Basic concepts
Let's start with the definitions even, odd and periodic functions.
Definition 2
An even function is a function that does not change its value when the sign of the independent variable changes:
Definition 3
A function that repeats its values at some regular interval of time:
T is the period of the function.
Even and odd trigonometric functions
Consider the following figure (Fig. 1):
Picture 1.
Here $\overrightarrow(OA_1)=(x_1,y_1)$ and $\overrightarrow(OA_2)=(x_2,y_2)$ are vectors of unit length symmetric with respect to the $Ox$ axis.
Obviously, the coordinates of these vectors are related by the following relations:
Since the trigonometric functions of sine and cosine can be determined using a unit trigonometric circle, we get that the sine function will be odd, and the cosine function will be an even function, that is:
Periodicity of trigonometric functions
Consider the following figure (Fig. 2).
Figure 2.
Here $\overrightarrow(OA)=(x,y)$ is a vector of unit length.
Let's make a full turn by the vector $\overrightarrow(OA)$. That is, let's rotate the given vector by $2\pi $ radians. After that, the vector will completely return to its original position.
Since the trigonometric functions of sine and cosine can be defined using the unit trigonometric circle, we get that
That is, the sine and cosine functions are periodic functions with the smallest period $T=2\pi $.
Consider now the functions of tangent and cotangent. Since $tgx=\frac(sinx)(cosx)$, then
Since $сtgx=\frac(cosx)(sinx)$, then
Examples of problems on the use of even, odd and periodicity of trigonometric functions
Example 1
Prove the following assertions:
a) $tg(385)^0=tg(25)^0$
c) $sin((-721)^0)=-sin1^0$
a) $tg(385)^0=tg(25)^0$
Since the tangent is a periodic function with a minimum period of $(360)^0$, we get
b) $(cos \left(-13\pi \right)\ )=-1$
Since the cosine is an even and periodic function with a minimum period of $2\pi $, we get
\[(cos \left(-13\pi \right)\ )=(cos 13\pi \ )=(cos \left(\pi +6\cdot 2\pi \right)=cos\pi \ )=- 1\]
c) $sin((-721)^0)=-sin1^0$
Since the sine is an odd and periodic function with a minimum period of $(360)^0$, we get
If we construct a unit circle centered at the origin and set an arbitrary value of the argument x0 and count from the axis Ox corner x 0, then this angle on the unit circle corresponds to some point A(Fig. 1) and its projection onto the axis Oh there will be a point M. Cut length OM equal to the absolute value of the abscissa of the point A. given argument value x0 mapped function value y= cos x 0 as the abscissa of a point A. Accordingly, the point IN(x 0 ;at 0) belongs to the function graph at= cos X(Fig. 2). If point A located to the right of the axis OU, the tocosine will be positive, if to the left it will be negative. But in any case, the point A cannot leave the circle. Therefore, the cosine ranges from -1 to 1:
-1 = cos x = 1.
Additional rotation to any angle, multiple of 2 p, returns a point A to the same place. Therefore, the function y= cos xp:
cos ( x+ 2p) = cos x.
If we take two values of the argument that are equal in absolute value but opposite in sign, x And - x, find corresponding points on the circle A x And A-x. As seen in fig. 3 their projection onto the axis Oh is the same point M. That's why
cos(- x) = cos( x),
those. cosine is an even function, f(–x) = f(x).
So, we can explore the properties of the function y= cos X on the segment , and then take into account its parity and periodicity.
At X= 0 point A lies on the axis Oh, its abscissa is 1, and therefore cos 0 = 1. With an increase X dot A moves around the circle up and to the left, its projection, of course, only to the left, and for x = p/2 cosine becomes 0. Point A at this moment it rises to the maximum height, and then continues to move to the left, but already descending. Its abscissa keeps decreasing until it reaches the smallest value equal to -1 at X= p. Thus, on the segment, the function at= cos X decreases monotonically from 1 to –1 (Fig. 4, 5).
It follows from the parity of the cosine that on the interval [– p, 0], the function increases monotonically from –1 to 1, taking on a zero value at x =–p/2. If you take several periods, you get a wavy curve (Fig. 6).
So the function y= cos x takes zero values at points X= p/2 + kp, Where k- any integer. Maximums equal to 1 are reached at points X= 2kp, i.e. with step 2 p, and the minima equal to –1 at the points X= p + 2kp.
Function y \u003d sin x.
On the unit circle x 0 corresponds to point A(Fig. 7), and its projection onto the axis OU there will be a point N.Z function value y 0 = sin x0 defined as the ordinate of a point A. Dot IN(corner x 0 ,at 0) belongs to the function graph y= sin x(Fig. 8). It is clear that the function y= sin x periodic, its period is 2 p:
sin( x+ 2p) = sin ( x).
For two argument values, X And - , projections of their corresponding points A x And A-x per axle OU located symmetrically about the point ABOUT. That's why
sin(- x) = –sin ( x),
those. sine is an odd function, f(– x) = –f( x) (Fig. 9).
If the point A rotate about a point ABOUT on the corner p/2 counterclockwise (in other words, if the angle X increase by p/2), then its ordinate in the new position will be equal to the abscissa in the old one. Which means
sin( x+ p/2) = cos x.
Otherwise, the sine is the cosine, "belated" by p/2, since any cosine value will "repeate" in the sine when the argument increases by p/2. And to build a sine graph, it is enough to shift the cosine graph by p/2 to the right (Fig. 10). An extremely important property of the sine is expressed by the equality
The geometric meaning of equality can be seen from Fig. 11. Here X - this is half of the arc AB, and sin X - half of the corresponding chord. Obviously, as the points approach A And IN the length of the chord is getting closer and closer to the length of the arc. From the same figure, it is easy to extract the inequality
|sin x| x|, valid for any X.
The formula (*) is called the wonderful limit by mathematicians. From it, in particular, it follows that sin X» X at small X.
Functions at=tg x, y=ctg X. Two other trigonometric functions - tangent and cotangent are easiest to define as ratios of the sine and cosine already known to us:
Like sine and cosine, tangent and cotangent are periodic functions, but their periods are equal p, i.e. they are half that of sine and cosine. The reason for this is clear: if the sine and cosine both change signs, then their ratio will not change.
Since there is a cosine in the denominator of the tangent, the tangent is not defined at those points where the cosine is 0 - when X= p/2 +kp. At all other points it increases monotonically. Direct X= p/2 + kp for the tangent are the vertical asymptotes. At points kp tangent and slope are 0 and 1, respectively (Fig. 12).
The cotangent is not defined where the sine is 0 (when x = kp). At other points it decreases monotonically, and the lines x = kp – its vertical asymptotes. At points x = p/2 +kp the cotangent turns to 0, and the slope at these points is -1 (Fig. 13).
Parity and periodicity.
A function is called even if f(–x) = f(x). The cosine and secant functions are even, and the sine, tangent, cotangent and cosecant functions are odd:
| sin(-α) = -sinα | tg (–α) = –tg α |
| cos(-α) = cosα | ctg(-α) = -ctgα |
| sec(-α) = secα | cosec (–α) = – cosec α |
The parity properties follow from the symmetry of the points P a and R-a (Fig. 14) about the axis X. With such a symmetry, the ordinate of the point changes sign (( X;at) goes to ( X; -y)). All functions - periodic, sine, cosine, secant and cosecant have a period of 2 p, and tangent and cotangent - p:
| sin (α + 2 kπ) = sinα | cos (α + 2 kπ) = cosα |
| tan (α + kπ) = tgα | ctg(α + kπ) = ctgα |
| sec (α + 2 kπ) = sec | cosec (α + 2 kπ) = cosecα |
The periodicity of the sine and cosine follows from the fact that all points P a + 2 kp, Where k= 0, ±1, ±2,…, coincide, and the periodicity of the tangent and cotangent is due to the fact that the points P a + kp alternately fall into two diametrically opposite points of the circle, giving the same point on the axis of tangents.
The main properties of trigonometric functions can be summarized in a table:
| Function | Domain | Many values | Parity | Areas of monotonicity ( k= 0, ± 1, ± 2,…) |
| sin x | –Ґ x Ґ | [–1, +1] | odd | increases with x O((4 k – 1) p /2, (4k + 1) p/2), decreases as x O((4 k + 1) p /2, (4k + 3) p/2) |
| cos x | –Ґ x Ґ | [–1, +1] | even | Increases with x O((2 k – 1) p, 2kp), decreases at x Oh (2 kp, (2k + 1) p) |
| tg x | x № p/2 + p k | (–Ґ , +Ґ ) | odd | increases with x O((2 k – 1) p /2, (2k + 1) p /2) |
| ctg x | x № p k | (–Ґ , +Ґ ) | odd | decreases at x ABOUT ( kp, (k + 1) p) |
| sec x | x № p/2 + p k | (–Ґ , –1] AND [+1, +Ґ ) | even | Increases with x Oh (2 kp, (2k + 1) p), decreases at x O((2 k– 1) p , 2 kp) |
| cause x | x № p k | (–Ґ , –1] AND [+1, +Ґ ) | odd | increases with x O((4 k + 1) p /2, (4k + 3) p/2), decreases as x O((4 k – 1) p /2, (4k + 1) p /2) |
Casting formulas.
According to these formulas, the value of the trigonometric function of the argument a, where p/2 a p , can be reduced to the value of the function of the argument a , where 0 a p /2, both the same and additional to it.
| Argument b |  – a |
+a | p– a | p+a | +a | +a | 2p– a |
| sin b | cos a | cos a | sin a | –sin a | -cos a | -cos a | –sin a |
| cosb | sin a | –sin a | -cos a | -cos a | –sin a | sin a | cos a |
Therefore, in the tables of trigonometric functions, values \u200b\u200bare given only for acute angles, and it is enough to confine ourselves, for example, to sine and tangent. The table contains only the most commonly used formulas for sine and cosine. From them it is easy to obtain formulas for tangent and cotangent. When casting a function from an argument of the form kp/2 ± a , where k is an integer, to a function from the argument a :
1) the name of the function is saved if k even, and changes to "complementary" if k odd;
2) the sign on the right side coincides with the sign of the reducible function at the point kp/2 ± a if the angle a is acute.
For example, when casting ctg (a - p/2) make sure that a - p/2 at 0 a p /2 lies in the fourth quadrant, where the cotangent is negative, and, according to rule 1, we change the name of the function: ctg (a - p/2) = –tg a .
Addition formulas.
Multiple angle formulas.
These formulas are derived directly from the addition formulas:
sin 2a \u003d 2 sin a cos a;
cos 2a \u003d cos 2 a - sin 2 a \u003d 2 cos 2 a - 1 \u003d 1 - 2 sin 2 a;
sin 3a \u003d 3 sin a - 4 sin 3 a;
cos 3a \u003d 4 cos 3 a - 3 cos a;
The formula for cos 3a was used by Francois Viet when solving a cubic equation. He was the first to find expressions for cos n a and sin n a , which were later obtained in a simpler way from De Moivre's formula.
If you replace a with a /2 in double argument formulas, they can be converted to half angle formulas:
Universal substitution formulas.
Using these formulas, an expression involving different trigonometric functions from the same argument can be rewritten as a rational expression from a single function tg (a / 2), this is useful when solving some equations:
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Formulas for converting sums to products and products to sums.
Before the advent of computers, these formulas were used to simplify calculations. Calculations were made using logarithmic tables, and later - a slide rule, because. logarithms are best suited for multiplying numbers, so all the original expressions were reduced to a form convenient for logarithms, i.e. for works such as:
2 sin a sin b = cos( a-b) – cos( a+b);
2 cos a cos b= cos ( a-b) + cos ( a+b);
2 sin a cos b= sin ( a-b) + sin ( a+b).
The formulas for the tangent and cotangent functions can be obtained from the above.
Degree reduction formulas.
From the formulas of a multiple argument, formulas are derived:
| sin 2 a \u003d (1 - cos 2a) / 2; | cos 2 a = (1 + cos 2a )/2; |
| sin 3 a \u003d (3 sin a - sin 3a) / 4; | cos 3 a = (3 cos a + cos3 a )/4. |
With the help of these formulas, trigonometric equations can be reduced to equations of lower degrees. In the same way, reduction formulas for higher powers of sine and cosine can be derived.
| Derivatives and integrals of trigonometric functions | |
| (sin x)` = cos x; | (cos x)` = -sin x; |
| (tg x)` = ; | (ctg x)` = – ; |
| t sin x dx= -cos x + C; | t cos x dx= sin x + C; |
| t tg x dx= –ln |cos x| + C; | t ctg x dx = ln|sin x| + C; |
Every trigonometric function at every point of its domain of definition is continuous and infinitely differentiable. Moreover, the derivatives of trigonometric functions are trigonometric functions, and when integrated, trigonometric functions or their logarithms are also obtained. Integrals of rational combinations of trigonometric functions are always elementary functions.
Representation of trigonometric functions in the form of power series and infinite products.
All trigonometric functions can be expanded into power series. In this case, the functions sin x b cos x appear in rows. convergent for all values x:
These series can be used to obtain approximate expressions for sin x and cos x for small values x:
at | x| p/2;
at 0x| p
(B n are Bernoulli numbers).
sin functions x and cos x can be represented as infinite products:
Trigonometric system 1, cos x, sin x, cos 2 x, sin 2 x, ¼, cos nx, sin nx, ¼, forms on the interval [– p, p] orthogonal system of functions, which makes it possible to represent functions in the form of trigonometric series.
are defined as analytic continuations of the corresponding trigonometric functions of a real argument into the complex plane. Yes, sin z and cos z can be defined using series for sin x and cos x, if instead of x put z:
These series converge over the entire plane, so sin z and cos z are entire functions.
Tangent and cotangent are determined by the formulas:
tg functions z and ctg z are meromorphic functions. Poles tg z and sec z are simple (1st order) and are located at points z=p/2 + pn, ctg poles z and cosec z are also simple and are located at points z = p n, n = 0, ±1, ±2,…
All formulas that are valid for trigonometric functions of a real argument are also valid for a complex one. In particular,
sin(- z) = -sin z,
cos(- z) = cos z,
tg(- z) = –tg z,
ctg (- z) = -ctg z,
those. even and odd parity are preserved. The formulas are also saved
sin( z + 2p) = sin z, (z + 2p) = cos z, (z + p) = tg z, (z + p) = ctg z,
those. the periodicity is also preserved, and the periods are the same as for functions of a real argument.
Trigonometric functions can be expressed in terms of an exponential function of a purely imaginary argument:
Back, e iz expressed in terms of cos z and sin z according to the formula:
e iz= cos z + i sin z
These formulas are called the Euler formulas. Leonhard Euler introduced them in 1743.
Trigonometric functions can also be expressed in terms of hyperbolic functions:
z = –i sh iz, cos z = ch iz, z = –i th iz.
where sh, ch and th are hyperbolic sine, cosine and tangent.
Trigonometric functions of complex argument z = x + iy, Where x And y- real numbers, can be expressed in terms of trigonometric and hyperbolic functions of real arguments, for example:
sin( x+iy) = sin x ch y + i cos x sh y;
cos ( x+iy) = cos x ch y + i sin x sh y.
The sine and cosine of a complex argument can take real values greater than 1 in absolute value. For example:
If an unknown angle enters the equation as an argument of trigonometric functions, then the equation is called trigonometric. Such equations are so common that their methods the solutions are very detailed and carefully designed. WITH using various methods and formulas, trigonometric equations are reduced to equations of the form f(x)= a, Where f- any of the simplest trigonometric functions: sine, cosine, tangent or cotangent. Then express the argument x this function through its known value A.
Since trigonometric functions are periodic, the same A from the range of values there are infinitely many values of the argument, and the solution of the equation cannot be written as a single function of A. Therefore, in the domain of definition of each of the main trigonometric functions, a section is selected in which it takes all its values, each only once, and a function is found that is inverse to it in this section. Such functions are denoted by attributing the prefix arc (arc) to the name of the original function, and are called inverse trigonometric functions or just arc functions.
Inverse trigonometric functions.
For sin X, cos X, tg X and ctg X inverse functions can be defined. They are designated respectively arcsin X(read "arxine x"), arcos x, arctg x and arcctg x. By definition, arcsin X there is such a number y, What
sin at = X.
The same is true for other inverse trigonometric functions. But this definition suffers from some inaccuracy.
If we reflect sin X, cos X, tg X and ctg X relative to the bisector of the first and third quadrants of the coordinate plane, then the functions become ambiguous due to their periodicity: the same sine (cosine, tangent, cotangent) corresponds to an infinite number of angles.
To get rid of the ambiguity, a section of the curve with a width of p, while it is necessary that a one-to-one correspondence be observed between the argument and the value of the function. Areas near the origin are selected. For the sinus as the "interval of one-to-one" is taken the segment [- p/2, p/2], on which the sine monotonically increases from –1 to 1, for the cosine - the segment , for the tangent and cotangent, respectively, the intervals (– p/2, p/2) and (0, p). Each curve in the interval is reflected about the bisector and now you can define inverse trigonometric functions. For example, let the argument value be given x 0 , such that 0 J x 0 Ј 1. Then the value of the function y 0 = arcsin x 0 will be the only value at 0 , such that - p/2 J at 0 Ј p/2 and x 0 = sin y 0 .
Thus, the arcsine is a function of arcsin A, defined on the interval [–1, 1] and equal for each A such a value a , – p/2 a p /2 that sin a = A. It is very convenient to represent it using a unit circle (Fig. 15). When | a| 1 there are two points on the circle with an ordinate a, symmetrical about the axis y. One of them is the angle a= arcsin A, and the other is the angle p - a. WITH taking into account the periodicity of the sine, the solution of the equation sin x= A is written as follows:
x =(–1)n arc sin a + 2p n,
Where n= 0, ±1, ±2,...
Other simple trigonometric equations are also solved:
cos x = a, –1 =a= 1;
x=±arcos a + 2p n,
Where P= 0, ±1, ±2,... (Fig. 16);
tg X = a;
x= arctg a + p n,
Where n = 0, ±1, ±2,... (Fig. 17);
ctg X= A;
X= arcctg a + p n,
Where n = 0, ±1, ±2,... (Fig. 18).
The main properties of inverse trigonometric functions:
arc sin X(Fig. 19): the domain of definition is the segment [–1, 1]; range - [- p/2, p/2], a monotonically increasing function;
arccos X(Fig. 20): the domain of definition is the segment [–1, 1]; range of values - ; monotonically decreasing function;
arctg X(Fig. 21): domain of definition - all real numbers; range of values – interval (– p/2, p/2); monotonically increasing function; straight at= –p/2 and y \u003d p / 2 - horizontal asymptotes;
arcctg X(Fig. 22): domain of definition - all real numbers; range of values - interval (0, p); monotonically decreasing function; straight y= 0 and y = p are the horizontal asymptotes.
Because trigonometric functions of complex argument sin z and cos z(in contrast to the functions of a real argument) take all complex values, then the equations sin z = a and cos z = a have solutions for any complex a x And y are real numbers, there are inequalities
½| e\ey–e-y| ≤|sin z|≤½( e y +e-y),
½| e y–e-y| ≤|cos z|≤½( e y +e -y),
of which y® Ґ asymptotic formulas follow (uniformly with respect to x)
|sin z| » 1/2 e |y| ,
|cos z| » 1/2 e |y| .
Trigonometric functions arose for the first time in connection with research in astronomy and geometry. The ratios of segments in a triangle and a circle, which are essentially trigonometric functions, are found already in the 3rd century. BC e. in the works of mathematicians of Ancient Greece – Euclid, Archimedes, Apollonius of Perga and others, however, these ratios were not an independent object of study, so they did not study trigonometric functions as such. They were originally considered as segments and in this form were used by Aristarchus (late 4th - 2nd half of the 3rd centuries BC), Hipparchus (2nd century BC), Menelaus (1st century AD). ) and Ptolemy (2nd century AD) when solving spherical triangles. Ptolemy compiled the first table of chords for acute angles through 30 "with an accuracy of 10 -6. This was the first table of sines. As a ratio, the function sin a is already found in Ariabhata (end of the 5th century). The functions tg a and ctg a are found in al- Battani (2nd half of the 9th - early 10th centuries) and Abul-Wefa (10th century), who also uses sec a and cosec a... Aryabhata already knew the formula (sin 2 a + cos 2 a) \u003d 1, as well as half-angle sin and cos formulas, with the help of which he built tables of sines for angles through 3 ° 45 "; based on the known values of trigonometric functions for the simplest arguments. Bhaskara (12th century) gave a method for constructing tables through 1 using addition formulas. Formulas for converting the sum and difference of trigonometric functions of various arguments into a product were derived by Regiomontanus (15th century) and J. Napier in connection with the latter's invention of logarithms (1614). Regiomontanus gave a table of sine values through 1 ". The expansion of trigonometric functions into power series was obtained by I. Newton (1669). L. Euler (18th century) brought the theory of trigonometric functions into a modern form. He owns their definition for real and complex arguments, adopted now symbolism, establishing a connection with the exponential function and orthogonality of the system of sines and cosines.
|BD| - the length of the arc of a circle centered at point A.
α is the angle expressed in radians.
Tangent ( tgα) is a trigonometric function depending on the angle α between the hypotenuse and the leg of a right triangle, equal to the ratio of the length of the opposite leg |BC| to the length of the adjacent leg |AB| .
Cotangent ( ctgα) is a trigonometric function depending on the angle α between the hypotenuse and the leg of a right triangle, equal to the ratio of the length of the adjacent leg |AB| to the length of the opposite leg |BC| .
Tangent
Where n- whole.
In Western literature, the tangent is denoted as follows:
.
;
;
.
Graph of the tangent function, y = tg x
Cotangent
Where n- whole.
In Western literature, the cotangent is denoted as follows:
.
The following notation has also been adopted:
;
;
.
Graph of the cotangent function, y = ctg x
Properties of tangent and cotangent
Periodicity
Functions y= tg x and y= ctg x are periodic with period π.
Parity
The functions tangent and cotangent are odd.
Domains of definition and values, ascending, descending
The functions tangent and cotangent are continuous on their domain of definition (see the proof of continuity). The main properties of the tangent and cotangent are presented in the table ( n- integer).
| y= tg x | y= ctg x | |
| Scope and continuity | ||
| Range of values | -∞ < y < +∞ | -∞ < y < +∞ |
| Ascending | - | |
| Descending | - | |
| Extremes | - | - |
| Zeros, y= 0 | ||
| Points of intersection with the y-axis, x = 0 | y= 0 | - |
Formulas
Expressions in terms of sine and cosine
;
;
;
;
;
Formulas for tangent and cotangent of sum and difference
The rest of the formulas are easy to obtain, for example
Product of tangents
The formula for the sum and difference of tangents
This table shows the values of tangents and cotangents for some values of the argument. 
Expressions in terms of complex numbers
Expressions in terms of hyperbolic functions
;
;
Derivatives
; .
.
Derivative of the nth order with respect to the variable x of the function :
.
Derivation of formulas for tangent > > > ; for cotangent > > >
Integrals
Expansions into series
To get the expansion of the tangent in powers of x, you need to take several terms of the expansion in a power series for the functions sin x And cos x and divide these polynomials into each other , . This results in the following formulas.
At .
at .
Where B n- Bernoulli numbers. They are determined either from the recurrence relation:
;
;
Where .
Or according to the Laplace formula: 
Inverse functions
The inverse functions to tangent and cotangent are arctangent and arccotangent, respectively.
Arctangent, arctg
, Where n- whole.
Arc tangent, arcctg
, Where n- whole.
References:
I.N. Bronstein, K.A. Semendyaev, Handbook of Mathematics for Engineers and Students of Higher Educational Institutions, Lan, 2009.
G. Korn, Handbook of Mathematics for Researchers and Engineers, 2012.
Basic concepts
Let's start with the definitions even, odd and periodic functions.
Definition 2
An even function is a function that does not change its value when the sign of the independent variable changes:
Definition 3
A function that repeats its values at some regular interval of time:
T is the period of the function.
Even and odd trigonometric functions
Consider the following figure (Fig. 1):
Picture 1.
Here $\overrightarrow(OA_1)=(x_1,y_1)$ and $\overrightarrow(OA_2)=(x_2,y_2)$ are symmetric with respect to the $Ox$ axis vectors single length.
Obviously, the coordinates of these vectors are related by the following relations:
Since the trigonometric functions of sine and cosine can be determined using a unit trigonometric circle, we get that the sine function will be odd, and the cosine function will be an even function, that is:
Periodicity of trigonometric functions
Consider the following figure (Fig. 2).
Figure 2.
Here $\overrightarrow(OA)=(x,y)$ is a vector of unit length.
Let's make a full turn by the vector $\overrightarrow(OA)$. That is, let's rotate the given vector by $2\pi $ radians. After that, the vector will completely return to its original position.
Since the trigonometric functions of sine and cosine can be defined using the unit trigonometric circle, we get that
That is, the sine and cosine functions are periodic functions with the smallest period $T=2\pi $.
Consider now the functions of tangent and cotangent. Since $tgx=\frac(sinx)(cosx)$, then
Since $сtgx=\frac(cosx)(sinx)$, then
Examples of problems on the use of even, odd and periodicity of trigonometric functions
Example 1
Prove the following assertions:
a) $tg(385)^0=tg(25)^0$
c) $sin((-721)^0)=-sin1^0$
a) $tg(385)^0=tg(25)^0$
Since the tangent is a periodic function with a minimum period of $(360)^0$, we get
b) $(cos \left(-13\pi \right)\ )=-1$
Since the cosine is an even and periodic function with a minimum period of $2\pi $, we get
\[(cos \left(-13\pi \right)\ )=(cos 13\pi \ )=(cos \left(\pi +6\cdot 2\pi \right)=cos\pi \ )=- 1\]
c) $sin((-721)^0)=-sin1^0$
Since the sine is an odd and periodic function with a minimum period of $(360)^0$, we get
