What is the graph of Y 3sin2x

What is the sine function?

......A function is a clear assignment. With the sine function, the ratio of the opposite cathetus to the hypotenuse is assigned to the angle in the right-angled triangle. The ratio is called the sine value or sine for short.
In the formula language this is called [alpha sin (alpha) with sin (alpha) = a / c].
For this first definition, the domain is D = {alpha | 0
Of the six aspect ratios that can be read off the right-angled triangle and that lead to six trigonometric functions, this website is essentially only about the sine.
At the end all functions are presented.

Sine values ​​in a right triangle Top

......Since the leg a of a right triangle is always smaller than the hypotenuse c, the sine values ​​a / c are smaller than 1.

Certain sine values ​​are obtained from special triangles.

......If the angle alpha approaches the zero angle in a right-angled triangle, then a / c approaches 0.
If the angle alpha approaches the right angle, a / c approaches 1.
Therefore, it makes sense to define sin (0 °) = 0 and sin (90 °) = 1.

The calculations can be summarized in a table of values.
alpha
30°
45°
60°
90°
sin (alpha)
0
1/2
(1/2) sqrt (2)
(1/2) sqrt (3)
1

The five function values ​​are easy to remember. The terms are of the same shape.
alpha
30°
45°
60°
90°
sin (alpha)
(1/2) sqrt (0)
(1/2) sqrt (1)
(1/2) sqrt (2)
(1/2) sqrt (3)
(1/2) sqrt (4)

Root terms (irrational numbers) as function values ​​are exceptions. In general, the sine values ​​are transcendent numbers that can be approximated as decimal fractions. These are obtained via convergent series, but then with any precision.
Today it is no problem to get rounded sine values ​​using the pocket calculator. For example, you can determine sin (52 °) with the TI 30 using the key sequence (5) (2) (SIN). The result is sin (52 °) = 0.788010754. However, it is pointless to take all 9 decimals from the calculator. That is too precise, because the angle is only given to the nearest two digits. According to a rule of thumb, two valid digits are sufficient for the sine value, sin (52 °) = 0.79.
It is common practice to round the sine value to four decimal places. This then includes an angular accuracy of about 1 '= 1/60 °.
So it says sin (52 °) = 0.7880.
You can use the pocket calculator to obtain additional values ​​for a table of values ​​and to draw a graph. However, the following graph was created with the Winplot program (URL below).

Cosine values Top
The cosine function is closely related to the sine function.

...The cosine value is the ratio of the adjacent to the hypotenuse.
In the formula language this is called cos (alpha) = b / c.

Then cos (alpha) = sin (beta) = sin (90 ° -alpha).


Since beta = 90 ° -alpha applies, the table of values ​​and graph look like this.

 
alpha
30°
45°
60°
90°
sin (alpha)
1
(1/2) sqrt (3)
(1/2) sqrt (2)
1/2
0

Basically, the cosine function is a modification of the sine function.
This statement will be made clearer below.

Extension of the definition range Top
D = {alpha | 0 <= alpha <= 360 °}
An extension of the definition range can be achieved with the following definition, which must not and does not contradict the considerations above.

......In a Cartesian coordinate system, one specifies a circle with radius 1 and a point P (x | y) on the circular line. If you connect this point with the zero point O, the angle alpha arises with the positive x-axis.

The sine of the angle alpha is given by the y-value of the point P, because in the right-angled triangle OP'P is sin (alpha) = PP '/ 1 = PP'.

The sine here is therefore a distance that is measured in the unit r = 1, insofar as it is also an aspect ratio.

......If the point P moves counterclockwise on the unit circle beyond the angle of 90 °, the definition PP '= sin (alpha) remains.

On the left it is shown how to add the sine values ​​of the angles between 90 ° and 180 ° to the sine values Sharpener Angle alpha returns.

The following applies: sin (180-alpha) = sin (alpha).


......On the left it is shown how to add the sine values ​​of the angles between 180 ° and 270 ° to the sine values Sharpener Angle alpha returns.

The following applies: sin (180 ° + alpha) = - sin (alpha).


......On the left it is shown how to add the sine values ​​of the angles between 270 ° and 360 ° to the sine values Sharpener Angle alpha returns.

The following applies sin (360 ° -alpha) = - sin (alpha)


......The facts are shown again in an animation.
(The radians of an angle are anticipated.)

This explains the sine for all angles between 0 ° and 360 °.

D = {alpha | alpha any}
If point P moves counterclockwise from the x-axis, the sine values ​​are repeated: sin (alpha + 360 °) = sin (alpha)
more general: sin (alpha + n * 360 °) = sin (alpha), n = 1, 2, 3, ...

If the point P moves away from the x-axis in the Clockwise, the angles will be negative. Here, too, the sine value is defined in the same way: sin (-alpha) = - sin (alpha).

This defines the sine for all angles.
A graph shows this situation again clearly.

D = | R.
It is common and also useful to specify the angles of the sine function not in degrees but in radians. Then we write f (x) = sin (x) instead of f (alpha) = sin (alpha).
The domain is D = {x |}.
The specification of the unit 1 radian (1 rad) is unnecessary, as described under Radians can be read on my circle page.

......The radian measure is also useful because it can also be found in the unit circle. The size of an angle is determined by the length of the arc that belongs to the angle. The radius of the circle is used as the unit of length measurement.
The radians determine the shape of the graph. In the case of the degree, it is not clear where, for example, the angle 360 ​​° lies.
For example, stencils use radians to draw the sine curve.


Cosine function for D = | R.

As mentioned above, the cosine function cos (alpha) = sin (90 ° -alpha).
In this context, the expansion of the cosine function is called cos (x) = sin (pi / 2-x) = sin (x + pi / 2) with D = | R. The graph emerges from the graph of the sine by shifting it by pi / 2 in the -x direction.


Determination of the sine values Top
Function values ​​with the calculator
It was already described above how to get sin (52.0 °) = 0.7880 with the pocket calculator.
You have to make sure that this is in degree mode. This can be recognized by the fact that DEG appears in the display. This is the case after switching on.
If you want to determine the sine value of an angle, which is given in radians, you switch on the radian mode RAD with the DRG key. Then, for example, sin (1 rad) = 0.8415.


Functional values ​​with the table
......In the days before the pocket calculator, the function values ​​were available in tables ("logarithm tables").

On the left one recognizes sin (52 °) = 0.7880.

The tables contain sine values ​​for angles accurate to 1 '= 1/60 °.
For this accuracy you need columns of numbers like on the right.
 

Extensive tables containing the logarithms of trigonometric functions were more useful than the excerpt from the table. In the past, with the help of logarithms, multiplication was avoided and replaced by simpler addition, albeit at the expense of accuracy.

Calculation of the sine values
The sine values ​​can be calculated using the Taylor series.
sin (x) = x - (1/3!) x3 + (1/5!) X5 - (1/7!) X7 + ...

To derive the formula
The Taylor series is generally f (x) = f (a) + (1/1!) F '(a) (x-a) + (1/2!) F' '(a) (x-a)2 + (1/3!) F '' '(a) (x-a)3+ ...
Here is f (x) = sin x, f '(x) = cos (x), f' '(x) = - sin (x), f' '' (x) = - cos (x), .. .
This implies f (0) = 0, f '(0) = 1, f' '(0) = 0, f' '' (0) = - 1, f(4)(0) = 0, f(5)(0)=1 ...
Then sin (x) = x - (1/3!) X3 + (1/5!) X5 - (1/7!) X7+ (1/9!) X9 - + ..., wzbw.

Numerical example
The calculator delivers sin (52 °) = sin (0.9076 rad) = 0.7880.
The series gives sin (0.9076 rad) = 0.90757-0.12459 + 0.00513-0.00010 + 0.00000 = 0.7880.
The series is rapidly converging.

Properties of the sine function Top
graph


Definition and value range
Largest possible definition range
Range of values
-1 <= sin (x) <= 1

period
The following applies: sin (alpha + 360 °) = sin (alpha) or sin (x + 2pi) = sin (x).
The sine function is thus a periodic function with the (smallest) period 360 ° or 2pi rad.

amplitude
The amplitude is 1.
Special points
zeropoint
xN= k * pi
Extremes
xE.= (1/2) pi + k * pi
Turning points
xW.= k * pi
The variable k stands for whole numbers.

symmetry
The sine function is point-symmetric with respect to the zero point. The following applies: sin (x) = - sin (-x).

pitch
......How big is the slope of the sine curve at its points?
The incline is accessible at certain points.
You can see that the slope is described by the cosine function.

This is shown by the following calculation.
The following formulas are assumed.
lim [sin (x) / x] = 1 for x towards 0
sin (alpha) -sin (beta) = 2sin (beta / 2) cos (alpha / 2)

The difference quotient is [sin (x + h) -sin (x)] / h = {2sin (h / 2) cos [(2x + h) / 2]} / h = {sin [(h / 2)] / h} cos (x + h / 2)
If h tends towards 0, then sin (h / 2) / (h / 2) tends towards 1 and cos (x + h / 2) towards cos (x).
Result: The difference quotient tends towards cos (x), f '(x) = cos (x).

surface
......What is the area of ​​the area between the sine curve and the x-axis?
Assumption: Compared to the red unit square, it should be around 2 units of area.

solution
Well appreciated.

Curve length
......How long is the segment of the sine curve between two zeros?
Assumption: Compared to the unit distance on the y-axis, it is about 4 units of length.

solution
The approach is
The integral cannot be calculated elementarily as an elliptical integral, but approximately over a series.
Result: The length is s = pi (1 + 1 / 4-3 / 64 + 5 / 256-175 / 16384 + -...) = 3.808 LE, as demonstrated in (2), page 469f.

Some trigonometric formulasTop
Some formulas are used on this page and proven here.
I will limit myself to an examination in the first quadrant.
sin² (alpha) + cos² (alpha) = 1

...According to the Pythagorean theorem, sin² (alpha) + cos² (alpha) = 1.

sin (2alpha) = 2sin (alpha) cos (alpha)
......The angles alpha are drawn in the figure of the unit circle and the figure is completed as on the left.
In the yellow triangle, sin (2alpha) = x + y applies.
......In the right-angled triangle ODC, OD = cos (alpha) applies, in the triangle OGD, sin (alpha) = x / OD applies.
Then x = OD * sin (alpha) = sin (alpha) cos (alpha).
In the right-angled triangle ODC, DC = sin (alpha), in the triangle FDC, cos (alpha) = y / DC.
Then y = DC * cos (alpha) = sin (alpha) cos (alpha).
This means that sin (2alpha) = x + y = 2sin (alpha) cos (alpha).

cos (2alpha) = cos² (alpha) -sin² (alpha)
......The angles alpha are drawn in the figure of the unit circle and the figure is completed as on the left.
In the yellow triangle, cos (2alpha) = u-v applies.

......In the right-angled triangle ODC, OD = cos (alpha), in the triangle OGD, cos (alpha) = u / OD.
Then u = OD * cos (alpha) = cos² (alpha).
In the right-angled triangle ODC, DC = sin (alpha), in the triangle FDC, sin (alpha) = v / DC.
Then v = DC * sin (alpha) = sin² (alpha).
This means that cos (2alpha) = u-v = cos² (alpha) -sin² (alpha).

Source: (3), page 272

General sine function Top
From the sine function comes the general sine function emerge. It has the form f (x) = a * sin (bx + c), where a, b and c are real numbers. It makes sense to require that they not take the value 0.
The question arises as to what effect the parameters a, b and c have, based on the functional equation f (x) = sin (x)?


> If one specifies f (x) = a * sin (x), | a |> 1 causes a stretching, | a | <1 a compression of the sinusoid in the y-axis direction.
This is illustrated by the following four graphs.
......f (x) = sin (x), f (x) = 2sin (x), f (x) = (1/2) sin (x), f (x) = - sin (x) ..

> If one specifies f (x) = sin (bx), then | b | a change in the period 2pi to 2pi / | b |.
This is illustrated by the following four graphs.
.......f (x) = sin (x), f (x) = sin (2x), f (x) = sin [(1/2) x], f (x) = sin (-x) ....

> If one specifies f (x) = sin (x + c), then c causes a shift of the graph by c in the x-axis directions.
This is illustrated by the following four graphs.
......f (x) = sin (x), f (x) = sin (x + 2), f (x) = sin (x + 1/2), f (x) = sin (x-1)

example
......f (x) = sin (x)
f (x) = 3sin (2x + 1) or f (x) = 3sin [2 (x + 1/2)]

The amplitude is 3, the period is 2 and
the shift in the -x direction is 1/2.
...................................................................................


Combination of sine waves Top
sin (x) + cos (x)

......The red curves are the graphs of the sine and cosine. The black curve is created by superimposing both curves and is a sine curve. One can read off: The period is 2pi, the peak value sqrt (2) and the shift in the -x direction is pi / 2.
This results in the equation sin (x) + cos (x) = sqrt (2) sin (x + pi / 2).

sin (x) cos (x)
......The red curves are the graphs of the sine and cosine. The black curve is created by superimposing both curves and is a sine curve. One can read off: The period is pi and the peak value is 1/2.
This results in the equation sin (x) cos (x) = (1/2) sin (2x), a well-known trigonometric formula.

sin (x) sin (x)
......The given function term is f (x) = sin² (x).

As can be seen in the drawing,
sin² (x) = 1 / 2- (1/2) cos (2x).

The equation follows from the trigonometric formulas sin² (x) + cos² (x) = 1 and cos (2x) = cos² (x) -sin² (x).
Beat
......f (x) = sin (12x).
The graph of f (x) = sin (13x) is almost identical.

......f (x) = sin (12x) + sin (13x)
f1(x) = + cos (x / 2) and f2(x) = + cos (x / 2).
Two sine curves are superimposed, the periods of which differ only slightly.
The trigonometric formula sin (x) + sin (y) = 2cos [(x-y) / 2] sin [(x + y) / 2] applies.
In this special case, sin (13x) + sin (12x) = 2cos (x / 2) sin (25x / 2).

Representation of periodic functions
A rectangular curve can be represented approximately by superimposing sine curves.
Here is an example.

f (x) = sin (x) + 0.5sin (3x) + 0.3sin (5x)

The Fourier series is in the background. A periodic function can be approximated by a number of sine functions with any precision.

Arcsine Top

......The sine function as a whole is not reversible, because e.g. for y = 0 there are any number of x-values, namely the zeros. If, on the other hand, the definition range is restricted to - (1/2) pi <= x <= (1/2) pi, there is exactly one y-value for each x-value.
So in this area it is reversible.

......The graph of the inverse function is obtained by mirroring the sine curve at the first bisector.
For the functional term of the inverse function there are the symbols arc sin (x) or sin-1(x)
The domain is D = {x | -1 <= x <= 1},
the range of values ​​W = {y | (1/2) pi <= y <= (1/2) pi}.

Angle with the calculator
If you enter a sine value between -1 and +1 in the calculator, only an angle between - (1/2) pi and + (1/2) pi or -90 ° and + 90 ° is output according to the inverse function .
So for the sine value 0.7880 you get the angle 52 ° via the key sequence (2nd) (sin) ...
One writes arc sin (0.7880) = 52 °.
Two other examples are arc sin (0.3333) = 19.5 ° and arc sin (-0.3333) = -19.5 °.

Figures in the sinus lens Top
Largest rectangle
The graphs of the functions f (x) = sin (x) and g (x) = - sin (x) form a lens in the range 0 <= x <= pi.

......If you fit rectangles into the lens so that the corner points are on the sine curves as on the left, the question arises as to the rectangle with the largest area.

solution
The approach is A = (pi-2x) (2y)
With y = sin (x) we have A (x) = 2pi * sin (x) -4x * sin (x).
The derivative is A '(x) = 2pi * cos (x) -4 * sin (x) -4x * cos (x).
A '(x) = 0 leads to pi * cos (x) -2sin (x) -2x * cos (x) = 0.
This is a transcendent equation, which i.a. can only be solved approximately.
......The solution of the equation pi * cos (x) -2sin (x) -2x * cos (x) = 0 can be obtained, for example, by taking the zero of the function f (x) = pi * cos (x) -2sin (x) -2x * cos (x) = 0 determined.

For this purpose, the graph is drawn with the Winplot drawing program and the zero is read: x1= 0.72. - Also the second zero x2= 2.42 = pi-x1 fits the problem.
Result: the rectangle 1.70 times 1.32 is the biggest.


square
......If the red rectangle is to become a square from above, it must
2y = pi-2x or 2sin (x) -pi + 2x = 0 apply. As with the largest rectangle problem, the equation is solved graphically.
The result is x = 0.84 and then as the side of the square
a = pi-2x = 1.46 (LE).
......

Five more figures

Overview of the trigonometric functions Top
In addition to the sine function, there are five other trigonometric functions that assign different aspect ratios to an angle in a right-angled triangle.

Sine
sin (x) = a / c
cosine
cos (x) = b / c
tangent
tan (x) = a / b
cotangent
cot (x) = b / a
Secans
sec (x) = c / a
cosecant
csc (x) = c / b
The following applies: 0 <= x <= (1/2) pi.

All function values ​​can be found on the unit circle as lines, shown here in the 1st quadrant.
sin (x) = PP '
cos (x) = QP
tan (x) = TR
cot (x) = US
sec (x) = OR
csc (x) = OS
It applies.

All graphs in one picture
sin (x) and cos (x) in red

tan (x) and cot (x) in blue

sec (x) and csc (x) in green


All function values ​​can be traced back to sine values, because the following formulas apply.
sin (x)
given
cos (x)
= sqrt [1-sin² (x)]
tan (x)
= sin (x) / sqrt [1-sin² (x)]
cot (x)
= sqrt [(1-sin² (x)] / sin (x)
sec (x)
= 1 / sqrt [1-sin² (x)]
csc (x)
= 1 / sin (x)

The formula sin² (x) + cos² (x) = 1 is used in the table.

Only the sine, the cosine and the tangent are used in schools today.
The cotangent has largely disappeared from textbooks.
I first discovered the secans and the cosecans in an English textbook.

Areas in space of the form z = f (x, y) Top



Figures from sine lines Top


The sine in other places on my homepage Top
Sine law

...a: b: c = sin (alpha): sin (beta): sin (gamma)
More on my General Triangle page

Parametric graphs of curves
x (t) = r * cos (t), y (t) = r * sin (t)

More on my circle page
x (t) = at * cos (t), y (t) = at * sin (t)

More on my spirals page
x = a * sqrt [cos (2t)] cos (t)
y = a * sqrt [cos (2t)] sin (t)


More on my page eight curves


x (t) = 4r * cos (t / 3) -a * cos (4t / 3)
y (t) = 4r * sin (t / 3) -a * sin (4t / 3)
More on my page three-jet figures

x (t) = 5r * cos (t / 4) -a * cos (5t / 4)
y (t) = 5r * sin (t / 4) -a * sin (5t / 4)

More on my page four-pointed figures


Representations in polar coordinates
More on my page three-jet figures


Egg chains

More on my express lines page.

Creation of a heart shape

More on my heart curves page


A sine wave
More on my page A vibration through friction

Sin function on the internet Top

German

leifiphysik (Rupprecht-Gymnasium in Munich)
The sine function

Wikibooks
Differentiation of the sine function

Wikipedia
Sine and cosine, arcsine and arccosine, sine versus and cosine versus, SOHCAHTOA, trigonometry formula collection, Fourier series, sine tone, tone 440Hz - audible, beat

English

Eric W. Weisstein (MathWorld)
Sine, Inverse Sine, Inverse trigonometric functions, Simple Harmonic Motion, Fourier Series

Richard Parris (freeware program WINPLOT)
The official website is closed. Download the German program e.g. from heise

Wikipedia
Sine, Inverse trigonometric function, Sine quadrant, Versine, Beat (acoustics),
List of trigonometric identities


credentials Top
(1) F.G. Gauß: four-digit logarithmic and trigonometric tables, Stuttgart 1953
(2) Georg Ulrich / Paul Hoffmann: Differential and integral calculus for self-teaching, Hollfeld
[ISBN 3 8044 0575 4]
(3) Association of authors: Algebra and Geometry for Engineers, Frankfurt / M Zurich 1966
[ISBN 978-3-87144-107-3]


Feedback: Email address on my main page

URL of my homepage:
http://www.mathematische-basteleien.de/

© 2011 Jürgen Köller

Top