Why are trig functions defined for the unit circle?

Question

Why did we ever need to define the trig functions of angles greater than 90 degrees or less than 0 degrees? What is the use of applying trig functions to such angles?

If we apply the trig functions on a regular right triangle, it makes sense. We can get the ratio of two sides and find out an unknown side if there is a known side (and the other way around).

Let's say that I have a right triangle in which an angle x is 30 degrees and the hypotenuse is 20 cm. I have to find the length of side AY, which is opposite to angle x . Well I can use the function sin(30 degrees), which comes out to be 1/2 . Now 1/2 = AY / 20 . And after solving it we get AY = 10.

Or let's say that I have a right triangle in which I have to find an angle x. The side opposite to x is 10cm and the hypotenuse is 20 cm. Then 10/20 = 1/2. What is the arcsin of 1/2? 30 degrees. Angle x is 30 degrees.

But what use is it to take the sine of an angle 120 degrees of an obtuse triangle? We are not getting a ratio of the sides or anything if we apply it to a non-right triangle.

Answer 1

If you think of the graph of $\sin(x)$ it's a nice periodic function, the graph is a wave. It's very useful in physics (for example) to have functions that model wave behavior. For this you need to allow the angle to go for multiple cycles. Without such functions things like Fourier analysis would be impossible.

Answer 2

Your question actually has a lot of answers.

1. Because we can! Having unrestricted $\sin$ and $\cos$ functions does no harm to people who only need the restricted versions, so why not?

2. As noted by other answers and comments, they do work fine for obtuse, or reversed, triangles, even if it is possible to manage without them. For a thought experiment, you should also note that by your definition, they are also “useless” outside the interval $[0, \pi/4]$ (I am using radians, so $\pi/4 = 45°$), since you can always use the formula $\sin (\pi/2-x) = \cos(x)$ to reduce to this case.

3. The trigonometric functions are not useful only for triangles. For example, imagine a point moving regularly in a circle. The position of the point at time $t$ is given by $(\cos (t), \sin(t))$. Using restricted-trigonometric functions, it would be something with lots of separate cases like $$\def\p{\frac{\pi}{2}}\begin{cases} (\cos t, \sin t)& \text{for $t \in [0, \p[$,}\\ (-\sin (t-\p), \cos(t-\p)) & \text{for $t \in [\p, \pi[$,}\\ (-\cos (t-\pi), -\sin (t-\pi)) & \text{for $t \in [\pi, 3\p[$,}\\ (\sin(t-3\p), -\cos(t-3\p)) & \text{for $t \in [3\p, 2\pi[$,}\\ \dots \end{cases}$$ which is ugly.

4. The trigonometric functions are not useful only for geometry. For example, we have the celebrated Euler formula, which is very very deep: $$ \exp (i\, t) = \cos (t) + i \sin (t). $$ Restricting this one to $[0, \pi/2[$ would be extremely weird.

5. Another analysis example: the trigonometric functions are solutions of differential equations such as $d^2\cos(t)/dt^2 + \cos(t) = 0$ (with initial condition $\cos(0) = 1, \cos'(0) = 0$). As such, they have a natural domain of definition, which is the domain in which the above solution exists, which is the full set $\mathbb R$.

6. Related to 4. (and 3.): these functions are also defined by way of power series, such as $$ \cos t = \sum_{n \geq 0} (-1)^n \frac{t^{2n}}{(2n)!}.$$ These sums are convergent for every $t \in \mathbb R$.

Answer 3

What is the use of applying trig functions to such angles?

Who said that we should apply them only to angles ? For instance, the concept of number started historically with natural numbers $($positive integers$)$, meant for counting. Soon enough, it spread to rationals, which, for the most part, do not exactly “count” anything. Nevertheless, they do measure. So the $($initial$)$ concept was further enriched, by gaining a whole new meaning. The same applies here to trigonometry: although initially intended for purely geometric purposes, its use later extended to completely unrelated areas $($e.g., to signal processing, among countless others$)$. Hope this helps.