Introduction to the fourth autumn series
The topic of this year’s fourth series is trigonometric functions. To help you solve the problems from this series, here are some basic facts and properties that might come in handy.
First of all, let’s define the basic trig functions.1 We will start with a right-angled triangleABC:
A B
C
α
a
c b
Sidec, opposite to the right angle, is called thehypotenuse, sidesaandbare calledopposite and adjacent (to angleα), respectively. We define the sine, cosine and tangent functions as follows:
sinα=a
c, cosα=b
c, tanα=a b.
Using these definitions and basic geometry, we obtain the following properties:
(i) tanα=cossinαα, (ii) sin2α+ cos2α= 1, (iii) sinα= cos(π/2−α), (iv) cosα= sin(π/2−α).
The above definitions only make sense forα∈(0, π/2), but it’s often useful to define trigono- metric functions for other values ofαas well. For this definition, we use a circle with a radius of 1 centered at the origin (the unit circle). For a given angleα, we draw a line through the origin at an angle ofα, like in the following picture, and we define cosαand sinαas the xandy coordinates of the intersection point of the line with the unit circle:
α cosα
sinα
1Note that in English-written literature, tangent is denoted tan instead of tg. Furthermore, square brackets [0, π] are used for denoting intervals instead of angle bracketsh0, πi.
Note that this defines the sine and cosine function for anyα∈R. We can also define the tangent function as tanα= sinα/cosαwhereα6=π/2 +kπ,k∈Z.
As can be seen from the picture, when αreaches 2π (a full rotation), the values of the three functions start to repeat. Therefore, they areperiodicwith a period of 2π. The tangent function even has a period ofπ.
We can also observe that cosine isstrictly monotone on all intervals in the form [kπ, π+kπ]
and sine is strictly monotone on [π/2 +kπ,3π/2 +kπ] fork∈Z. The tangent function isstrictly increasing on (−π/2 +kπ, π/2 +kπ).
Other properties include that cosα= cos(−α) (cosine is aneven function), sinα=−sin(−α) and tanα=−tan(−α) for any α (sine and tangent areodd). Also, it is worth noting that for α >0, sinα < α.
The best way to get an insight into the various properties is to draw a graph. Here, sine (from
−2πto 2π) is plotted with a solid line, and cosine with a dashed one:
0
−1 1
π 2
π 3π
2
− 2π
π
−π 2
−
3π
−2π 2
And here is a graph of tanxfrom−2πto 2π:
1 2 3 4
−1
−2
−3
−4 0
π 2π
−π
−2π
You might also find useful the following formulas:
sin(α+β) = sinαcosβ+ cosαsinβ, cos(α+β) = cosαcosβ−sinαsinβ.
As mentioned above, sine is monotone on [−π/2, π/2], cosine on [0, π] and tangent on (−π/2, π/2).
Restricting their domains to these intervals, we can define their inverse functions. They are called arcsine, arccosine and arctangent and denoted arcsin, arccos and arctan. So for example arccos(0) = π/2.
