What is the value of tan1

What is the tangent function?

......A point P (x | y) is given in a Cartesian coordinate system.
If you connect it with the zero point O (0 | 0), the angle alpha arises between the connecting line and the positive x-axis.
With the tangent function, the ratio of the ordinate to the abscissa is assigned to the angle alpha.
In the formula language this is called [alpha tan (alpha) with tan (alpha) = y / x].

The mathematician Thomas Fincke (1561–1656) used the term “tangent” in his book Geometriae rotundi (1583) a.
Source: http://www-history.mcs.st-andrews.ac.uk/Biographies/Fincke.html

Tangent values ​​for acute angles Top
The basic values ​​of the tangent function relate to acute angles. They are called basic values ​​because they are repeated in all areas in which the tangent is defined.

......The best way to get an overview of them is to represent the tangent as a segment. To do this, expand the drawing above. A circle with the radius r = 1 LE is drawn around the zero point. It intersects the x-axis at point A. Draw the tangent to the circle at point A.
It intersects the straight line OP at point B. AB = tan (alpha) applies.
The distance AB is measured in the unit r = 1, insofar as it is also an aspect ratio.
Derivation
According to the 2nd ray sentence, AB: PQ = OA: OQ or AB = PQ * OA: OQ = (y * 1): x = y: x = tan (alpha).

It is so easy to see that tan (0 °) = 0 and that tan (alpha) goes beyond all limits when alpha approaches 90 °. This means that tan (90 °) is not defined. The point alpha = 90 ° is a pole. It can also be assumed that the tangent values ​​steadily increase monotonically with increasing angles. More details emerge from the following considerations.

Certain tangent values ​​are obtained from special triangles.
......If alpha = 30 °, a 30-60-90 triangle is created.
The following applies: tan (30 °) = 1 / sqrt (3) = (1/3) sqrt (3).

......If alpha = 60 °, a 30-60-90 triangle is created and tan (60 °) = sqrt (3).

The numbers can be summarized in a table of values.
alpha
30°
45°
60°
90°
tan (alpha)
0
(1/3) sqrt (3)
1
sqrt (3)
-

Root terms or irrational numbers as function values ​​are exceptions. In general, the tangent values ​​are transcendent numbers that can be approximated as decimal fractions, but then with any precision. These are obtained via convergent series.
Today it is no problem to get rounded tangent values ​​using the calculator. For example, you can determine tan (30 °) with the TI 30 using the key sequence (3) (0) (TAN). The result is tan (30 °) = 0.577350269. However, there is no point in taking all 9 decimals from the calculator. That is too precise, because the angle is only given with an accuracy of two digits. According to a rule of thumb, two valid digits are sufficient for the tangent value, tan (30 °) = 0.58.
However, it is common practice to round the tangent value to four decimal places. This then includes an angular accuracy of about 1 '= 1/60 °. So it is called tan (30.0 °) = 0.5774.
For angles greater than 45 ° and especially for angles close to 90 °, it is better to use the term of the applicable digits. For example, according to the pocket calculator, tan (88 °) = 28.63625328. That is 10 valid digits. You should round to 4 valid digits. That means tan (88 °) = 28.64.
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).

Cotangent function Top
The cotangent function is closely related to the tangent function.

...The kotang value is the ratio of the adjacent to the opposite side.
In the formula language this is called cot (alpha) = b / a.

Then cot (alpha) = tan (beta) = tan (90 ° -alpha).


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

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

Basically, the cotangent function is just a variant of the tangent function.

Tangent values ​​for further angles
Above it is allowed that the starting point P does not necessarily lie in the first quadrant as previously assumed. The tangent is defined for any angle.
The following drawings represent tan (alpha) for angles greater than 90 ° and show that the tangent values ​​of these angles can be applied to the tangent values Sharpener Angle alpha returns.


tan (90 ° + alpha) = tan (alpha).

tan (180 ° + alpha) = - tan (alpha).

tan (360 ° -alpha) = - tan (alpha)

Without loss of generality one can assume that the point P lies on the unit circle.

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

If the point P moves from the x-axis counterclockwise, the tangent values ​​are repeated: tan (alpha + 360 °) = tan (alpha).
If the signs are taken into account, the tangent values ​​repeat themselves after 180 °.
tan (alpha + 180) = tan (alpha), more generally: tan (alpha + n * 180) = tan (alpha), n = 1, 2, 3, ...

If the point P moves away from the x-axis in the Clockwise, the angles will be negative.
Then tan (alpha-n * 180) = tan (alpha), n = 1, 2, 3, ...

A graph shows this situation again.

It is common and also useful to specify the angles of the tangent function not in degrees but in radians. Then we write f (x) = tan (x) or y = tan (x) instead of f (alpha) = tan (alpha).
The domain is D = | R \ {[(2k-1) / 2] pi, k = ...- 2, -1, 0, 1, 2, ...}.
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 suitable 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.

Cotangent function for D = | R.

As mentioned above, the cotangent function cot (alpha) = tan (90 ° -alpha).
The extension of the cotangent function for any angle in radians is then cot (x) = tan (-x + pi / 2) = - tan (x-p / 2).
The graph of the cotangent function is derived from that of the tangent function by mirroring on the y-axis and shifting by pi / 2 rad in the x-axis direction.


Determination of the tangent values Top
Function values ​​with the calculator
It was already described above how to get tan (30.0 °) = 0.5774 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 the computer.
If you want to determine the tangent value of an angle, which is given in radians, you switch on the radian mode RAD with the DRG key. Then, for example, tan (1 rad) = 1.5574.


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

tan (30 ° 0 ') = 0.5774.

The tables contain tangent 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. Because with the help of logarithms, multiplication was previously avoided and replaced by simpler addition, albeit at the expense of accuracy.


Calculation of the tangent values
The tangent values ​​can be calculated for angles | x |  
Thereby Bn the Bernoulli numbers.
The variable n stands for the natural numbers.

The formula becomes more readable if you limit yourself to the first few terms.
tan (x) = x + (1/3) x3 + (2/15) x5 + (17/315) x7 + (62/2835) x9 +...

The calculator delivers tan (30.0 °) = tan (pi / 6 rad) = 0.5774.
The series gives tan (pi / 6 rad) = 0.52360 + 0.04785 + 0.00525 + 0.00058 = 0.5773.

Formulas Top
Relationship to the cotangent
It applies tan (alpha) = 1 / cot (alpha)
Derivation for acute angles: tan (alpha) = y / x = 1 / (x / y) = cot (alpha).


Relationship to sine and cosine
It applies tan (alpha) = sin (alpha) / cos (alpha)
Derivation for acute angles: tan (alpha) = y / x = (y / r) / (x / r) = sin (alpha) / cos (alpha).

Addition theorem
It applies tan (alpha + beta) = [tan (alpha) + tan (beta)] / [1-tan (alpha) tan (beta)]
Derivation
tan (alpha + beta)
= sin (alpha + beta) / cos (alpha + beta)
= [sin (alpha) cos (beta) + cos (alpha) sin (beta)] / [cos (alpha) cos (beta) -sin (alpha) sin (beta)]
= [sin (alpha) / cos (alpha) + sin (beta) / cos (beta)] / [1-sin (alpha) / cos (alpha) sin (beta) cos (beta)]
= [tan (alpha) + tan (beta)] / [1-tan (alpha) tan (beta)]

Tan (n * alpha)
If alpha = beta, the formula is tan (2alpha) = 2tan (alpha) / [1-tan² (alpha)].
The generalization is tan (n * alpha) = {tan [(n-1) alpha] + tan (alpha)} / {[1 + tan [(n-1) alpha)] tan (alpha)}.
Proof by complete induction
Induction start (n = 2)
tan (2alpha) = [tan (alpha) + tan (alpha)] / [1 + tan (alpha) tan (alpha)] = 2tan (alpha) / [1-tan² (alpha)]
Induction assumption (correct for n)
tan (n * alpha) = {tan [(n-1) alpha] + tan (alpha)} / {[1 + tan [(n-1) alpha] tan (alpha)}
Induction reasoning (proof for n + 1)
tan [(n + 1) alpha] = tan (n * alpha + alpha) = [tan (n * alpha) + tan (alpha)] / [1-tan (n * alpha) tan (alpha)], wzbw.

The formulas for Tangent theorem are on my website General Triangle.

Further formulas for the tangent can be found on the website Tangent from MathWorld (url below).

Properties of the tangent function Top
Again the graph of the function f (x) = tan (x)


Definition and value range
Largest possible definition range
D = {x |} \ {[(2k-1) / 2] * pi, k = ...- 2, -1, 0, 1, 2, 3, ...}
Range of values
W = | R

period
The following applies: tan (alpha + 180 °) = tan (alpha) or tan (x + pi) = tan (x).
The tangent function is thus a periodic function with the (smallest) period 180 ° or pi rad.

Special points
Zeroing and turning points at the same time
xN= k * pi
Poles
xP.= [(2k-1) / 2] pi
The variable k stands for whole numbers.

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

Derivation
The following applies: f '(x) = 1 / cos² (x).
Derivation
According to the quotient rule, [tan (x)] '= [sin (x) / cos (x)]' = [cos² (x) + sin² (x)] / cos² (x) = 1 / cos² (x), wzbw ..
Below in the arc function chapter, the equation cos² (x) = 1 / [1 + tan² (x)] is derived, so that also applies
f '(x) = 1 + tan² (x).

integral
...
Substitution: cos (x) = u and dx = -du / sin (x).

Known tasks Top
Slope of a straight line

......In a planar, Cartesian coordinate system, straight lines are represented by Ax + By + C = 0.
If B = 0, the associated straight lines are parallel to the y-axis.
If B is not equal to 0, the other straight lines have the normal representation y = mx + b.
The question is: How big is the slope angle alpha?
solution
The variable m is the slope of the straight line. The following applies: m = tan (alpha).
In the drawing, the normal form is y = (1/2) x + 1 and the slope angle alpha = 26.6 °.

Angle in the square pyramid
There are two notable angles in a pyramid, namely the angle alpha1 between the base and a side surface and the angle alpha2 between the base and a side edge.
They can be calculated from the base side a and the height h with the help of the tangent function.
The angles are clear when the base and the side edge are the same size. Let us consider this case. For this special case, the pyramid is called the square pyramid.
For them we have h = (1/2) sqrt (2) a.

alpha1
......tan (alpha1) = 2h / a = sqrt (2) or alpha1=54,7°. 

alpha2
......tan (alpha2) = 2h / [(1/2) sqrt (2) a] = 2 or alpha2=63,4°. 
The pairs of images allow a three-dimensional view of the pyramid.

Slope of a road
There are road sections that are so steep that it is advisable to show them to road users beforehand. There are also traffic signs on which there is a percentage as a measure of the gradient, e.g. 10%.
......The steep stretch of road is the route c. You can specify a gradient triangle for it.
The slope m is the ratio of the two legs, i.e. m = a: b, then expressed in percent. This is the tangent of the slope angle alpha.

......If m = 10% = 1/10, the angle of incline is
alpha = arc tan (1/10) = 5.7 °.

......It is graphically well done to fit the gradient triangle into a "danger sign".

Mathematically speaking, the representation is not as good. The triangle is upside down. The uphill section of the road is horizontal.


......It remains to be added that there is a traffic sign for oncoming traffic that warns of the slope.

annotation
The Gauseköte in the Teutoburg Forest has the greatest gradient here in the Lippe district with a gradient of up to 16%.

Height of a tree
......The following applies: tan (45 °) = 1.

This fact is used with the (right-angled) forester triangle to estimate the height of a tree.

One looks for a distance from the tree so that one can locate the top of the tree along the hypotenuse of the triangle.
Then the distance e from the tree is as great as the height h.
The height of the tree is then h + a.
 


Arctangent Top
definition

......The tangent function as a whole is not reversible, because there are, for example, any number of x values ​​for y = 0, namely the zeros. For a function, however, the assignment must be unique.

On the other hand, if you restrict the domain of definition
D = {x | - (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 tangent curve at the first bisector.
For the functional term of the inverse function are the symbols
arc tan (x) or tan-1(x) common.
The domain of definition is D = | R,
the range of values ​​W = {y | - (1/2) pi rad <= y <= + (1/2) pi rad}.

Arctangent values ​​with the calculator
If you enter a tangent value into the calculator, an angle between - (1/2) pi and + (1/2) pi or -90 ° and + 90 ° is output according to the inverse function.
So you get the angle 30.0 ° for the tangent value 0.5774 via the key sequence (2nd) (tan) ...
One writes arc tan (0.5774) = 30.0 °.
Two other examples are arc tan (0.3333) = 18.4 ° and arc tan (-0.3333) = -18.4 °.

Derivation
Claim: The derivative of the function g (x) = arc tan (x) is g '(x) = 1 / (1 + x²).
Derivation
The equation cos² (x) = 1 / [1 + tan² (x)] is determined beforehand.
From tan² (x) = sin² (x) / cos² (x) follows cos² (x) tan² (x) = 1-cos² (x) or cos² (x) = 1 / [1 + tan² (x)].
Below you need cos² (y) = 1 / [1 + tan² (y)].

Since g with g (x) = arc tan (x) is the inverse function of f with f (x) = tan (x), the equation x = tan (y) follows from y = arc tan (x).

Then the derivatives g '(x) = 1 / f' (y).
Then [arc tan (x)] '= 1 / tan' (y) = 1 / [1 / cos² (y)] = cos² (y) = 1 / [1 + tan² (y)] = 1 / (1 + x²).

Indefinite integral
Claim:
Derivation
The product rule is applied.her name is
One sets u (x) = arc tan (x) and v (x) = x, then u '(x) = 1 / (1 + x²) and v' (x) = 1.
It follows

It is used that H (x) = (1/2) ln (1 + x²) is the antiderivative of h (x) = x / (1 + x²), as can be shown by deriving H (x).

line
As with the tangent function, there is also a convergent series for the arctangent function.
It reads arc tan (x) = x - (1/3) x3 + (1/5) x5 - (1/7) x7+ (1/9) x9 - +... 

The series for the special case x = 1 is always astonishing.
The turn is arc tan (1) = 1 - 1/3 + 1 / 5- 1/7 + 1/9 - ...+... 
Since arc tan (1) = pi / 4 applies, there is a formula for calculating pi.
pi / 4 = 1 - 1/3 + 1 / 5- 1/7 + 1/9 - ... + ...
However, the convergence behavior is poor.
The first six summands lead to pi = 3.35. There are better approximations to pi.

Tangent curves Top
Antiderivative and derivative of f (x) = tan (x)

......
Antiderivative F (x) = - ln (| cos (x) |)

Function: f (x) = tan (x)

Derivative: f '(x) = 1 / cos² (x)


Antiderivative and derivative of g (x) = arc tan (x)
......Antiderivative G (x) = x arc tan (x) - (1/2) ln (1 + x²)

Function: g (x) = arc tan (x)

Derivative: g '(x) = 1 / (1 + x²)

 


Four curves in comparison
......f1(x) = tan (x)

f2(x) = x³

f3(x) = tan³ (x)

f4(x) = arc sin (x)

The curves have a turning point at the zero point.
 


Mirror images to the tangent
......y = tan (x)
y = tan (-x)
y = arc tan (x)
y = arc tan (-x)

.......................................................................................


The tangent in the polar coordinate system
......The graph opposite has the simple representation r = tan (t) with 0 <= t <= 2pi in the polar coordinate system.
In the Cartesian coordinate system this is the graph of the relation x² (x² + y²) = y².
Derivation
The following applies: r² = x² + y², sin (t) = y / r and cos (t) = x / r. This leads to with r² = tan² (t)
x² + y² = sin² (t) / cos² (t) = (y² / r²) / (x² / r²) = y² / x² or x² (x² + y²) = y² or y2= x4/ (1-x2).

Overview of the trigonometric functions Top
In addition to the tangent function, there are five other trigonometric functions in which the other aspect ratios are assigned to an angle in a right 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
We have 0
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)

tan (x) and cot (x)

sec (x) and csc (x)


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

Tangent function in the internet Top

German

André Mössner
The definitions of the angle function tangent,
Angle calculation with the tangent or arctangent
(Applets)

Matroid
The relationships of sine and cosine

Underneath
sin [arc tan (alpha)] = alpha / sqrt [(alpha) ² + 1]
cos [arc tan (alpha)] = 1 / sqrt [(alpha) ² + 1]

Walter Fendt
Sine, cosine and tangent of an angle (applet)

Werner Brefeld
The tangent of the three interior angles in the triangle

Wikipedia
Tangent and cotangent, trigonometric function, arctangent and arctangent,
Hyperbolic tangent and hyperbolic cotangent, tangent theorem, Förster triangle, Gausekote

English

School of Mathematics and Statistics - University of St Andrews, Scotland
Thomas Fincke

Eric W. Weisstein (MathWorld)
Tangent, Cotangent, SOHCAHTOA, Inverse Tangent, Hyperbolic Tangent

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

Wikipedia
Trigonometric functions, hyperbolic function, law of tangents

Wolfram Matematica
ONLINE INTEGRATOR


credentials Top
(1) F. G. Gauß: four-digit logarithmic and trigonometric tables, Stuttgart 1953
(2) 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

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