In this article we will take a comprehensive look. Basic trigonometric identities are equalities that establish a connection between the sine, cosine, tangent and cotangent of one angle, and allow one to find any of these trigonometric functions through a known other.
Let us immediately list the main trigonometric identities that we will analyze in this article. Let's write them down in a table, and below we'll give the output of these formulas and provide the necessary explanations.
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Relationship between sine and cosine of one angle
Sometimes they do not talk about the main trigonometric identities listed in the table above, but about one single basic trigonometric identity kind 
. The explanation for this fact is quite simple: the equalities are obtained from the main trigonometric identity after dividing both of its parts by and, respectively, and the equalities 
And 
follow from the definitions of sine, cosine, tangent and cotangent. We'll talk about this in more detail in the following paragraphs.
That is, it is the equality that is of particular interest, which was given the name of the main trigonometric identity.
Before proving the main trigonometric identity, we give its formulation: the sum of the squares of the sine and cosine of one angle is identically equal to one. Now let's prove it.
The basic trigonometric identity is very often used when converting trigonometric expressions. It allows the sum of the squares of the sine and cosine of one angle to be replaced by one. No less often, the basic trigonometric identity is used in the reverse order: unit is replaced by the sum of the squares of the sine and cosine of any angle.
Tangent and cotangent through sine and cosine
Identities connecting tangent and cotangent with sine and cosine of one angle of view and 
follow immediately from the definitions of sine, cosine, tangent and cotangent. Indeed, by definition, sine is the ordinate of y, cosine is the abscissa of x, tangent is the ratio of the ordinate to the abscissa, that is, 
, and the cotangent is the ratio of the abscissa to the ordinate, that is, 
.
Thanks to such obviousness of the identities and Tangent and cotangent are often defined not through the ratio of abscissa and ordinate, but through the ratio of sine and cosine. So the tangent of an angle is the ratio of the sine to the cosine of this angle, and the cotangent is the ratio of the cosine to the sine.
In conclusion of this point, it should be noted that the identities and take place for all angles at which the trigonometric functions included in them make sense. So the formula is valid for any , other than (otherwise the denominator will have zero, and we did not define division by zero), and the formula - for all , different from , where z is any .
Relationship between tangent and cotangent
An even more obvious trigonometric identity than the previous two is the identity connecting the tangent and cotangent of one angle of the form . It is clear that it holds for any angles other than , otherwise either the tangent or the cotangent are not defined.
Proof of the formula very simple. By definition and from where 
. The proof could have been carried out a little differently. Since , That 
.
So, the tangent and cotangent of the same angle at which they make sense are .
We continue our conversation about the most used formulas in trigonometry. The most important of them are addition formulas.
Definition 1
Addition formulas allow you to express functions of the difference or sum of two angles using trigonometric functions of those angles.
To begin with, we will give full list addition formulas, then we will prove them and analyze several illustrative examples.
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Basic addition formulas in trigonometry
There are eight basic formulas: sine of the sum and sine of the difference of two angles, cosines of the sum and difference, tangents and cotangents of the sum and difference, respectively. Below are their standard formulations and calculations.
1. The sine of the sum of two angles can be obtained as follows:
We calculate the product of the sine of the first angle and the cosine of the second;
Multiply the cosine of the first angle by the sine of the first;
Add up the resulting values.
The graphical writing of the formula looks like this: sin (α + β) = sin α · cos β + cos α · sin β
2. The sine of the difference is calculated in almost the same way, only the resulting products should not be added, but subtracted from each other. Thus, we calculate the products of the sine of the first angle by the cosine of the second and the cosine of the first angle by the sine of the second and find their difference. The formula is written like this: sin (α - β) = sin α · cos β + sin α · sin β
3. Cosine of the sum. For it, we find the products of the cosine of the first angle by the cosine of the second and the sine of the first angle by the sine of the second, respectively, and find their difference: cos (α + β) = cos α · cos β - sin α · sin β
4. Cosine of the difference: calculate the products of sines and cosines of these angles, as before, and add them. Formula: cos (α - β) = cos α cos β + sin α sin β
5. Tangent of the sum. This formula is expressed as a fraction, the numerator of which is the sum of the tangents of the required angles, and the denominator is a unit from which the product of the tangents of the desired angles is subtracted. Everything is clear from its graphical notation: t g (α + β) = t g α + t g β 1 - t g α · t g β
6. Tangent of the difference. We calculate the values of the difference and product of the tangents of these angles and proceed with them in a similar way. In the denominator we add to one, and not vice versa: t g (α - β) = t g α - t g β 1 + t g α · t g β
7. Cotangent of the sum. To calculate using this formula, we will need the product and the sum of the cotangents of these angles, which we proceed as follows: c t g (α + β) = - 1 + c t g α · c t g β c t g α + c t g β
8. Cotangent of the difference . The formula is similar to the previous one, but the numerator and denominator are minus, not plus c t g (α - β) = - 1 - c t g α · c t g β c t g α - c t g β.
You probably noticed that these formulas are similar in pairs. Using the signs ± (plus-minus) and ∓ (minus-plus), we can group them for ease of recording:
sin (α ± β) = sin α · cos β ± cos α · sin β cos (α ± β) = cos α · cos β ∓ sin α · sin β t g (α ± β) = t g α ± t g β 1 ∓ t g α · t g β c t g (α ± β) = - 1 ± c t g α · c t g β c t g α ± c t g β
Accordingly, we have one recording formula for the sum and difference of each value, just in one case we pay attention to the upper sign, in the other – to the lower one.
Definition 2
We can take any angles α and β, and the addition formulas for cosine and sine will work for them. If we can correctly determine the values of the tangents and cotangents of these angles, then the addition formulas for tangent and cotangent will also be valid for them.
Like most concepts in algebra, addition formulas can be proven. The first formula we will prove is the difference cosine formula. The rest of the evidence can then be easily deduced from it.
Let's clarify the basic concepts. We will need unit circle. It will work if we take a certain point A and rotate the angles α and β around the center (point O). Then the angle between the vectors O A 1 → and O A → 2 will be equal to (α - β) + 2 π · z or 2 π - (α - β) + 2 π · z (z is any integer). The resulting vectors form an angle that is equal to α - β or 2 π - (α - β), or it may differ from these values by an integer number of full revolutions. Take a look at the picture:
We used the reduction formulas and got the following results:
cos ((α - β) + 2 π z) = cos (α - β) cos (2 π - (α - β) + 2 π z) = cos (α - β)
Result: the cosine of the angle between the vectors O A 1 → and O A 2 → is equal to the cosine of the angle α - β, therefore, cos (O A 1 → O A 2 →) = cos (α - β).
Let us recall the definitions of sine and cosine: sine is a function of angle, equal to the ratio the leg of the opposite angle to the hypotenuse, cosine is the sine of the complementary angle. Therefore, the points A 1 And A 2 have coordinates (cos α, sin α) and (cos β, sin β).
We get the following:
O A 1 → = (cos α, sin α) and O A 2 → = (cos β, sin β)
If it is not clear, look at the coordinates of the points located at the beginning and end of the vectors.
The lengths of the vectors are equal to 1, because We have a unit circle.
Let's look at it now dot product vectors O A 1 → and O A 2 → . In coordinates it looks like this:
(O A 1 → , O A 2) → = cos α · cos β + sin α · sin β
From this we can derive the equality:
cos (α - β) = cos α cos β + sin α sin β
Thus, the difference cosine formula is proven.
Now we will prove the following formula - the cosine of the sum. This is easier because we can use the previous calculations. Let's take the representation α + β = α - (- β) . We have:
cos (α + β) = cos (α - (- β)) = = cos α cos (- β) + sin α sin (- β) = = cos α cos β + sin α sin β
This is the proof of the cosine sum formula. The last line uses the property of sine and cosine of opposite angles.
The formula for the sine of a sum can be derived from the formula for the cosine of a difference. Let's take the reduction formula for this:
of the form sin (α + β) = cos (π 2 (α + β)). So
sin (α + β) = cos (π 2 (α + β)) = cos ((π 2 - α) - β) = = cos (π 2 - α) cos β + sin (π 2 - α) sin β = = sin α cos β + cos α sin β
And here is the proof of the difference sine formula:
sin (α - β) = sin (α + (- β)) = sin α cos (- β) + cos α sin (- β) = = sin α cos β - cos α sin β
Note the use of the sine and cosine properties of opposite angles in the last calculation.
Next we need proofs of the addition formulas for tangent and cotangent. Let's remember the basic definitions (tangent is the ratio of sine to cosine, and cotangent is vice versa) and take the formulas already derived in advance. We got this:
t g (α + β) = sin (α + β) cos (α + β) = sin α cos β + cos α sin β cos α cos β - sin α sin β
We have a complex fraction. Next, we need to divide its numerator and denominator by cos α · cos β, given that cos α ≠ 0 and cos β ≠ 0, we get:
sin α · cos β + cos α · sin β cos α · cos β cos α · cos β - sin α · sin β cos α · cos β = sin α · cos β cos α · cos β + cos α · sin β cos α · cos β cos α · cos β cos α · cos β - sin α · sin β cos α · cos β
Now we reduce the fractions and get the following formula: sin α cos α + sin β cos β 1 - sin α cos α · s i n β cos β = t g α + t g β 1 - t g α · t g β.
We got t g (α + β) = t g α + t g β 1 - t g α · t g β. This is the proof of the tangent addition formula.
The next formula that we will prove is the tangent of the difference formula. Everything is clearly shown in the calculations:
t g (α - β) = t g (α + (- β)) = t g α + t g (- β) 1 - t g α t g (- β) = t g α - t g β 1 + t g α t g β
Formulas for cotangent are proved in a similar way:
c t g (α + β) = cos (α + β) sin (α + β) = cos α · cos β - sin α · sin β sin α · cos β + cos α · sin β = = cos α · cos β - sin α · sin β sin α · sin β sin α · cos β + cos α · sin β sin α · sin β = cos α · cos β sin α · sin β - 1 sin α · cos β sin α · sin β + cos α · sin β sin α · sin β = = - 1 + c t g α · c t g β c t g α + c t g β
Next:
c t g (α - β) = c t g (α + (- β)) = - 1 + c t g α c t g (- β) c t g α + c t g (- β) = - 1 - c t g α c t g β c t g α - c t g β
Cosine of the sum and difference of two angles
In this section the following two formulas will be proved:
cos (α + β) = cos α cos β - sin α sin β, (1)
cos (α - β) = cos α cos β + sin α sin β. (2)
The cosine of the sum (difference) of two angles is equal to the product of the cosines of these angles minus (plus) the product of the sines of these angles.
It will be more convenient for us to start with the proof of formula (2). For simplicity of presentation, let us first assume that the angles α And β satisfy the following conditions:
1) each of these angles is non-negative and less 2π:
0 < α <2π, 0< β < 2π;
2) α > β .
Let the positive part of the 0x axis be the common starting side of the angles α And β .
We denote the end sides of these angles by 0A and 0B, respectively. Obviously the angle α - β can be considered as the angle by which beam 0B needs to be rotated around point 0 counterclockwise so that its direction coincides with the direction of beam 0A.
On rays 0A and 0B we mark points M and N, located at a distance of 1 from the origin of coordinates 0, so that 0M = 0N = 1.
In the x0y coordinate system, point M has coordinates ( cos α, sin α), and point N is the coordinates ( cos β, sin β). Therefore, the square of the distance between them is:
d 1 2 = (cos α - cos β) 2 + (sin α - sin β) 2 = cos 2 α - 2 cos α cos β +
+ cos 2 β + sin 2 α - 2sin α sin β + sin 2 β = .
In our calculations we used the identity
sin 2 φ + cos 2 φ = 1.
Now consider another coordinate system B0C, which is obtained by rotating the 0x and 0y axes around point 0 counterclockwise by an angle β .
In this coordinate system, point M has coordinates (cos ( α - β ), sin ( α - β )), and the point N is coordinates (1,0). Therefore, the square of the distance between them is:
d 2 2 = 2 + 2 = cos 2 (α - β) - 2 cos (α - β) + 1 +
+ sin 2 (α - β) = 2 .
But the distance between points M and N does not depend on which coordinate system we are considering these points in relation to. That's why
d 1 2 = d 2 2
2 (1 - cos α cos β - sin α sin β) = 2 .
This is where formula (2) follows.
Now we should remember the two restrictions that we imposed for simplicity of presentation on the angles α And β .
The requirement that each of the corners α And β was non-negative, not really significant. After all, to any of these angles you can add an angle that is a multiple of 2, which will not affect the validity of formula (2). In the same way, from each of these angles you can subtract an angle that is a multiple of 2π. Therefore we can assume that 0 < α < 2π, 0 < β < 2π.
The condition also turns out to be insignificant α > β . Indeed, if α < β , That β >α ; therefore, given the parity of the function cos X , we get:
cos (α - β) = cos (β - α) = cos β cos α + sin β sin α,
which essentially coincides with formula (2). So the formula
cos (α - β) = cos α cos β + sin α sin β
true for all angles α And β . In particular, replacing in it β on - β and given that the function cosX is even, and the function sinX odd, we get:
cos (α + β) = cos [α - (- β)] = cos α cos (-β) + sin α sin (-β) =
= cos α cos β - sin α sin β,
which proves formula (1).
So, formulas (1) and (2) are proven.
Examples.
1) cos 75° = cos (30° + 45°) = cos 30° cos 45°-sin 30°-sin 45° =
2) cos 15° = cos (45° - 30°) = cos 45° cos 30° + sin 45° sin 30° =
Exercises
1 . Calculate without using trigonometric tables:
a) cos 17° cos 43° - sin 17° sin 43°;
b) sin 3° sin 42° - cos 39° cos 42°;
c) cos 29° cos 74° + sin 29° sin 74°;
d) sin 97° sin 37° + cos 37° cos 97°;
e) cos 3π / 8 cos π / 8 + sin 3π / 8 sin π / 8 ;
e) sin 3π / 5 sin 7π / 5 - cos 3π / 5 cos 7π / 5 .
2.Simplify expressions:
a). cos( α + π/3 ) + cos(π/3 - α ) .
b). cos (36° + α ) cos (24° - α ) + sin (36° + α ) sin ( α - 24°).
V). sin(π/4 - α ) sin (π / 4 + α ) - cos (π / 4 + α ) cos (π / 4 - α )
d) cos 2 α + tg α sin 2 α .
3 . Calculate :
a) cos(α - β), If
cos α = - 2 / 5 , sin β = - 5 / 13 ;
90°< α < 180°, 180° < β < 270°;
b) cos ( α + π / 6), if cos α = 0,6;
3π/2< α < 2π.
4 . Find cos(α + β) and cos (α - β) ,if it is known that sin α = 7 / 25, cos β = - 5 / 13 and both angles ( α And β ) end in the same quarter.
5 .Calculate:
A). cos [ arcsin 1 / 3 + arccos 2 / 3 ]
b). cos [ arcsin 1 / 3 - arccos (- 2 / 3)] .
V). cos [ arctan 1 / 2 + arccos (- 2) ]
Formulas for the sum and difference of sines and cosines for two angles α and β allow us to move from the sum of these angles to the product of angles α + β 2 and α - β 2. Let us immediately note that you should not confuse the formulas for the sum and difference of sines and cosines with the formulas for sines and cosines of the sum and difference. Below we list these formulas, give their derivation and show examples of application for specific tasks.
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Formulas for the sum and difference of sines and cosines
Let's write down what the sum and difference formulas look like for sines and cosines
Sum and difference formulas for sines
sin α + sin β = 2 sin α + β 2 cos α - β 2 sin α - sin β = 2 sin α - β 2 cos α + β 2
Sum and difference formulas for cosines
cos α + cos β = 2 cos α + β 2 cos α - β 2 cos α - cos β = - 2 sin α + β 2 cos α - β 2 , cos α - cos β = 2 sin α + β 2 · β - α 2
These formulas are valid for any angles α and β. The angles α + β 2 and α - β 2 are called the half-sum and half-difference of the angles alpha and beta, respectively. Let us give the formulation for each formula.
Definitions of formulas for sums and differences of sines and cosines
Sum of sines of two angles is equal to twice the product of the sine of the half-sum of these angles and the cosine of the half-difference.
Difference of sines of two angles is equal to twice the product of the sine of the half-difference of these angles and the cosine of the half-sum.
Sum of cosines of two angles is equal to twice the product of the cosine of the half-sum and the cosine of the half-difference of these angles.
Difference of cosines of two angles is equal to twice the product of the sine of the half-sum and the cosine of the half-difference of these angles, taken with a negative sign.
Deriving formulas for the sum and difference of sines and cosines
To derive formulas for the sum and difference of the sine and cosine of two angles, addition formulas are used. Let's list them below
sin (α + β) = sin α · cos β + cos α · sin β sin (α - β) = sin α · cos β - cos α · sin β cos (α + β) = cos α · cos β - sin α sin β cos (α - β) = cos α cos β + sin α sin β
Let’s also imagine the angles themselves as a sum of half-sums and half-differences.
α = α + β 2 + α - β 2 = α 2 + β 2 + α 2 - β 2 β = α + β 2 - α - β 2 = α 2 + β 2 - α 2 + β 2
We proceed directly to the derivation of the sum and difference formulas for sin and cos.
Derivation of the formula for the sum of sines
In the sum sin α + sin β, we replace α and β with the expressions for these angles given above. We get
sin α + sin β = sin α + β 2 + α - β 2 + sin α + β 2 - α - β 2
Now we apply the addition formula to the first expression, and to the second - the formula for the sine of angle differences (see formulas above)
sin α + β 2 + α - β 2 = sin α + β 2 cos α - β 2 + cos α + β 2 sin α - β 2 sin α + β 2 - α - β 2 = sin α + β 2 cos α - β 2 - cos α + β 2 sin α - β 2 sin α + β 2 + α - β 2 + sin α + β 2 - α - β 2 = sin α + β 2 cos α - β 2 + cos α + β 2 sin α - β 2 + sin α + β 2 cos α - β 2 - cos α + β 2 sin α - β 2 Open the brackets, add similar terms and get the required formula
sin α + β 2 cos α - β 2 + cos α + β 2 sin α - β 2 + sin α + β 2 cos α - β 2 - cos α + β 2 sin α - β 2 = = 2 sin α + β 2 cos α - β 2
The steps to derive the remaining formulas are similar.
Derivation of the formula for the difference of sines
sin α - sin β = sin α + β 2 + α - β 2 - sin α + β 2 - α - β 2 sin α + β 2 + α - β 2 - sin α + β 2 - α - β 2 = sin α + β 2 cos α - β 2 + cos α + β 2 sin α - β 2 - sin α + β 2 cos α - β 2 - cos α + β 2 sin α - β 2 = = 2 sin α - β 2 cos α + β 2
Derivation of the formula for the sum of cosines
cos α + cos β = cos α + β 2 + α - β 2 + cos α + β 2 - α - β 2 cos α + β 2 + α - β 2 + cos α + β 2 - α - β 2 = cos α + β 2 cos α - β 2 - sin α + β 2 sin α - β 2 + cos α + β 2 cos α - β 2 + sin α + β 2 sin α - β 2 = = 2 cos α + β 2 cos α - β 2
Derivation of the formula for the difference of cosines
cos α - cos β = cos α + β 2 + α - β 2 - cos α + β 2 - α - β 2 cos α + β 2 + α - β 2 - cos α + β 2 - α - β 2 = cos α + β 2 cos α - β 2 - sin α + β 2 sin α - β 2 - cos α + β 2 cos α - β 2 + sin α + β 2 sin α - β 2 = = - 2 sin α + β 2 sin α - β 2
Examples of solving practical problems
First, let's check one of the formulas by substituting specific angle values into it. Let α = π 2, β = π 6. Let us calculate the value of the sum of the sines of these angles. First, we will use the table of basic values of trigonometric functions, and then we will apply the formula for the sum of sines.
Example 1. Checking the formula for the sum of sines of two angles
α = π 2, β = π 6 sin π 2 + sin π 6 = 1 + 1 2 = 3 2 sin π 2 + sin π 6 = 2 sin π 2 + π 6 2 cos π 2 - π 6 2 = 2 sin π 3 cos π 6 = 2 3 2 3 2 = 3 2
Let us now consider the case when the angle values differ from the basic values presented in the table. Let α = 165°, β = 75°. Let's calculate the difference between the sines of these angles.
Example 2. Application of the difference of sines formula
α = 165 °, β = 75 ° sin α - sin β = sin 165 ° - sin 75 ° sin 165 - sin 75 = 2 sin 165 ° - sin 75 ° 2 cos 165 ° + sin 75 ° 2 = = 2 sin 45° cos 120° = 2 2 2 - 1 2 = 2 2
Using the formulas for the sum and difference of sines and cosines, you can move from the sum or difference to the product of trigonometric functions. Often these formulas are called formulas for moving from a sum to a product. Formulas for the sum and difference of sines and cosines are widely used in solving trigonometric equations and when converting trigonometric expressions.
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Reference data for tangent (tg x) and cotangent (ctg x). Geometric definition, properties, graphs, formulas. Table of tangents and cotangents, derivatives, integrals, series expansions. Expressions through complex variables. Connection with hyperbolic functions.
Geometric definition
|BD| - length of the arc of a circle with center at point A.
α is the angle expressed in radians.
Tangent ( tan α) - This trigonometric function, depending on the angle α between the hypotenuse and the leg right triangle, equal to the ratio of the length of the opposite side |BC| to the length of the adjacent leg |AB| .
Cotangent ( ctg α) 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, tangent is denoted as follows:
.
;
;
.
Graph of the tangent function, y = tan x
Cotangent
Where n- whole.
In Western literature, cotangent is denoted as follows:
.
The following notations are also accepted:
;
;
.
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 tangent and cotangent functions are odd.
Areas of definition and values, increasing, decreasing
The tangent and cotangent functions are continuous in their domain of definition (see proof of continuity). The main properties of tangent and cotangent are presented in the table ( n- whole).
| y = tg x | y = ctg x | |
| Scope and continuity | ||
| Range of values | -∞ < y < +∞ | -∞ < y < +∞ |
| Increasing | - | |
| Descending | - | |
| Extremes | - | - |
| Zeros, y = 0 | ||
| Intercept points with the ordinate axis, x = 0 | y = 0 | - |
Formulas
Expressions using sine and cosine
;
;
;
;
;
Formulas for tangent and cotangent from sum and difference
The remaining formulas are easy to obtain, for example
Product of tangents
Formula for the sum and difference of tangents
This table presents the values of tangents and cotangents for certain values of the argument. 
Expressions using complex numbers
Expressions through hyperbolic functions
;
;
Derivatives
; .
.
Derivative of the nth order with respect to the variable x of the function:
.
Deriving formulas for tangent > > > ; for cotangent > > >
Integrals
Series expansions
To obtain the expansion of the tangent in powers of x, you need to take several terms of the expansion in power series for functions sin x And cos x and divide these polynomials by each other, . This produces the following formulas.
At .
at .
Where Bn- Bernoulli numbers. They are determined either from the recurrence relation:
;
;
Where .
Or according to Laplace's formula: 
Inverse functions
Inverse functions to tangent and cotangent are arctangent and arccotangent, respectively.
Arctangent, arctg
, Where n- whole.
Arccotangent, arcctg
, Where n- whole.
Used literature:
I.N. Bronstein, K.A. Semendyaev, Handbook of mathematics for engineers and college students, “Lan”, 2009.
G. Korn, Handbook of Mathematics for Scientists and Engineers, 2012.