4.9 Trigonometric functions

As trigonometric functions we name the functions sine ($\sin$), cosine ($\cos$), tangent ($\tg$) and cotangent ($\ctg$). Furthermore, in this chapter we will mention theirs appropriately chosen inverse functions, which are the functions arcsine ($\arcsin$), arccosine ($\arccos$) and arctangent ($\arctg$).

The functions sine and cosine are defined by using the following geometric construction or algorithm. The input is the angle $alpha$ and the output is given by $\sin(\alpha)$ and $\cos(\alpha)$. While reading the algorithm, it is advisable to look at the picture 4.10.

1. Consider an orthogonal coordinate system with coordinate axis $x$ and $y$ and construct a unit circle $K$ (i.e. a unit circle with radius $1$ in the given axes units) and center point at the origin $(0,0)$.

2. Clockwise from the positive direction of the $x$ axes we measure the angle⁴⁴ $\alpha$. One side of this angle is the positive $x$ axis and we denote the other side by $p$.

3. Let $A$ be the point at the intersection of $p$ and $K$. Then we construct the point $P$ as the intersection with the $y$ axis of the line passing through $A$ and parallel to $x$. In this way, we obtain the rectangular triangle $OPA$.

4. The length of the (oriented) side $OP$ represents $\cos(\alpha)$ and the other (oriented) side $PA$ represents $\sin(\alpha)$.

Of course, the accuracy of the result depends on the accuracy of our drawing tools. Infinite accuracy can only be achieved with infinitely accurate tools (here ruler, compass and protractor). Obviously, this „calculation method“ is not very practical. In the course  BIE-ZMA we will show how to effectively evaluate functional values of (not only) these functions.

The basic values of the sine and cosine functions are summarized in the following table and you can remind yourself of their graphs in figure 4.11.

From the construction of sine and cosine, it follows immediately the following equality

This equation is a consequence of the Pitagora's theorem applied to the triangle $OPA$ with hypotenuse of lenght $1$ and sides of lenght $\sin(\alpha)$ and $\cos(\alpha)$ (see the construction above and picture 4.10). Furthermore, it is evident from the construction that the sine function is odd and the cosine function is even, i.e.

For the domain of these function we have

Their image set is

To conclude, both functions are periodic with period $2\pi$, both functions are well defined on $\mathbb{R}$ and for every $x \in \mathbb{R}$ the equality $\sin(x+2\pi)=\sin(x)$ and $\cos(x+2\pi)=\cos(x)$ holds.

Very useful are the so-called addition formulas for the functions sine and cosine: for any real $\alpha$ and $\beta$, it holds

and

These formulas could be most easily derived using the property of multiplication of complex numbers by their goniometric expression.

By using the fact that sine is and odd function and cosine is an even function, from formulas (4.13) a (4.14) we immediately get analogous formulas for the difference of angles:

Similar formulas can be derived for both the tangent and cotangent functions. The meaning of these formulas and their use is obvious: if we have information about the values of $\sin \alpha$ and $\cos \beta$, then they allow us to get information for example on the value of $\sin(\alpha + \beta)$.

From formulas (4.13) and (4.14), we obtain the double-angle formulas, which are very often used:

and

By using the relationships (4.12) and (4.15), we immediately derive formulas for the sine and cosine of half angle,

If we want to get rid of absolute values in these formulas, we have to decide the sign of expressions based on the angle $\alpha$, more precisely to which of the four quadrants in the Cartesian plane it belongs.

Through the functions $\sin$ and $\cos$ we define the functions tangent $\tg$ and cotangent $\ctg$ as

Their image set is the whole $\R$. For the sake of clarity, we present their graphs in picture 4.11.

Neither of the previously introduced trigonometric functions is injective on its domain. If you choose any $y$ in the image set of $\sin$, there exists infinite $x$ in the domain of $\sin$ such that $\sin(x) = y$ (see picture 4.11). Therefore, for any given $y\in H_{\sin}$, it cannot be unambiguously specified $x\in D_{\sin}$ satisfying $y = \sin(x)$. The same observation applies to $\cos$, $\tg$ and $\ctg$. Trigonometric functions are not injective on their natural domain and therefore can not have inverse functions… unless we restrict them appropriately, that is, we reduce their domain. In accordance with the established convention, we define

• arcsine, $\arcsin$, as the inverse function of $\sin$ restricted on the interval $[-\frac{\pi}{2}, \frac{\pi}{2}]$,

• arccosine, $\arccos$, as the inverse function of $\cos$ restricted on the interval $[0,\pi]$,

• arctangent, $\arctg$, as the inverse function of $\tg$ restricted on the interval $(-\frac{\pi}{2}, \frac{\pi}{2})$,

• arccotangent, $\arcctg$, as the inverse function of $\ctg$ restricted on the interval $(0,\pi)$.