In trigonometry, many formulas are easier to deduce than to memorize. The cosine of a double angle is a wonderful formula! It allows you to get the reduction formulas and half angle formulas.
So, we need the cosine of the double angle and the trigonometric unit:
They are even similar: in the formula of the cosine of a double angle - the difference between the squares of the cosine and sine, and in the trigonometric unit - their sum. If we express the cosine from the trigonometric unit:
and substitute it into the cosine of the double angle, we get:
This is another formula for the cosine of a double angle:
This formula is the key to getting the reduction formula:
So, the formula for lowering the degree of the sine is:
If in it the angle alpha is replaced by a half angle alpha in half, and the double angle two alpha is replaced by the angle alpha, then we get the formula for the half angle for the sine:
Now, from the trigonometric unit, we express the sine:
Substitute this expression into the formula for the cosine of a double angle:
We got another formula for the cosine of a double angle:
This formula is the key to finding the cosine reduction and half angle formula for cosine.
Thus, the formula for lowering the degree of cosine is:
If we replace α by α/2 in it, and 2α by α, then we get the formula for the half argument for the cosine:
Since tangent is the ratio of sine to cosine, the formula for tangent is:
Cotangent is the ratio of cosine to sine. So the formula for the cotangent is:
Of course, in the process of simplifying trigonometric expressions, there is no point in deriving half-angle formulas or lowering the degree every time. It is much easier to put a sheet of formulas in front of you. And simplification will advance faster, and visual memory will turn on for memorization.
But it is still worth deriving these formulas several times. Then you will be absolutely sure that during the exam, when there is no way to use a cheat sheet, you can easily get them if the need arises.
The sine values are in the range [-1; 1], i.e. -1 ≤ sin α ≤ 1. Therefore, if |a| > 1, then the equation sin x = a has no roots. For example, the equation sin x = 2 has no roots.
Let's turn to some tasks.
Solve the equation sin x = 1/2.
Decision.
Note that sin x is the ordinate of the point of the unit circle, which is obtained as a result of the rotation of the point Р (1; 0) by the angle x around the origin.
An ordinate equal to ½ is present at two points of the circle M 1 and M 2.
Since 1/2 \u003d sin π / 6, then the point M 1 is obtained from the point P (1; 0) by turning through the angle x 1 \u003d π / 6, as well as through the angles x \u003d π / 6 + 2πk, where k \u003d +/-1, +/-2, …
The point M 2 is obtained from the point P (1; 0) as a result of turning through the angle x 2 = 5π/6, as well as through the angles x = 5π/6 + 2πk, where k = +/-1, +/-2, ... , i.e. at angles x = π – π/6 + 2πk, where k = +/-1, +/-2, ….
So, all the roots of the equation sin x = 1/2 can be found by the formulas x = π/6 + 2πk, x = π - π/6 + 2πk, where k € Z.
These formulas can be combined into one: x \u003d (-1) n π / 6 + πn, where n € Z (1).
Indeed, if n is an even number, i.e. n = 2k, then from formula (1) we obtain х = π/6 + 2πk, and if n is an odd number, i.e. n = 2k + 1, then from formula (1) we obtain х = π – π/6 + 2πk.
Answer. x \u003d (-1) n π / 6 + πn, where n € Z.
Solve the equation sin x = -1/2.
Decision.
The ordinate -1/2 have two points of the unit circle M 1 and M 2, where x 1 = -π/6, x 2 = -5π/6. Therefore, all the roots of the equation sin x = -1/2 can be found by the formulas x = -π/6 + 2πk, x = -5π/6 + 2πk, k ∈ Z.
We can combine these formulas into one: x \u003d (-1) n (-π / 6) + πn, n € Z (2).
Indeed, if n = 2k, then by formula (2) we obtain x = -π/6 + 2πk, and if n = 2k – 1, then by formula (2) we find x = -5π/6 + 2πk.
Answer. x \u003d (-1) n (-π / 6) + πn, n € Z.
Thus, each of the equations sin x = 1/2 and sin x = -1/2 has an infinite number of roots.
On the segment -π/2 ≤ x ≤ π/2, each of these equations has only one root:
x 1 \u003d π / 6 - the root of the equation sin x \u003d 1/2 and x 1 \u003d -π / 6 - the root of the equation sin x \u003d -1/2.
The number π/6 is called the arcsine of the number 1/2 and is written: arcsin 1/2 = π/6; the number -π/6 is called the arcsine of the number -1/2 and they write: arcsin (-1/2) = -π/6.
In general, the equation sin x \u003d a, where -1 ≤ a ≤ 1, on the segment -π / 2 ≤ x ≤ π / 2 has only one root. If a ≥ 0, then the root is enclosed in the interval; if a< 0, то в промежутке [-π/2; 0). Этот корень называют арксинусом числа а и обозначают arcsin а.
Thus, the arcsine of the number a € [–1; 1] such a number is called a € [–π/2; π/2], whose sine is a.
arcsin a = α if sin α = a and -π/2 ≤ x ≤ π/2 (3).
For example, arcsin √2/2 = π/4, since sin π/4 = √2/2 and – π/2 ≤ π/4 ≤ π/2;
arcsin (-√3/2) = -π/3, since sin (-π/3) = -√3/2 and – π/2 ≤ 
– π/3 ≤ π/2.
Similarly to how it was done when solving problems 1 and 2, it can be shown that the roots of the equation sin x = a, where |a| ≤ 1 are expressed by the formula
x \u003d (-1) n arcsin a + πn, n € Z (4).
We can also prove that for any a € [-1; 1] the formula arcsin (-a) = -arcsin a is valid.
From formula (4) it follows that the roots of the equation
sin x \u003d a for a \u003d 0, a \u003d 1, a \u003d -1 can be found using simpler formulas:
sin x \u003d 0 x \u003d πn, n € Z (5)
sin x \u003d 1 x \u003d π / 2 + 2πn, n € Z (6)
sin x \u003d -1 x \u003d -π / 2 + 2πn, n € Z (7)
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|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.
The simplest trigonometric identities
The quotient of dividing the sine of the angle alpha by the cosine of the same angle is equal to the tangent of this angle (Formula 1). See also the proof of the correctness of the transformation of the simplest trigonometric identities.
The quotient of dividing the cosine of the angle alpha by the sine of the same angle is equal to the cotangent of the same angle (Formula 2)
The secant of an angle is equal to one divided by the cosine of the same angle (Formula 3)
The sum of the squares of the sine and cosine of the same angle is equal to one (Formula 4). see also the proof of the sum of squares of cosine and sine.
The sum of the unit and the tangent of the angle is equal to the ratio of the unit to the square of the cosine of this angle (Formula 5)
The unit plus the cotangent of the angle is equal to the quotient of dividing the unit by the sine square of this angle (Formula 6)
The product of the tangent and the cotangent of the same angle is equal to one (Formula 7).
Converting negative angles of trigonometric functions (even and odd)
In order to get rid of the negative value of the degree measure of the angle when calculating the sine, cosine or tangent, you can use the following trigonometric transformations (identities) based on the principles of even or odd trigonometric functions.
As seen, cosine and secant is even function, sine, tangent and cotangent are odd functions.
The sine of a negative angle is equal to the negative value of the sine of that same positive angle (minus the sine of alpha).
The cosine "minus alpha" will give the same value as the cosine of the angle alpha.
Tangent minus alpha is equal to minus tangent alpha.
Double angle reduction formulas (sine, cosine, tangent and cotangent of a double angle)
If you need to divide the angle in half, or vice versa, go from a double angle to a single angle, you can use the following trigonometric identities:
Double Angle Conversion (double angle sine, double angle cosine and double angle tangent) into a single one occurs according to the following rules:
Sine of a double angle is equal to twice the product of the sine and the cosine of a single angle
Cosine of a double angle is equal to the difference between the square of the cosine of a single angle and the square of the sine of this angle
Cosine of a double angle equal to twice the square of the cosine of a single angle minus one
Cosine of a double angle equals one minus the double sine square of a single angle
Double angle tangent is equal to a fraction whose numerator is twice the tangent of a single angle, and whose denominator is equal to one minus the tangent of the square of a single angle.
Double angle cotangent is equal to a fraction whose numerator is the square of the cotangent of a single angle minus one, and the denominator is equal to twice the cotangent of a single angle
Universal Trigonometric Substitution Formulas
The conversion formulas below can be useful when you need to divide the argument of the trigonometric function (sin α, cos α, tg α) by two and bring the expression to the value of half the angle. From the value of α we get α/2 .These formulas are called formulas of the universal trigonometric substitution. Their value lies in the fact that the trigonometric expression with their help is reduced to the expression of the tangent of half an angle, regardless of what trigonometric functions (sin cos tg ctg) were originally in the expression. After that, the equation with the tangent of half an angle is much easier to solve.
Trigonometric half-angle transformation identities
The following are the formulas for the trigonometric conversion of half the value of an angle to its integer value.The value of the argument of the trigonometric function α/2 is reduced to the value of the argument of the trigonometric function α.
Trigonometric formulas for adding angles
cos (α - β) = cos α cos β + sin α sin β
sin (α + β) = sin α cos β + sin β cos α
sin (α - β) = sin α cos β - sin β cos α
cos (α + β) = cos α cos β - sin α sin β
Tangent and cotangent of the sum of angles alpha and beta can be converted according to the following rules for converting trigonometric functions:
Tangent of sum of angles is equal to a fraction whose numerator is the sum of the tangent of the first and the tangent of the second angle, and the denominator is one minus the product of the tangent of the first angle and the tangent of the second angle.
Angle difference tangent is equal to a fraction, the numerator of which is equal to the difference between the tangent of the reduced angle and the tangent of the angle to be subtracted, and the denominator is one plus the product of the tangents of these angles.
Cotangent of sum of angles is equal to a fraction whose numerator is equal to the product of the cotangents of these angles plus one, and the denominator is equal to the difference between the cotangent of the second angle and the cotangent of the first angle.
Cotangent of angle difference is equal to a fraction whose numerator is the product of the cotangents of these angles minus one, and the denominator is equal to the sum of the cotangents of these angles.
These trigonometric identities are convenient to use when you need to calculate, for example, the tangent of 105 degrees (tg 105). If it is represented as tg (45 + 60), then you can use the given identical transformations of the tangent of the sum of the angles, after which you simply substitute the tabular values of the tangent of 45 and the tangent of 60 degrees.
Formulas for converting the sum or difference of trigonometric functions
Expressions representing the sum of the form sin α + sin β can be converted using the following formulas:
Triple angle formulas - convert sin3α cos3α tg3α to sinα cosα tgα
Sometimes it is necessary to convert the triple value of the angle so that the angle α becomes the argument of the trigonometric function instead of 3α.In this case, you can use the formulas (identities) for the transformation of the triple angle:
Formulas for transforming the product of trigonometric functions
If it becomes necessary to convert the product of sines of different angles of cosines of different angles, or even the product of sine and cosine, then you can use the following trigonometric identities:
In this case, the product of the sine, cosine or tangent functions of different angles will be converted to a sum or difference.
Formulas for reducing trigonometric functions
You need to use the cast table as follows. In the line, select the function that interests us. The column is an angle. For example, the sine of the angle (α+90) at the intersection of the first row and the first column, we find out that sin (α+90) = cos α .
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