Where the tasks for solving a right-angled triangle were considered, I promised to present a technique for memorizing the definitions of sine and cosine. Using it, you will always quickly remember which leg belongs to the hypotenuse (adjacent or opposite). I decided not to put it off indefinitely, the necessary material is below, please read it 😉

The fact is that I have repeatedly observed how students in grades 10-11 have difficulty remembering these definitions. They remember very well that the leg refers to the hypotenuse, but which one- forget and confused. The price of a mistake, as you know in the exam, is a lost score.

The information that I will present directly to mathematics has nothing to do. It is associated with figurative thinking, and with the methods of verbal-logical connection. That's right, I myself, once and for all remembereddefinition data. If you still forget them, then with the help of the presented techniques it is always easy to remember.

Let me remind you the definitions of sine and cosine in a right triangle:

Cosine acute angle in a right triangle is the ratio of the adjacent leg to the hypotenuse:

Sinus acute angle in a right triangle is the ratio of the opposite leg to the hypotenuse:

So, what associations does the word cosine evoke in you?

Probably everyone has their ownRemember the link:

Thus, you will immediately have an expression in your memory -

«… ratio of ADJACENT leg to hypotenuse».

The problem with the definition of cosine is solved.

If you need to remember the definition of the sine in a right triangle, then remembering the definition of the cosine, you can easily establish that the sine of an acute angle in a right triangle is the ratio of the opposite leg to the hypotenuse. After all, there are only two legs, if the adjacent leg is “occupied” by the cosine, then only the opposite side remains for the sine.

What about tangent and cotangent? Same confusion. Students know that this is the ratio of legs, but the problem is to remember which one refers to which - either opposite to adjacent, or vice versa.

Definitions:

Tangent an acute angle in a right triangle is the ratio of the opposite leg to the adjacent one:

Cotangent acute angle in a right triangle is the ratio of the adjacent leg to the opposite:

How to remember? There are two ways. One also uses a verbal-logical connection, the other - a mathematical one.

MATHEMATICAL METHOD

There is such a definition - the tangent of an acute angle is the ratio of the sine of an angle to its cosine:

* Remembering the formula, you can always determine that the tangent of an acute angle in a right triangle is the ratio of the opposite leg to the adjacent one.

Likewise.The cotangent of an acute angle is the ratio of the cosine of an angle to its sine:

So! Remembering these formulas, you can always determine that:

- the tangent of an acute angle in a right triangle is the ratio of the opposite leg to the adjacent

- the cotangent of an acute angle in a right triangle is the ratio of the adjacent leg to the opposite one.

VERBAL-LOGICAL METHOD

About tangent. Remember the link:

That is, if you need to remember the definition of the tangent, using this logical connection, you can easily remember what it is

"... the ratio of the opposite leg to the adjacent"

If it comes to cotangent, then remembering the definition of tangent, you can easily voice the definition of cotangent -

"... the ratio of the adjacent leg to the opposite"

There is an interesting technique for memorizing tangent and cotangent on the site " Mathematical tandem " , look.

METHOD UNIVERSAL

You can just grind.But as practice shows, thanks to verbal-logical connections, a person remembers information for a long time, and not only mathematical.

I hope the material was useful to you.

Sincerely, Alexander Krutitskikh

P.S: I would be grateful if you tell about the site in social networks.

Trigonometry, as a science, originated in the Ancient East. The first trigonometric ratios were developed by astronomers to create an accurate calendar and orientate by the stars. These calculations related to spherical trigonometry, while in the school course they study the ratio of the sides and angle of a flat triangle.

Trigonometry is a branch of mathematics dealing with the properties of trigonometric functions and the relationship between the sides and angles of triangles.

During the heyday of culture and science in the 1st millennium AD, knowledge spread from the Ancient East to Greece. But the main discoveries of trigonometry are the merit of the men of the Arab Caliphate. In particular, the Turkmen scientist al-Marazvi introduced such functions as tangent and cotangent, compiled the first tables of values ​​for sines, tangents and cotangents. The concept of sine and cosine was introduced by Indian scientists. A lot of attention is devoted to trigonometry in the works of such great figures of antiquity as Euclid, Archimedes and Eratosthenes.

Basic quantities of trigonometry

The basic trigonometric functions of a numerical argument are sine, cosine, tangent, and cotangent. Each of them has its own graph: sine, cosine, tangent and cotangent.

The formulas for calculating the values ​​of these quantities are based on the Pythagorean theorem. It is better known to schoolchildren in the formulation: “Pythagorean pants, equal in all directions,” since the proof is given on the example of an isosceles right triangle.

Sine, cosine and other dependencies establish a relationship between acute angles and sides of any right triangle. We give formulas for calculating these quantities for angle A and trace the relationship of trigonometric functions:

As you can see, tg and ctg are inverse functions. If we represent the leg a as the product of sin A and the hypotenuse c, and the leg b as cos A * c, then we get the following formulas for the tangent and cotangent:

trigonometric circle

Graphically, the ratio of the mentioned quantities can be represented as follows:

The circle, in this case, represents all possible values ​​of the angle α - from 0° to 360°. As can be seen from the figure, each function takes a negative or positive value depending on the angle. For example, sin α will be with a “+” sign if α belongs to the I and II quarters of the circle, that is, it is in the range from 0 ° to 180 °. With α from 180° to 360° (III and IV quarters), sin α can only be a negative value.

Let's try to build trigonometric tables for specific angles and find out the meaning of the quantities.

The values ​​of α equal to 30°, 45°, 60°, 90°, 180° and so on are called special cases. The values ​​of trigonometric functions for them are calculated and presented in the form of special tables.

These angles were not chosen by chance. The designation π in the tables is for radians. Rad is the angle at which the length of a circular arc corresponds to its radius. This value was introduced in order to establish a universal relationship; when calculating in radians, the actual length of the radius in cm does not matter.

The angles in the tables for trigonometric functions correspond to radian values:

So, it is not difficult to guess that 2π is a full circle or 360°.

Properties of trigonometric functions: sine and cosine

In order to consider and compare the basic properties of sine and cosine, tangent and cotangent, it is necessary to draw their functions. This can be done in the form of a curve located in a two-dimensional coordinate system.

Consider a comparative table of properties for a sine wave and a cosine wave:

sinusoid	cosine wave
y = sin x	y = cos x
ODZ [-1; 1]	ODZ [-1; 1]
sin x = 0, for x = πk, where k ϵ Z	cos x = 0, for x = π/2 + πk, where k ϵ Z
sin x = 1, for x = π/2 + 2πk, where k ϵ Z	cos x = 1, for x = 2πk, where k ϵ Z
sin x = - 1, at x = 3π/2 + 2πk, where k ϵ Z	cos x = - 1, for x = π + 2πk, where k ϵ Z
sin (-x) = - sin x, i.e. odd function	cos (-x) = cos x, i.e. the function is even
	the function is periodic, the smallest period is 2π
sin x › 0, with x belonging to quarters I and II or from 0° to 180° (2πk, π + 2πk)	cos x › 0, with x belonging to quarters I and IV or from 270° to 90° (- π/2 + 2πk, π/2 + 2πk)
sin x ‹ 0, with x belonging to quarters III and IV or from 180° to 360° (π + 2πk, 2π + 2πk)	cos x ‹ 0, with x belonging to quarters II and III or from 90° to 270° (π/2 + 2πk, 3π/2 + 2πk)
increases on the interval [- π/2 + 2πk, π/2 + 2πk]	increases on the interval [-π + 2πk, 2πk]
decreases on the intervals [ π/2 + 2πk, 3π/2 + 2πk]	decreases in intervals
derivative (sin x)' = cos x	derivative (cos x)’ = - sin x

Determining whether a function is even or not is very simple. It is enough to imagine a trigonometric circle with signs of trigonometric quantities and mentally “fold” the graph relative to the OX axis. If the signs are the same, the function is even; otherwise, it is odd.

The introduction of radians and the enumeration of the main properties of the sinusoid and cosine wave allow us to bring the following pattern:

It is very easy to verify the correctness of the formula. For example, for x = π/2, the sine is equal to 1, as is the cosine of x = 0. Checking can be done by looking at tables or by tracing function curves for given values.

Properties of tangentoid and cotangentoid

The graphs of the tangent and cotangent functions differ significantly from the sinusoid and cosine wave. The values ​​tg and ctg are inverse to each other.

1. Y = tgx.
2. The tangent tends to the values ​​of y at x = π/2 + πk, but never reaches them.
3. The smallest positive period of the tangentoid is π.
4. Tg (- x) \u003d - tg x, i.e., the function is odd.
5. Tg x = 0, for x = πk.
6. The function is increasing.
7. Tg x › 0, for x ϵ (πk, π/2 + πk).
8. Tg x ‹ 0, for x ϵ (— π/2 + πk, πk).
9. Derivative (tg x)' = 1/cos 2 ⁡x .

Consider the graphical representation of the cotangentoid below in the text.

The main properties of the cotangentoid:

1. Y = ctgx.
2. Unlike the sine and cosine functions, in the tangentoid Y can take on the values ​​of the set of all real numbers.
3. The cotangentoid tends to the values ​​of y at x = πk, but never reaches them.
4. The smallest positive period of the cotangentoid is π.
5. Ctg (- x) \u003d - ctg x, i.e., the function is odd.
6. Ctg x = 0, for x = π/2 + πk.
7. The function is decreasing.
8. Ctg x › 0, for x ϵ (πk, π/2 + πk).
9. Ctg x ‹ 0, for x ϵ (π/2 + πk, πk).
10. Derivative (ctg x)' = - 1/sin 2 ⁡x Fix

Table of values ​​of trigonometric functions

Note. This table of values ​​for trigonometric functions uses the √ sign to denote the square root. To denote a fraction - the symbol "/".

see also useful materials:

For determining the value of a trigonometric function, find it at the intersection of the line indicating the trigonometric function. For example, a sine of 30 degrees - we are looking for a column with the heading sin (sine) and we find the intersection of this column of the table with the line "30 degrees", at their intersection we read the result - one second. Similarly, we find cosine 60 degrees, sine 60 degrees (once again, at the intersection of the sin (sine) column and the 60 degree row, we find the value sin 60 = √3/2), etc. In the same way, the values ​​of sines, cosines and tangents of other "popular" angles are found.

Sine of pi, cosine of pi, tangent of pi and other angles in radians

The table of cosines, sines and tangents below is also suitable for finding the value of trigonometric functions whose argument is given in radians. To do this, use the second column of angle values. Thanks to this, you can convert the value of popular angles from degrees to radians. For example, let's find the 60 degree angle in the first line and read its value in radians under it. 60 degrees is equal to π/3 radians.

The number pi uniquely expresses the dependence of the circumference of a circle on the degree measure of the angle. So pi radians equals 180 degrees.

Any number expressed in terms of pi (radian) can be easily converted to degrees by replacing the number pi (π) with 180.

Examples:
1. sine pi.
sin π = sin 180 = 0
thus, the sine of pi is the same as the sine of 180 degrees and is equal to zero.

2. cosine pi.
cos π = cos 180 = -1
thus, the cosine of pi is the same as the cosine of 180 degrees and is equal to minus one.

3. Tangent pi
tg π = tg 180 = 0
thus, the tangent of pi is the same as the tangent of 180 degrees and is equal to zero.

Table of sine, cosine, tangent values ​​for angles 0 - 360 degrees (frequent values)

angle α (degrees)	angle α in radians (via pi)	sin (sinus)	cos (cosine)	tg (tangent)	ctg (cotangent)	sec (secant)	cause (cosecant)
0	0	0	1	0	-	1	-
15	π/12			2 - √3	2 + √3
30	π/6	1/2	√3/2	1/√3	√3	2/√3	2
45	π/4	√2/2	√2/2	1	1	√2	√2
60	π/3	√3/2	1/2	√3	1/√3	2	2/√3
75	5π/12			2 + √3	2 - √3
90	π/2	1	0	-	0	-	1
105	7π/12		-	- 2 - √3	√3 - 2
120	2π/3	√3/2	-1/2	-√3	-√3/3
135	3π/4	√2/2	-√2/2	-1	-1	-√2	√2
150	5π/6	1/2	-√3/2	-√3/3	-√3
180	π	0	-1	0	-	-1	-
210	7π/6	-1/2	-√3/2	√3/3	√3
240	4π/3	-√3/2	-1/2	√3	√3/3
270	3π/2	-1	0	-	0	-	-1
360	2π	0	1	0	-	1	-

If in the table of values ​​of trigonometric functions, instead of the value of the function, a dash is indicated (tangent (tg) 90 degrees, cotangent (ctg) 180 degrees), then for a given value of the degree measure of the angle, the function does not have a definite value. If there is no dash, the cell is empty, so we have not yet entered the desired value. We are interested in what requests users come to us for and supplement the table with new values, despite the fact that the current data on the values ​​of cosines, sines and tangents of the most common angle values ​​is enough to solve most problems.

Table of values ​​of trigonometric functions sin, cos, tg for the most popular angles
0, 15, 30, 45, 60, 90 ... 360 degrees
(numerical values ​​"as per Bradis tables")

angle value α (degrees)	value of angle α in radians	sin (sine)	cos (cosine)	tg (tangent)	ctg (cotangent)
0	0
15		0,2588	0,9659	0,2679
30		0,5000		0,5774
45		0,7071
		0,7660
60		0,8660	0,5000	1,7321
	7π/18

The concepts of sine, cosine, tangent and cotangent are the main categories of trigonometry - a branch of mathematics, and are inextricably linked with the definition of an angle. Possession of this mathematical science requires memorization and understanding of formulas and theorems, as well as developed spatial thinking. That is why trigonometric calculations often cause difficulties for schoolchildren and students. To overcome them, you should become more familiar with trigonometric functions and formulas.

Concepts in trigonometry

To understand the basic concepts of trigonometry, you must first decide what a right triangle and an angle in a circle are, and why all basic trigonometric calculations are associated with them. A triangle in which one of the angles is 90 degrees is a right triangle. Historically, this figure was often used by people in architecture, navigation, art, astronomy. Accordingly, studying and analyzing the properties of this figure, people came to the calculation of the corresponding ratios of its parameters.

The main categories associated with right triangles are the hypotenuse and the legs. The hypotenuse is the side of a triangle that is opposite the right angle. The legs, respectively, are the other two sides. The sum of the angles of any triangle is always 180 degrees.

Spherical trigonometry is a section of trigonometry that is not studied at school, but in applied sciences such as astronomy and geodesy, scientists use it. A feature of a triangle in spherical trigonometry is that it always has a sum of angles greater than 180 degrees.

Angles of a triangle

In a right triangle, the sine of an angle is the ratio of the leg opposite the desired angle to the hypotenuse of the triangle. Accordingly, the cosine is the ratio of the adjacent leg and the hypotenuse. Both of these values ​​always have a value less than one, since the hypotenuse is always longer than the leg.

The tangent of an angle is a value equal to the ratio of the opposite leg to the adjacent leg of the desired angle, or sine to cosine. The cotangent, in turn, is the ratio of the adjacent leg of the desired angle to the opposite cactet. The cotangent of an angle can also be obtained by dividing the unit by the value of the tangent.

unit circle

A unit circle in geometry is a circle whose radius is equal to one. Such a circle is constructed in the Cartesian coordinate system, with the center of the circle coinciding with the point of origin, and the initial position of the radius vector is determined by the positive direction of the X axis (abscissa axis). Each point of the circle has two coordinates: XX and YY, that is, the coordinates of the abscissa and ordinate. Selecting any point on the circle in the XX plane, and lowering the perpendicular from it to the abscissa axis, we get a right triangle formed by a radius to the selected point (let us denote it by the letter C), a perpendicular drawn to the X axis (the intersection point is denoted by the letter G), and a segment the abscissa axis between the origin (the point is denoted by the letter A) and the intersection point G. The resulting triangle ACG is a right triangle inscribed in a circle, where AG is the hypotenuse, and AC and GC are the legs. The angle between the radius of the circle AC and the segment of the abscissa axis with the designation AG, we define as α (alpha). So, cos α = AG/AC. Given that AC is the radius of the unit circle, and it is equal to one, it turns out that cos α=AG. Similarly, sin α=CG.

In addition, knowing these data, it is possible to determine the coordinate of point C on the circle, since cos α=AG, and sin α=CG, which means that point C has the given coordinates (cos α; sin α). Knowing that the tangent is equal to the ratio of the sine to the cosine, we can determine that tg α \u003d y / x, and ctg α \u003d x / y. Considering angles in a negative coordinate system, one can calculate that the sine and cosine values ​​of some angles can be negative.

Calculations and basic formulas

Values ​​of trigonometric functions

Having considered the essence of trigonometric functions through the unit circle, we can derive the values ​​of these functions for some angles. The values ​​are listed in the table below.

The simplest trigonometric identities

Equations in which there is an unknown value under the sign of the trigonometric function are called trigonometric. Identities with the value sin x = α, k is any integer:

1. sin x = 0, x = πk.
2. 2. sin x \u003d 1, x \u003d π / 2 + 2πk.
3. sin x \u003d -1, x \u003d -π / 2 + 2πk.
4. sin x = a, |a| > 1, no solutions.
5. sin x = a, |a| ≦ 1, x = (-1)^k * arcsin α + πk.

Identities with the value cos x = a, where k is any integer:

1. cos x = 0, x = π/2 + πk.
2. cos x = 1, x = 2πk.
3. cos x \u003d -1, x \u003d π + 2πk.
4. cos x = a, |a| > 1, no solutions.
5. cos x = a, |a| ≦ 1, х = ±arccos α + 2πk.

Identities with the value tg x = a, where k is any integer:

1. tg x = 0, x = π/2 + πk.
2. tg x \u003d a, x \u003d arctg α + πk.

Identities with value ctg x = a, where k is any integer:

1. ctg x = 0, x = π/2 + πk.
2. ctg x \u003d a, x \u003d arcctg α + πk.

Cast formulas

This category of constant formulas denotes methods by which you can go from trigonometric functions of the form to functions of the argument, that is, convert the sine, cosine, tangent and cotangent of an angle of any value to the corresponding indicators of the angle of the interval from 0 to 90 degrees for greater convenience of calculations.

The formulas for reducing functions for the sine of an angle look like this:

• sin(900 - α) = α;
• sin(900 + α) = cos α;
• sin(1800 - α) = sin α;
• sin(1800 + α) = -sin α;
• sin(2700 - α) = -cos α;
• sin(2700 + α) = -cos α;
• sin(3600 - α) = -sin α;
• sin(3600 + α) = sin α.

For the cosine of an angle:

• cos(900 - α) = sin α;
• cos(900 + α) = -sin α;
• cos(1800 - α) = -cos α;
• cos(1800 + α) = -cos α;
• cos(2700 - α) = -sin α;
• cos(2700 + α) = sin α;
• cos(3600 - α) = cos α;
• cos(3600 + α) = cos α.

The use of the above formulas is possible subject to two rules. First, if the angle can be represented as a value (π/2 ± a) or (3π/2 ± a), the value of the function changes:

• from sin to cos;
• from cos to sin;
• from tg to ctg;
• from ctg to tg.

The value of the function remains unchanged if the angle can be represented as (π ± a) or (2π ± a).

Secondly, the sign of the reduced function does not change: if it was initially positive, it remains so. The same is true for negative functions.

Addition Formulas

These formulas express the values ​​of the sine, cosine, tangent, and cotangent of the sum and difference of two rotation angles in terms of their trigonometric functions. Angles are usually denoted as α and β.

The formulas look like this:

1. sin(α ± β) = sin α * cos β ± cos α * sin.
2. cos(α ± β) = cos α * cos β ∓ sin α * sin.
3. tan(α ± β) = (tan α ± tan β) / (1 ∓ tan α * tan β).
4. ctg(α ± β) = (-1 ± ctg α * ctg β) / (ctg α ± ctg β).

These formulas are valid for any angles α and β.

Double and triple angle formulas

The trigonometric formulas of a double and triple angle are formulas that relate the functions of the angles 2α and 3α, respectively, to the trigonometric functions of the angle α. Derived from addition formulas:

1. sin2α = 2sinα*cosα.
2. cos2α = 1 - 2sin^2α.
3. tg2α = 2tgα / (1 - tg^2 α).
4. sin3α = 3sinα - 4sin^3α.
5. cos3α = 4cos^3α - 3cosα.
6. tg3α = (3tgα - tg^3 α) / (1-tg^2 α).

Transition from sum to product

Considering that 2sinx*cosy = sin(x+y) + sin(x-y), simplifying this formula, we obtain the identity sinα + sinβ = 2sin(α + β)/2 * cos(α − β)/2. Similarly, sinα - sinβ = 2sin(α - β)/2 * cos(α + β)/2; cosα + cosβ = 2cos(α + β)/2 * cos(α − β)/2; cosα - cosβ = 2sin(α + β)/2 * sin(α − β)/2; tgα + tgβ = sin(α + β) / cosα * cosβ; tgα - tgβ = sin(α - β) / cosα * cosβ; cosα + sinα = √2sin(π/4 ∓ α) = √2cos(π/4 ± α).

Transition from product to sum

These formulas follow from the identities for the transition of the sum to the product:

• sinα * sinβ = 1/2*;
• cosα * cosβ = 1/2*;
• sinα * cosβ = 1/2*.

Reduction formulas

In these identities, the square and cubic powers of the sine and cosine can be expressed in terms of the sine and cosine of the first power of a multiple angle:

• sin^2 α = (1 - cos2α)/2;
• cos^2α = (1 + cos2α)/2;
• sin^3 α = (3 * sinα - sin3α)/4;
• cos^3 α = (3 * cosα + cos3α)/4;
• sin^4 α = (3 - 4cos2α + cos4α)/8;
• cos^4 α = (3 + 4cos2α + cos4α)/8.

Universal substitution

The universal trigonometric substitution formulas express trigonometric functions in terms of the tangent of a half angle.

• sin x \u003d (2tgx / 2) * (1 + tg ^ 2 x / 2), while x \u003d π + 2πn;
• cos x = (1 - tg^2 x/2) / (1 + tg^2 x/2), where x = π + 2πn;
• tg x \u003d (2tgx / 2) / (1 - tg ^ 2 x / 2), where x \u003d π + 2πn;
• ctg x \u003d (1 - tg ^ 2 x / 2) / (2tgx / 2), while x \u003d π + 2πn.

Special cases

Particular cases of the simplest trigonometric equations are given below (k is any integer).

Private for sine:

sin x value	x value
0	pk
1	π/2 + 2πk
-1	-π/2 + 2πk
1/2	π/6 + 2πk or 5π/6 + 2πk
-1/2	-π/6 + 2πk or -5π/6 + 2πk
√2/2	π/4 + 2πk or 3π/4 + 2πk
-√2/2	-π/4 + 2πk or -3π/4 + 2πk
√3/2	π/3 + 2πk or 2π/3 + 2πk
-√3/2	-π/3 + 2πk or -2π/3 + 2πk

Cosine quotients:

cos x value	x value
0	π/2 + 2πk
1	2πk
-1	2 + 2πk
1/2	±π/3 + 2πk
-1/2	±2π/3 + 2πk
√2/2	±π/4 + 2πk
-√2/2	±3π/4 + 2πk
√3/2	±π/6 + 2πk
-√3/2	±5π/6 + 2πk

Private for tangent:

tg x value	x value
0	pk
1	π/4 + πk
-1	-π/4 + πk
√3/3	π/6 + πk
-√3/3	-π/6 + πk
√3	π/3 + πk
-√3	-π/3 + πk

Cotangent quotients:

ctg x value	x value
0	π/2 + πk
1	π/4 + πk
-1	-π/4 + πk
√3	π/6 + πk
-√3	-π/3 + πk
√3/3	π/3 + πk
-√3/3	-π/3 + πk

Theorems

Sine theorem

There are two versions of the theorem - simple and extended. Simple sine theorem: a/sin α = b/sin β = c/sin γ. In this case, a, b, c are the sides of the triangle, and α, β, γ are the opposite angles, respectively.

Extended sine theorem for an arbitrary triangle: a/sin α = b/sin β = c/sin γ = 2R. In this identity, R denotes the radius of the circle in which the given triangle is inscribed.

Cosine theorem

The identity is displayed in this way: a^2 = b^2 + c^2 - 2*b*c*cos α. In the formula, a, b, c are the sides of the triangle, and α is the angle opposite side a.

Tangent theorem

The formula expresses the relationship between the tangents of two angles, and the length of the sides opposite them. The sides are labeled a, b, c, and the corresponding opposite angles are α, β, γ. The formula of the tangent theorem: (a - b) / (a+b) = tg((α - β)/2) / tg((α + β)/2).

Cotangent theorem

Associates the radius of a circle inscribed in a triangle with the length of its sides. If a, b, c are the sides of a triangle, and A, B, C, respectively, are their opposite angles, r is the radius of the inscribed circle, and p is the half-perimeter of the triangle, the following identities hold:

• ctg A/2 = (p-a)/r;
• ctg B/2 = (p-b)/r;
• ctg C/2 = (p-c)/r.

Applications

Trigonometry is not only a theoretical science associated with mathematical formulas. Its properties, theorems and rules are used in practice by various branches of human activity - astronomy, air and sea navigation, music theory, geodesy, chemistry, acoustics, optics, electronics, architecture, economics, mechanical engineering, measuring work, computer graphics, cartography, oceanography, and many others.

Sine, cosine, tangent and cotangent are the basic concepts of trigonometry, with which you can mathematically express the relationship between angles and lengths of sides in a triangle, and find the desired quantities through identities, theorems and rules.

The basic formulas of trigonometry are formulas that establish relationships between basic trigonometric functions. Sine, cosine, tangent and cotangent are interconnected by many relationships. Below we give the main trigonometric formulas, and for convenience we group them according to their purpose. Using these formulas, you can solve almost any problem from the standard trigonometry course. We note right away that only the formulas themselves are given below, and not their derivation, to which separate articles will be devoted.

Basic identities of trigonometry

Trigonometric identities give a relationship between the sine, cosine, tangent, and cotangent of one angle, allowing one function to be expressed in terms of another.

Trigonometric identities

sin 2 a + cos 2 a = 1 t g α = sin α cos α , c t g α = cos α sin α t g α c t g α = 1 t g 2 α + 1 = 1 cos 2 α , c t g 2 α + 1 = 1 sin 2α

These identities follow directly from the definitions of the unit circle, sine (sin), cosine (cos), tangent (tg), and cotangent (ctg).

Cast formulas

Casting formulas allow you to move from working with arbitrary and arbitrarily large angles to working with angles ranging from 0 to 90 degrees.

Cast formulas

sin α + 2 π z = sin α , cos α + 2 π z = cos α t g α + 2 π z = t g α , c t g α + 2 π z = c t g α sin - α + 2 π z = - sin α , cos - α + 2 π z = cos α t g - α + 2 π z = - t g α , c t g - α + 2 π z = - c t g α sin π 2 + α + 2 π z = cos α , cos π 2 + α + 2 π z = - sin α t g π 2 + α + 2 π z = - c t g α , c t g π 2 + α + 2 π z = - t g α sin π 2 - α + 2 π z = cos α , cos π 2 - α + 2 π z = sin α t g π 2 - α + 2 π z = c t g α , c t g π 2 - α + 2 π z = t g α sin π + α + 2 π z = - sin α , cos π + α + 2 π z = - cos α t g π + α + 2 π z = t g α , c t g π + α + 2 π z = c t g α sin π - α + 2 π z = sin α , cos π - α + 2 π z = - cos α t g π - α + 2 π z = - t g α , c t g π - α + 2 π z = - c t g α sin 3 π 2 + α + 2 π z = - cos α , cos 3 π 2 + α + 2 π z = sin α t g 3 π 2 + α + 2 π z = - c t g α , c t g 3 π 2 + α + 2 π z = - t g α sin 3 π 2 - α + 2 π z = - cos α , cos 3 π 2 - α + 2 π z = - sin α t g 3 π 2 - α + 2 π z = c t g α , c t g 3 π 2 - α + 2 π z = t g α

The reduction formulas are a consequence of the periodicity of trigonometric functions.

Trigonometric addition formulas

The addition formulas in trigonometry allow you to express the trigonometric function of the sum or difference of angles in terms of the trigonometric functions of these angles.

Trigonometric addition formulas

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 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

Based on the addition formulas, trigonometric formulas for a multiple angle are derived.

Multiple angle formulas: double, triple, etc.

Double and triple angle formulas

sin 2 α \u003d 2 sin α cos α cos 2 α \u003d cos 2 α - sin 2 α, cos 2 α \u003d 1 - 2 sin 2 α, cos 2 α \u003d 2 cos 2 α - 1 t g 2 α \u003d 2 t g α 1 - t g 2 α with t g 2 α \u003d with t g 2 α - 1 2 with t g α sin 3 α \u003d 3 sin α cos 2 α - sin 3 α, sin 3 α \u003d 3 sin α - 4 sin 3 α cos 3 α = cos 3 α - 3 sin 2 α cos α , cos 3 α = - 3 cos α + 4 cos 3 α t g 3 α = 3 t g α - t g 3 α 1 - 3 t g 2 α c t g 3 α = c t g 3 α - 3 c t g α 3 c t g 2 α - 1

Half Angle Formulas

The half angle formulas in trigonometry are a consequence of the double angle formulas and express the relationship between the basic functions of the half angle and the cosine of the whole angle.

Half Angle Formulas

sin 2 α 2 = 1 - cos α 2 cos 2 α 2 = 1 + cos α 2 t g 2 α 2 = 1 - cos α 1 + cos α c t g 2 α 2 = 1 + cos α 1 - cos α

Reduction formulas

Reduction formulas

sin 2 α = 1 - cos 2 α 2 cos 2 α = 1 + cos 2 α 2 sin 3 α = 3 sin α - sin 3 α 4 cos 3 α = 3 cos α + cos 3 α 4 sin 4 α = 3 - 4 cos 2 α + cos 4 α 8 cos 4 α = 3 + 4 cos 2 α + cos 4 α 8

Often, in calculations, it is inconvenient to operate with cumbersome powers. Degree reduction formulas allow you to reduce the degree of a trigonometric function from arbitrarily large to the first. Here is their general view:

General form of reduction formulas

for even n

sin n α = C n 2 n 2 n + 1 2 n - 1 ∑ k = 0 n 2 - 1 (- 1) n 2 - k C k n cos ((n - 2 k) α) cos n α = C n 2 n 2 n + 1 2 n - 1 ∑ k = 0 n 2 - 1 C k n cos ((n - 2 k) α)

for odd n

sin n α = 1 2 n - 1 ∑ k = 0 n - 1 2 (- 1) n - 1 2 - k C k n sin ((n - 2 k) α) cos n α = 1 2 n - 1 ∑ k = 0 n - 1 2 C k n cos ((n - 2 k) α)

Sum and difference of trigonometric functions

The difference and sum of trigonometric functions can be represented as a product. Factoring the differences of sines and cosines is very convenient to use when solving trigonometric equations and simplifying expressions.

Sum and difference of trigonometric functions

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 cos α + cos β = 2 cos α + β 2 cos α - β 2 cos α - cos β \u003d - 2 sin α + β 2 sin α - β 2, cos α - cos β \u003d 2 sin α + β 2 sin β - α 2

Product of trigonometric functions

If the formulas for the sum and difference of functions allow you to go to their product, then the formulas for the product of trigonometric functions carry out the reverse transition - from the product to the sum. Formulas for the product of sines, cosines and sine by cosine are considered.

Formulas for the product of trigonometric functions

sin α sin β = 1 2 (cos (α - β) - cos (α + β)) cos α cos β = 1 2 (cos (α - β) + cos (α + β)) sin α cos β = 1 2 (sin (α - β) + sin (α + β))

Universal trigonometric substitution

All basic trigonometric functions - sine, cosine, tangent and cotangent - can be expressed in terms of the tangent of a half angle.

Universal trigonometric substitution

sin α = 2 t g α 2 1 + t g 2 α 2 cos α = 1 - t g 2 α 2 1 + t g 2 α 2 t g α = 2 t g α 2 1 - t g 2 α 2 c t g α = 1 - t g 2 α 2 2 t g α 2

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