, or tangents. (And of course there were no calculators that could compute trig functions for you--the only practical way to use trig functions until a few decades ago was to read their values from tables.)

In India, the story was a little different; Aryabhata computed tables of the sine function and the versed sine or versine in the 6th century CE.

Eventually some clever people realized that just having tables of chords of angles (or even sines and versines) was not always the most convenient way to solve a trigonometry problem, and they started to come up with tables of new functions that were more convenient to use, depending on what problem it was that you needed to solve. They came up with a lot of functions that you will not see in a typical trigonometry textbook.

These new functions also were defined in terms of measurements on a circle, in particular a circle of radius $1$ called the unit circle. (For some reason, many educators seem to like to start with the definition you have seen, $\sin \alpha = \frac ph,$ although it is only useful for angles less than a right angle; the circle-based definition is more general.) But the idea that the circle should be a unit circle is apparently a relatively modern one; Ptolemy's circle had radius $r = 60$.

Eventually the published tables of trigonometric functions settled (mostly) on the sine, cosine, and tangent. Apparently those were the most-demanded tables, or at least the publishers of tables thought they were.

So in the end it all comes down to (perceived) convenience and usefulness.