What is called the sine of an acute angle of a right triangle. Right triangle

Instruction

If you need to find the cosine angle in an arbitrary triangle, it is necessary to use the cosine theorem:
if the angle is acute: cos? = (a2 + b2 – c2)/(2ab);
if angle : cos? = (c2 - a2 - b2)/(2ab), where a, b are the lengths of the sides adjacent to the corner, c is the length of the side opposite the corner.

Helpful advice

The mathematical notation for cosine is cos.
The cosine value cannot be greater than 1 and less than -1.

Sources:

• how to calculate the cosine of an angle
• Trigonometric functions on the unit circle

Cosine is the basic trigonometric function of the angle. The ability to determine the cosine is useful in vector algebra when determining the projections of vectors on various axes.

Instruction

cos?=(b?+c?-a?)/(2*b*c)

There is a triangle with sides a, b, c equal to 3, 4, 5 mm, respectively.

Find cosine the angle enclosed between the large sides.

Let us denote the angle opposite to the side a through?, then, according to the formula derived above, we have:

cos?=(b?+c?-a?)/(2*b*c)=(4?+5?-3?)/(2*4*5)=(16+25-9)/40 =32/40=0.8

Answer: 0.8.

If the triangle is a right triangle, then to find cosine and it is enough to know the lengths of any two sides of the angle ( cosine right angle is 0).

Let there be a right triangle with sides a, b, c, where c is the hypotenuse.

Consider all options:

Find cos? if the lengths of the sides a and b (of a triangle) are known

Let's use additionally the Pythagorean theorem:

cos?=(b?+c?-a?)/(2*b*c)=(b?+b?+a?-a?)/(2*b*v(b?+a?)) =(2*b?)/(2*b*v(b?+a?))=b/v(b?+a?)

In order for the correctness of the resulting formula, we substitute into it from example 1, i.e.

Having done elementary calculations, we get:

Similarly, there is cosine in a rectangular triangle in other cases:

Known a and c (hypotenuse and opposite leg), find cos?

cos?=(b?+c?-a?)/(2*b*c)=(c?-a?+c?-a?)/(2*c*v(c?-a?)) =(2*s?-2*a?)/(2*s*v(s?-a?))=v(s?-a?)/s.

Substituting the values ​​a=3 and c=5 from the example, we get:

b and c are known (the hypotenuse and the adjacent leg).

Find sos?

Having performed similar transformations (shown in examples 2 and 3), we obtain that in this case cosine V triangle calculated using a very simple formula:

The simplicity of the derived formula is explained in an elementary way: in fact, adjacent to the corner? the leg is a projection of the hypotenuse, its length is equal to the length of the hypotenuse multiplied by cos?.

Substituting the values ​​b=4 and c=5 from the first example, we get:

So all our formulas are correct.

Tip 5: How to find an acute angle in a right triangle

Directly carbonic the triangle is probably one of the most famous, from a historical point of view, geometric shapes. Pythagorean "pants" can only compete with "Eureka!" Archimedes.

You will need

• - drawing of a triangle;
• - ruler;
• - protractor.

Instruction

The sum of the angles of a triangle is 180 degrees. in a rectangular triangle one angle (right) will always be 90 degrees, and the rest are acute, i.e. less than 90 degrees each. To determine which angle in a rectangular triangle is straight, measure the sides of the triangle with a ruler and determine the largest. It is the hypotenuse (AB) and is opposite the right angle (C). The remaining two sides form a right angle and legs (AC, BC).

Once you have determined which angle is acute, you can either use a protractor to calculate the angle, or calculate it using mathematical formulas.

To determine the value of the angle using a protractor, align its top (let's denote it with the letter A) with a special mark on the ruler in the center of the protractor, the AC leg must coincide with its upper edge. Mark on the semicircular part of the protractor the point through which the hypotenuse AB. The value at this point corresponds to the angle value in degrees. If 2 quantities are indicated on the protractor, then for acute angle you need to choose a smaller one, for a stupid one - a larger one.

Find the resulting value in the reference Bradis and determine which angle corresponds to the resulting numerical value. Our grandmothers used this method.

In ours, it is enough to take with the function of calculating trigonometric formulas. For example, the built-in Windows calculator. Launch the "Calculator" application, in the "View" menu item, select the "Engineering" item. Calculate the sine of the desired angle, for example, sin (A) = BC/AB = 2/4 = 0.5

Switch the calculator to inverse function mode by clicking on the INV button on the calculator display, then click on the arcsine function button (labeled sin to the minus one power on the display). The following inscription will appear in the calculation window: asind (0.5) = 30. That is, the desired angle is 30 degrees.

Sources:

• Bradis tables (sines, cosines)

The cosine theorem in mathematics is most often used when it is necessary to find the third side by an angle and two sides. However, sometimes the condition of the problem is set the other way around: it is required to find the angle for given three sides.

Instruction

Imagine that you are given a triangle with known lengths of two sides and the value of one angle. All angles of this triangle are not equal to each other, and its sides are also different in size. The angle γ lies opposite the side of the triangle, designated as AB, which is this figure. Through this angle, as well as through the remaining sides AC and BC, you can find that side of the triangle, which is unknown, using the cosine theorem, deriving the formula below on its basis:
a^2=b^2+c^2-2bc*cosγ, where a=BC, b=AB, c=AC
The cosine theorem is otherwise called the generalized Pythagorean theorem.

Now imagine that all three sides of the figure are given, but its angle γ is unknown. Knowing that the form a^2=b^2+c^2-2bc*cosγ, transform this expression so that the desired value is the angle γ: b^2+c^2=2bc*cosγ+a^2.
Then bring the above equation to a slightly different form: b^2+c^2-a^2=2bc*cosγ.
Then this expression should be transformed into the following: cosγ=√b^2+c^2-a^2/2bc.
It remains to substitute the numbers in the formula and carry out the calculations.

To find the cosine, denoted as γ, it must be expressed through the inverse trigonometric, called the inverse cosine. The arccosine of the number m is the value of the angle γ, for which the cosine of the angle γ is equal to m. The function y=arccos m is decreasing. Imagine, for example, that the cosine of the angle γ is one half. Then the angle γ can be defined in terms of the arc cosine as follows:
γ = arccos, m = arccos 1/2 = 60°, where m = 1/2.
Similarly, you can find the remaining angles of a triangle with two other unknown sides.

Sine and cosine are two trigonometric functions that are called "straight lines". It is they who have to be calculated more often than others, and today each of us has a considerable choice of options to solve this problem. Below are a few of the most simple ways.

Instruction

Use a protractor, pencil, and paper if other means of calculation are not available. One of the definitions of the cosine is given through acute angles in a right triangle - it is equal to the ratio between the length of the leg opposite this angle and the length. Draw a triangle where one of the angles is right (90°) and the other is the angle you want to calculate. The length of the sides does not matter - draw them in such a way that it is more convenient for you to measure. Measure the length of the desired leg and hypotenuse and divide the first by the second in any convenient way.

Seize the opportunity of value trigonometric functions using the calculator built into the Nigma search engine if you have internet access. For example, if you want to calculate the cosine of an angle of 20°, then by loading home page service http://nigma.ru type in the search query "cosine 20" and click the "Find!" button. You can omit “degrees”, and replace the word “cosine” with cos - in any case, the search engine will show the result with an accuracy of up to 15 decimal places (0.939692620785908).

Open the standard program - installed with the operating Windows system if there is no internet access. This can be done, for example, by simultaneously pressing the win and r keys, then entering the calc command and clicking the OK button. To calculate trigonometric functions, here is an interface called "engineering" or "scientific" (depending on the OS version) - select the desired item in the "View" section of the calculator menu. After that, enter the value of the angle in and click on the cos button in the program interface.

Related videos

Tip 8: How to determine angles in a right triangle

Rectangular is characterized by certain ratios between angles and sides. Knowing the values ​​of some of them, you can calculate others. For this, formulas are used, based, in turn, on the axioms and theorems of geometry.

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| - 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

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 sine is one of the basic trigonometric functions, the application of which is not limited to geometry alone. Tables for calculating trigonometric functions, like engineering calculators, are not always at hand, and the calculation of the sine is sometimes necessary to solve various problems. In general, the calculation of the sine will help to consolidate drawing skills and knowledge of trigonometric identities.

Ruler and pencil games

A simple task: how to find the sine of an angle drawn on paper? To solve, you need a regular ruler, a triangle (or a compass) and a pencil. The simplest way to calculate the sine of an angle is by dividing the far leg of a triangle with a right angle by the long side - the hypotenuse. Thus, first you need to complete the acute angle to the figure of a right triangle by drawing a line perpendicular to one of the rays at an arbitrary distance from the vertex of the angle. It will be necessary to observe an angle of exactly 90 °, for which we need a clerical triangle.

Using a compass is a bit more precise, but will take longer. On one of the rays, you need to mark 2 points at a certain distance, set a radius on the compass approximately equal to the distance between the points, and draw semicircles with centers at these points until these lines intersect. By connecting the points of intersection of our circles with each other, we will get a strict perpendicular to the ray of our angle, it remains only to extend the line until it intersects with another ray.

In the resulting triangle, you need to measure the side opposite the corner and the long side on one of the rays with a ruler. The ratio of the first measurement to the second will be the desired value of the sine of the acute angle.

Find the sine for an angle greater than 90°

For an obtuse angle, the task is not much more difficult. It is necessary to draw a ray from the vertex in the opposite direction using a ruler to form a straight line with one of the rays of the angle we are interested in. With the resulting acute angle, you should proceed as described above, the sines of adjacent angles, forming together a developed angle of 180 °, are equal.

Calculating the sine from other trigonometric functions

Also, the calculation of the sine is possible if the values ​​​​of other trigonometric functions of the angle or at least the length of the sides of the triangle are known. Trigonometric identities will help us with this. Let's look at common examples.

How to find the sine with a known cosine of an angle? The first trigonometric identity, coming from the Pythagorean theorem, says that the sum of the squares of the sine and cosine of the same angle is equal to one.

How to find the sine with a known tangent of an angle? The tangent is obtained by dividing the far leg by the near one or by dividing the sine by the cosine. Thus, the sine will be the product of the cosine and the tangent, and the square of the sine will be the square of this product. We replace the squared cosine with the difference between unity and the square sine according to the first trigonometric identity and, through simple manipulations, we bring the equation to calculate the square sine through the tangent, respectively, to calculate the sine, you will have to extract the root from the result obtained.

How to find the sine with a known cotangent of an angle? The value of the cotangent can be calculated by dividing the length of the near one from the leg angle by the length of the far one, and also dividing the cosine by the sine, that is, the cotangent is the inverse function of the tangent with respect to the number 1. To calculate the sine, you can calculate the tangent using the formula tg α \u003d 1 / ctg α and use the formula in the second option. You can also derive a direct formula by analogy with the tangent, which will look like this.

How to find the sine of the three sides of a triangle

There is a formula for finding the length of the unknown side of any triangle, not just a right triangle, given two known sides using the trigonometric function of the cosine of the opposite angle. She looks like this.

Well, the sine can be further calculated from the cosine according to the formulas above.

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 dropping 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, you can determine the coordinate of point C on the circle, since cos α \u003d AG, and sin α \u003d CG, which means that point C has 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 theoretical science associated with mathematical formulas. Its properties, theorems and rules are used in practice by various industries human activity– astronomy, air and sea navigation, music theory, geodesy, chemistry, acoustics, optics, electronics, architecture, economics, mechanical engineering, measuring works, 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.

What is the sine, cosine, tangent, cotangent of an angle will help you understand a right triangle.

What are the sides of a right triangle called? That's right, the hypotenuse and legs: the hypotenuse is the side that lies opposite the right angle (in our example, this is the side \ (AC \) ); the legs are the two remaining sides \ (AB \) and \ (BC \) (those that are adjacent to the right angle), moreover, if we consider the legs with respect to the angle \ (BC \) , then the leg \ (AB \) is adjacent leg, and the leg \ (BC \) is opposite. So, now let's answer the question: what are the sine, cosine, tangent and cotangent of an angle?

Sine of an angle- this is the ratio of the opposite (far) leg to the hypotenuse.

In our triangle:

\[ \sin \beta =\dfrac(BC)(AC) \]

Cosine of an angle- this is the ratio of the adjacent (close) leg to the hypotenuse.

In our triangle:

\[ \cos \beta =\dfrac(AB)(AC) \]

Angle tangent- this is the ratio of the opposite (far) leg to the adjacent (close).

In our triangle:

\[ tg\beta =\dfrac(BC)(AB) \]

Cotangent of an angle- this is the ratio of the adjacent (close) leg to the opposite (far).

In our triangle:

\[ ctg\beta =\dfrac(AB)(BC) \]

These definitions are necessary remember! To make it easier to remember which leg to divide by what, you need to clearly understand that in tangent And cotangent only the legs sit, and the hypotenuse appears only in sinus And cosine. And then you can come up with a chain of associations. For example, this one:

cosine→touch→touch→adjacent;

Cotangent→touch→touch→adjacent.

First of all, it is necessary to remember that the sine, cosine, tangent and cotangent as ratios of the sides of a triangle do not depend on the lengths of these sides (at one angle). Do not believe? Then make sure by looking at the picture:

Consider, for example, the cosine of the angle \(\beta \) . By definition, from a triangle \(ABC \) : \(\cos \beta =\dfrac(AB)(AC)=\dfrac(4)(6)=\dfrac(2)(3) \), but we can calculate the cosine of the angle \(\beta \) from the triangle \(AHI \) : \(\cos \beta =\dfrac(AH)(AI)=\dfrac(6)(9)=\dfrac(2)(3) \). You see, the lengths of the sides are different, but the value of the cosine of one angle is the same. Thus, the values ​​of sine, cosine, tangent and cotangent depend solely on the magnitude of the angle.

If you understand the definitions, then go ahead and fix them!

For the triangle \(ABC \) , shown in the figure below, we find \(\sin \ \alpha ,\ \cos \ \alpha ,\ tg\ \alpha ,\ ctg\ \alpha \).

\(\begin(array)(l)\sin \ \alpha =\dfrac(4)(5)=0.8\\\cos \ \alpha =\dfrac(3)(5)=0.6\\ tg\ \alpha =\dfrac(4)(3)\\ctg\ \alpha =\dfrac(3)(4)=0.75\end(array) \)

Well, did you get it? Then try it yourself: calculate the same for the angle \(\beta \) .

Answers: \(\sin \ \beta =0.6;\ \cos \ \beta =0.8;\ tg\ \beta =0.75;\ ctg\ \beta =\dfrac(4)(3) \).

Unit (trigonometric) circle

Understanding the concepts of degree and radian, we considered a circle with a radius equal to \ (1 \) . Such a circle is called single. It is very useful in the study of trigonometry. Therefore, we dwell on it in a little more detail.

As you can see, this circle is built in the Cartesian coordinate system. The radius of the circle is equal to one, while the center of the circle lies at the origin, the initial position of the radius vector is fixed along the positive direction of the \(x \) axis (in our example, this is the radius \(AB \) ).

Each point on the circle corresponds to two numbers: the coordinate along the axis \(x \) and the coordinate along the axis \(y \) . What are these coordinate numbers? And in general, what do they have to do with the topic at hand? To do this, remember about the considered right-angled triangle. In the figure above, you can see two whole right triangles. Consider the triangle \(ACG \) . It's rectangular because \(CG \) is perpendicular to the \(x \) axis.

What is \(\cos \ \alpha \) from the triangle \(ACG \) ? That's right \(\cos \ \alpha =\dfrac(AG)(AC) \). Besides, we know that \(AC \) is the radius of the unit circle, so \(AC=1 \) . Substitute this value into our cosine formula. Here's what happens:

\(\cos \ \alpha =\dfrac(AG)(AC)=\dfrac(AG)(1)=AG \).

And what is \(\sin \ \alpha \) from the triangle \(ACG \) ? Well, of course, \(\sin \alpha =\dfrac(CG)(AC) \)! Substitute the value of the radius \ (AC \) in this formula and get:

\(\sin \alpha =\dfrac(CG)(AC)=\dfrac(CG)(1)=CG \)

So, can you tell me what are the coordinates of the point \(C \) , which belongs to the circle? Well, no way? But what if you realize that \(\cos \ \alpha \) and \(\sin \alpha \) are just numbers? What coordinate does \(\cos \alpha \) correspond to? Well, of course, the coordinate \(x \) ! And what coordinate does \(\sin \alpha \) correspond to? That's right, the \(y \) coordinate! So the point \(C(x;y)=C(\cos \alpha ;\sin \alpha) \).

What then are \(tg \alpha \) and \(ctg \alpha \) ? That's right, let's use the appropriate definitions of tangent and cotangent and get that \(tg \alpha =\dfrac(\sin \alpha )(\cos \alpha )=\dfrac(y)(x) \), A \(ctg \alpha =\dfrac(\cos \alpha )(\sin \alpha )=\dfrac(x)(y) \).

What if the angle is larger? Here, for example, as in this picture:

What has changed in this example? Let's figure it out. To do this, we again turn to a right-angled triangle. Consider a right triangle \(((A)_(1))((C)_(1))G \) : an angle (as adjacent to the angle \(\beta \) ). What is the value of sine, cosine, tangent and cotangent for an angle \(((C)_(1))((A)_(1))G=180()^\circ -\beta \ \)? That's right, we adhere to the corresponding definitions of trigonometric functions:

\(\begin(array)(l)\sin \angle ((C)_(1))((A)_(1))G=\dfrac(((C)_(1))G)(( (A)_(1))((C)_(1)))=\dfrac(((C)_(1))G)(1)=((C)_(1))G=y; \\\cos \angle ((C)_(1))((A)_(1))G=\dfrac(((A)_(1))G)(((A)_(1)) ((C)_(1)))=\dfrac(((A)_(1))G)(1)=((A)_(1))G=x;\\tg\angle ((C )_(1))((A)_(1))G=\dfrac(((C)_(1))G)(((A)_(1))G)=\dfrac(y)( x);\\ctg\angle ((C)_(1))((A)_(1))G=\dfrac(((A)_(1))G)(((C)_(1 ))G)=\dfrac(x)(y)\end(array) \)

Well, as you can see, the value of the sine of the angle still corresponds to the coordinate \ (y \) ; the value of the cosine of the angle - the coordinate \ (x \) ; and the values ​​of tangent and cotangent to the corresponding ratios. Thus, these relations are applicable to any rotations of the radius vector.

It has already been mentioned that the initial position of the radius vector is along the positive direction of the \(x \) axis. So far we have rotated this vector counterclockwise, but what happens if we rotate it clockwise? Nothing extraordinary, you will also get an angle of a certain size, but only it will be negative. Thus, when rotating the radius vector counterclockwise, we get positive angles, and when rotating clockwise - negative.

So, we know that the whole revolution of the radius vector around the circle is \(360()^\circ \) or \(2\pi \) . Is it possible to rotate the radius vector by \(390()^\circ \) or by \(-1140()^\circ \) ? Well, of course you can! In the first case, \(390()^\circ =360()^\circ +30()^\circ \), so the radius vector will make one full rotation and stop at \(30()^\circ \) or \(\dfrac(\pi )(6) \) .

In the second case, \(-1140()^\circ =-360()^\circ \cdot 3-60()^\circ \), that is, the radius vector will make three complete revolutions and stop at the position \(-60()^\circ \) or \(-\dfrac(\pi )(3) \) .

Thus, from the above examples, we can conclude that angles that differ by \(360()^\circ \cdot m \) or \(2\pi \cdot m \) (where \(m \) is any integer ) correspond to the same position of the radius vector.

The figure below shows the angle \(\beta =-60()^\circ \) . The same image corresponds to the corner \(-420()^\circ ,-780()^\circ ,\ 300()^\circ ,660()^\circ \) etc. This list can be continued indefinitely. All these angles can be written with the general formula \(\beta +360()^\circ \cdot m \) or \(\beta +2\pi \cdot m \) (where \(m \) is any integer)

\(\begin(array)(l)-420()^\circ =-60+360\cdot (-1);\\-780()^\circ =-60+360\cdot (-2); \\300()^\circ =-60+360\cdot 1;\\660()^\circ =-60+360\cdot 2.\end(array) \)

Now, knowing the definitions of the basic trigonometric functions and using the unit circle, try to answer what the values ​​\u200b\u200bare equal to:

\(\begin(array)(l)\sin \ 90()^\circ =?\\\cos \ 90()^\circ =?\\\text(tg)\ 90()^\circ =? \\\text(ctg)\ 90()^\circ =?\\\sin \ 180()^\circ =\sin \ \pi =?\\\cos \ 180()^\circ =\cos \ \pi =?\\\text(tg)\ 180()^\circ =\text(tg)\ \pi =?\\\text(ctg)\ 180()^\circ =\text(ctg)\ \pi =?\\\sin \ 270()^\circ =?\\\cos \ 270()^\circ =?\\\text(tg)\ 270()^\circ =?\\\text (ctg)\ 270()^\circ =?\\\sin \ 360()^\circ =?\\\cos \ 360()^\circ =?\\\text(tg)\ 360()^ \circ =?\\\text(ctg)\ 360()^\circ =?\\\sin \ 450()^\circ =?\\\cos \ 450()^\circ =?\\\text (tg)\ 450()^\circ =?\\\text(ctg)\ 450()^\circ =?\end(array) \)

Here's a unit circle to help you:

Any difficulties? Then let's figure it out. So we know that:

\(\begin(array)(l)\sin \alpha =y;\\cos\alpha =x;\\tg\alpha =\dfrac(y)(x);\\ctg\alpha =\dfrac(x )(y).\end(array) \)

From here, we determine the coordinates of the points corresponding to certain measures of the angle. Well, let's start in order: the corner in \(90()^\circ =\dfrac(\pi )(2) \) corresponds to a point with coordinates \(\left(0;1 \right) \) , therefore:

\(\sin 90()^\circ =y=1 \) ;

\(\cos 90()^\circ =x=0 \) ;

\(\text(tg)\ 90()^\circ =\dfrac(y)(x)=\dfrac(1)(0)\Rightarrow \text(tg)\ 90()^\circ \)- does not exist;

\(\text(ctg)\ 90()^\circ =\dfrac(x)(y)=\dfrac(0)(1)=0 \).

Further, adhering to the same logic, we find out that the corners in \(180()^\circ ,\ 270()^\circ ,\ 360()^\circ ,\ 450()^\circ (=360()^\circ +90()^\circ)\ \ ) correspond to points with coordinates \(\left(-1;0 \right),\text( )\left(0;-1 \right),\text( )\left(1;0 \right),\text( )\left(0 ;1 \right) \), respectively. Knowing this, it is easy to determine the values ​​of trigonometric functions at the corresponding points. Try it yourself first, then check the answers.

Answers:

\(\displaystyle \sin \ 180()^\circ =\sin \ \pi =0 \)

\(\displaystyle \cos \ 180()^\circ =\cos \ \pi =-1 \)

\(\text(tg)\ 180()^\circ =\text(tg)\ \pi =\dfrac(0)(-1)=0 \)

\(\text(ctg)\ 180()^\circ =\text(ctg)\ \pi =\dfrac(-1)(0)\Rightarrow \text(ctg)\ \pi \)- does not exist

\(\sin \ 270()^\circ =-1 \)

\(\cos \ 270()^\circ =0 \)

\(\text(tg)\ 270()^\circ =\dfrac(-1)(0)\Rightarrow \text(tg)\ 270()^\circ \)- does not exist

\(\text(ctg)\ 270()^\circ =\dfrac(0)(-1)=0 \)

\(\sin \ 360()^\circ =0 \)

\(\cos \ 360()^\circ =1 \)

\(\text(tg)\ 360()^\circ =\dfrac(0)(1)=0 \)

\(\text(ctg)\ 360()^\circ =\dfrac(1)(0)\Rightarrow \text(ctg)\ 2\pi \)- does not exist

\(\sin \ 450()^\circ =\sin \ \left(360()^\circ +90()^\circ \right)=\sin \ 90()^\circ =1 \)

\(\cos \ 450()^\circ =\cos \ \left(360()^\circ +90()^\circ \right)=\cos \ 90()^\circ =0 \)

\(\text(tg)\ 450()^\circ =\text(tg)\ \left(360()^\circ +90()^\circ \right)=\text(tg)\ 90() ^\circ =\dfrac(1)(0)\Rightarrow \text(tg)\ 450()^\circ \)- does not exist

\(\text(ctg)\ 450()^\circ =\text(ctg)\left(360()^\circ +90()^\circ \right)=\text(ctg)\ 90()^ \circ =\dfrac(0)(1)=0 \).

Thus, we can make the following table:

There is no need to remember all these values. It is enough to remember the correspondence between the coordinates of points on the unit circle and the values ​​of trigonometric functions:

\(\left. \begin(array)(l)\sin \alpha =y;\\cos \alpha =x;\\tg \alpha =\dfrac(y)(x);\\ctg \alpha =\ dfrac(x)(y).\end(array) \right\)\ \text(Need to remember or be able to output!! \) !}

And here are the values ​​​​of the trigonometric functions of the angles in and \(30()^\circ =\dfrac(\pi )(6),\ 45()^\circ =\dfrac(\pi )(4) \) given in the table below, you must remember:

No need to be scared, now we will show one of the examples of a fairly simple memorization of the corresponding values:

To use this method, it is vital to remember the sine values ​​\u200b\u200bfor all three angle measures ( \(30()^\circ =\dfrac(\pi )(6),\ 45()^\circ =\dfrac(\pi )(4),\ 60()^\circ =\dfrac(\pi )(3) \)), as well as the value of the tangent of the angle in \(30()^\circ \) . Knowing these \(4\) values, it is quite easy to restore the entire table - the cosine values ​​are transferred in accordance with the arrows, that is:

\(\begin(array)(l)\sin 30()^\circ =\cos \ 60()^\circ =\dfrac(1)(2)\ \ \\\sin 45()^\circ = \cos \ 45()^\circ =\dfrac(\sqrt(2))(2)\\\sin 60()^\circ =\cos \ 30()^\circ =\dfrac(\sqrt(3 ))(2)\ \end(array) \)

\(\text(tg)\ 30()^\circ \ =\dfrac(1)(\sqrt(3)) \), knowing this, it is possible to restore the values ​​for \(\text(tg)\ 45()^\circ , \text(tg)\ 60()^\circ \). The numerator “\(1 \) ” will match \(\text(tg)\ 45()^\circ \ \) , and the denominator “\(\sqrt(\text(3)) \) ” will match \(\text (tg)\ 60()^\circ \ \) . Cotangent values ​​are transferred in accordance with the arrows shown in the figure. If you understand this and remember the scheme with arrows, then it will be enough to remember only \(4 \) values ​​from the table.

Coordinates of a point on a circle

Is it possible to find a point (its coordinates) on a circle, knowing the coordinates of the center of the circle, its radius and angle of rotation? Well, of course you can! Let's derive a general formula for finding the coordinates of a point. Here, for example, we have such a circle:

We are given that point \(K(((x)_(0));((y)_(0)))=K(3;2) \) is the center of the circle. The radius of the circle is \(1,5 \) . It is necessary to find the coordinates of the point \(P \) obtained by rotating the point \(O \) by \(\delta \) degrees.

As can be seen from the figure, the coordinate \ (x \) of the point \ (P \) corresponds to the length of the segment \ (TP=UQ=UK+KQ \) . The length of the segment \ (UK \) corresponds to the coordinate \ (x \) of the center of the circle, that is, it is equal to \ (3 \) . The length of the segment \(KQ \) can be expressed using the definition of cosine:

\(\cos \ \delta =\dfrac(KQ)(KP)=\dfrac(KQ)(r)\Rightarrow KQ=r\cdot \cos \ \delta \).

Then we have that for the point \(P \) the coordinate \(x=((x)_(0))+r\cdot \cos \ \delta =3+1,5\cdot \cos \ \delta \).

By the same logic, we find the value of the y coordinate for the point \(P\) . Thus,

\(y=((y)_(0))+r\cdot \sin \ \delta =2+1,5\cdot \sin \delta \).

So in general view point coordinates are determined by the formulas:

\(\begin(array)(l)x=((x)_(0))+r\cdot \cos \ \delta \\y=((y)_(0))+r\cdot \sin \ \delta \end(array) \), Where

\(((x)_(0)),((y)_(0)) \) - coordinates of the center of the circle,

\(r\) - circle radius,

\(\delta \) - rotation angle of the vector radius.

As you can see, for the unit circle we are considering, these formulas are significantly reduced, since the coordinates of the center are zero, and the radius is equal to one:

\(\begin(array)(l)x=((x)_(0))+r\cdot \cos \ \delta =0+1\cdot \cos \ \delta =\cos \ \delta \\y =((y)_(0))+r\cdot \sin \ \delta =0+1\cdot \sin \ \delta =\sin \ \delta \end(array) \)

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