Trig functions for any value of the argument

Our definition of trig functions such as sin($\theta)$ and cos($\theta$) comes from looking at right triangles where $\theta$ has to be less than 90^o (π/2 rad). But once we have defined the set of trig functions, we are free to use them for any value we want. Many important physical systems that have properties that are well modeled by trig functions so it's useful to consider what happens for all values of $\theta$, even ones larger than $2\pi$ rad.

We'll begin with the math and consider sines and cosines in the unit circle. Then, we can consider an object moving in a circle where its angle is proportional to time. 

Trig functions on the unit circle

The figure at the left represents a unit circle (a circle with radius = 1). The red dot shows a point with coordinate (x,y). The radius of the circle is 1 and the angle the line from the origin to (x,y) is up from the x-axis by an angle $\theta$. The length of the x-coordinate is shown in blue and that of the y-coordinate in green.

From the triangle and our knowledge of trigonometry, we can conclude that

$$\sin{\theta} = y$$
$$\cos{\theta} = x$$

[How did we get away with taking a length to be "1" without any units? If we took the radius to be $R$, the x and y components would then be multiplied by $R$ so when we constructed sines and cosines, which are ratios, the $R$ would always cancel out. Or, we could pretend we were just doing math.]

We can now let $\theta$ grow from 0 all the way up to $2\pi$ (the full circle). Our original definition of sine and cosine were defined from a triangle. They had only lengths in them so they were always positive. If we now think of sine and cosine as being defined by coordinates instead of lengths, then the sine becomes negative when the point goes below the x axis and the cosine becomes negative when the point is on the left of the y axis.

The result is shown in the graph below as a function of $\theta$ expressed in radians. 

This looks reasonable. Sine goes to zero when the angle is 180^o ($\pi$ radians) and cosine goes to zero at 90^o ($\pi$/2 radians) and 270^o (3$\pi$/2 radians). A look at the unit circle will confirm that this is correct.

Trig functions for arbitrary angle

Using the unit circle, we can now define trig functions for any arbitrary angle. Since adding $2\pi$ radians just brings us back to 0 on the circle, we can just start again with the same values of sine and cosine. So 

$$\sin{\theta} = \sin{(\theta + 2\pi)} =\sin{(\theta + 4\pi)} = \sin{(\theta + 6\pi)} = ...$$

$$\cos{\theta} = \cos{(\theta + 2\pi)} =\cos{(\theta + 4\pi)} = \cos{(\theta + 6\pi)} = ...$$

for any angle, no matter how large, we can keep subtracting 2π until we get down to an angle between 0 and π/2 where we can relate it directly to a triangle.

The figure below shows what the graph of sine and cosine looks like when we have wound round the unit circle almost two full times.

We can, of course, go out as far as we want in either direction; to any arbitrarily large positive or negative angle.

But why should I want to do this?

Do we really care? Why should we bother to define sines and cosines for arbitrarily large angles? It turns out that these are amazingly useful functions for physical applications.

The first and most obvious is circular motion. If I have an object going around in a circle of radius $R$, then its x and y coordinates are $x= R \cos(θ)$ and $y = R \sin(θ)$. If it's going around the circle at a uniform rate, then the angle increase at a uniform rate, so

$$θ = ω_0t$$

where $t$ is the time and $ω_0$ is a constant with units of radians/sec. We now have the position of the object (be it a planet or a centrifuging test tube) as functions of time:

$$x(t)= R \cos(ω_0t)$$

$$y(t) = R \sin(ω_0t)$$

We can do lots of useful things with this form, like differentiating them to find velocity and acceleration (and therefore to infer forces).

But what if I don't care about circular motion? Can I then forget this stuff? Afraid not! These kinds of patterns occur everywhere in physics that there is a vibration and it is crucial in the study of wave physics including sound, light, and quantum mechanics. Even if you don't look at oscillations, these functions allow you to make up any signal by a mathematical technique known as Fourier analysis. This is what underlies anything that's called spectroscopy, a powerful technique that lets us identify the component elements in stars and the polluting chemicals emitted in a smoke plume.

The key is to remember the principle that "if the equation is the same, the solution is the same." An important property of the sine and cosine functions is that their second derivatives are proportional to themselves (with a minus sign). The function whose first derivative is proportional to itself is the exponential and that appears in lots of examples in physics. Since we are often dealing with second derivatives (Newton's 2nd law is about acceleration, a second derivative), functions whose second derivative is proportional to themselves will behave like sines and cosines. (See, the derivatives of trig functions.) 

Exponentials and sines and cosines are closely related. You can see this from looking at their power series. This becomes a powerful mathematical tool in advanced physics and engineering when you add complex numbers to the mix.)

Joe Redish 3/21/18