What is x if pi cos x

Sine- and cosine function under the microscope

You have been working with functions for a long time: definition range, zeros, function values, ... and also Sine- and you already know cosine functions in the unit circle and in the right triangle.

Now you will learn more about the domain and zeros of Sine and cosine. :-) Because the functions are periodic, things look a little different here.

Here they come Sine- and the cosine function with the angle sizes on the x-axis:




You can imagine the angle sizes well, but radians are more practical for calculating and examining the function. It will look like that:




You can easily read off the definition range and range of values.
You can substitute all numbers for x, i.e. $$ D = RR $$.
The y-values ​​lie between $$ - 1 $$ and $$ 1 $$, i.e.
$$ W = {y in RR $$ and $$ - 1 le y le 1} $$.

The division with $$ pi $$ is certainly unfamiliar at first. But later it will become a matter of course for you. Always have in mind: $$ pi $$ corresponds to $$ 180 ^ ° $$.

zeropoint

Sinefunction

Up to now, zeros have always been very clear: a function either had no zeros at all or one or two.

And here? Are there an infinite number of zeros!
The function is periodic and continues infinitely to the left and right.




You can read off the zeros here:

$$ x_1 = -2pi $$
$$ x_2 = -pi $$
$$ x_3 = 0 $$
$$ x_4 = pi $$
$$ x_5 = 2pi $$
$$ x_6 = 3pi $$

How can you do that for all zeros of the Sinegeneralize function?

In words: all multiples of $$ pi $$
As a formula: $$ k * pi $$ with $$ k in ZZ $$

That means: $$ sin (k * pi) = 0 $$ for $$ k in ZZ $$

And the cosine function?

It works like this:




Read from:
$$ x_1 = -3 / 2pi $$ $$ x_2 = -pi / 2 $$ $$ x_3 = pi / 2 $$
$$ x_4 = 3 / 2pi $$ $$ x_5 = 5 / 2pi $$


Generally:
In words: add multiples of $$ pi $$ to $$ pi / 2 $$
As a formula: $$ pi / 2 + k * pi $$ with $$ k in ZZ $$

That means: $$ cos (pi / 2 + k * pi) = 0 $$ for $$ k in ZZ $$

A zero is a place $$ x $$ at which the function $$ f $$ has the $$ y $$ value $$ 0 $$. $$ f (x) = 0 $$ applies. The graph intersects the x-axis at the zero point.

$$ ZZ $$ are the whole numbers: $$ {…; -2; -1; 0; 1; 2;…} $$

Highs and lows

There has been a high or low point, if at all, in the functions that you have come across so far.
At the high point the function assumes the largest function value and at the low point the smallest. *

In the Sinefunction there are infinitely many high points. The largest function value is 1.
There are an infinite number of low points, the smallest function value is -1.




The high points have the coordinates
$$ (pi / 2 + 2pi * k | 1) $$ for $$ k in ZZ $$.
The low points have the coordinates
$$ (- pi / 2 + 2pi * k | -1) $$ for $$ k in ZZ $$.

Continue with cosine




The high points have the coordinates $$ (2pi * k | 1) $$ for $$ k in ZZ $$.
The low points have the coordinates $$ (pi + 2pi * k | -1) $$ for $$ k in ZZ $$.

* If you want to know exactly:

Mathematically, that's not entirely correct. There are functions (that you don't know yet) whose function graphs have high and low points (these hills or valleys in the graph) and also have infinitely large or small function values. (Exciting, huh? Take a look at $$ f (x) = x ^ 3 + 3x ^ 2-2 $$.)
It should be quite correct here: At the high point, the function takes on a specific environment the largest function value at and the smallest at the bottom.

As a reminder, 2 parables:

The high point is here (-3.25 | 2) and the low point (3.5 | 0.5)

Maxima are the highest points of the curves, so the "mountain peaks".
Minima are the lowest points of the curves, i.e. the bottom of the valley.

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Symmetry at Sine

The Sinefunction is point-symmetrical to the coordinate origin. Imagine flipping the right arm of the graph around (0 | 0).




For the function values, the point symmetry means:




In words: $$ sin (-x) $$ is $$ sin x $$ with the opposite sign.
As a formula: $$ sin (-x) = - sin x $$


Example:
$$ sin (pi / 4) = 0.71 $$
$$ sin (-pi / 4) = - 0.71 $$

General symmetry:
Axial symmetry: $$ f (x) = f (-x) $$
Point symmetry: $$ f (-x) = - f (x) $$

Symmetry at the cosine

The cosine function is axially symmetric.




For the function values, the axis symmetry means:




In words: An x-value and the negative x-value have the same cosine value.
As a formula: $$ cos (-x) = cos x $$


Example:
$$ cos (3 / 8pi) = 0.38 $$
$$ cos (-3 / 8pi) = 0.38 $$

General symmetry:
Axial symmetry: $$ f (x) = f (-x) $$
Point symmetry: $$ f (-x) = - f (x) $$

Sine- and cosine function in a nutshell

Sine cosine
y values -1 to +1 -1 to +1
Period length 2 $$ pi $$ or 360 ° 2 $$ pi $$ or 360 °
Position of the high points $$ pi / 2 $$, $$ (5pi) / 2, ... $$ $$ 0 $$, $$ 2pi $$, $$ 4pi, ... $$
Position of the low points $$ (3pi) / 2 $$, $$ (7pi) / 2, ... $$ $$ pi $$, $$ 3pi $$, ...
symmetry point symmetric to (0 | 0) axially symmetrical to the y-axis

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