Why is cos 0 equal to 1

Cosine function and its properties

In this learning text you will get an overview of the properties the Cosine function. We'll also tell you how to use the Cosine curve can move in the x or y direction.

General functional equation

The cosine function is one of the trigonometric functions and assigns each $ x $ its corresponding cosine value $ y $.

You can see a unit circle. It is so called because the length of its radius is ‘1.

The cosine function assigns a segment length to each angle. The length of the line drawn in blue belongs to the angle $ x $. For example, if $ x $ is given with $ 60 ° $, then the length of the blue line is $ 0.5 $. Therefore cos⁡ $ 60 ° = 0.5 $. Each angle has a length of the circular arc. It is shown here in purple as a bow. The length of this arc is also called the radian measure of the angle $ x $. If the radius is 1, then the circumference of the entire circle is $ U = \ pi \ cdot d = \ pi \ cdot 2r = \ pi \ cdot 2 \ cdot 1 = 2 \ pi $. So the entire circle has an arc length of $ 2π $. That's about $ 6.28 units (for example cm). So for the angle $ 360 ° $ the radian measure $ 2π $ belongs. Correspondingly, the $ \ frac {2 \ pi} {6} = \ frac {\ pi} {3} $ radian dimension belongs to the degree $ 60 ° $.

Click here to expand

$ y ~ = cos (x) $

The cosine function with the x-axis in radians.

The cosine function, just like the sine function, has some special features. The radian measure is usually used to scale the axis. It is important at this point whether the calculator should calculate with degrees or radians. This must be taken into account in the settings. As a rule, the calculator has the settings RAD (for radians) and DEG (for degrees).

Set of definitions and values ​​of the cosine function

All real numbers are allowed for the x-values ​​of the cosine function. The definition set is thus:

$ \ mathbb {D} = \ mathbb {R} $

As you can see in the figure, the y-values ​​can only assume values ​​from $ -1 $ to $ 1 $. The range of values ​​of the normal cosine function is thus:

$ W = [-1; 1] $

Period and symmetry behavior of the cosine curve

The cosine curve runs periodically, which means that a single section repeats itself over and over again. One can also say that the function values ​​$ y $ are repeated at the same distance. A wave movement above and below the x-axis corresponds to a smallest period of $ 2 \ pi $.

In addition, the cosine function is axially symmetrical to the y-axis. This can be proven mathematically:

$ cos (-x) = cos (x) $