What is sine cosine tangent cotangent equal to? What are sine and cosine

Sine and cosine originally arose from the need to calculate quantities in right triangles. It was noticed that if the degree measure of the angles in a right triangle is not changed, then the aspect ratio, no matter how much these sides change in length, always remains the same.

This is how the concepts of sine and cosine were introduced. The sine of an acute angle in a right triangle is the ratio of the opposite side to the hypotenuse, and the cosine is the ratio of the side adjacent to the hypotenuse.

Theorems of cosines and sines

But cosines and sines can be used for more than just right triangles. To find the value of an obtuse or acute angle or side of any triangle, it is enough to apply the theorem of cosines and sines.

The cosine theorem is quite simple: “The square of a side of a triangle is equal to the sum of the squares of the other two sides minus twice the product of those sides and the cosine of the angle between them.”

There are two interpretations of the sine theorem: small and extended. According to the minor: “In a triangle, the angles are proportional to the opposite sides.” This theorem is often expanded due to the property of the circumscribed circle of a triangle: “In a triangle, the angles are proportional to the opposite sides, and their ratio is equal to the diameter of the circumscribed circle.”

Derivatives

The derivative is a mathematical tool that shows how quickly a function changes relative to a change in its argument. Derivatives are used in geometry, and in a number of technical disciplines.

When solving problems, you need to know the tabular values of the derivatives of trigonometric functions: sine and cosine. The derivative of a sine is a cosine, and a cosine is a sine, but with a minus sign.

Application in mathematics

Sines and cosines are especially often used when solving right triangles and tasks associated with them.

The convenience of sines and cosines is also reflected in technology. Angles and sides were easy to evaluate using the cosine and sine theorems, breaking down complex shapes and objects into “simple” triangles. Engineers who often deal with calculations of aspect ratios and degree measures spent a lot of time and effort calculating the cosines and sines of non-tabular angles.

Then Bradis tables came to the rescue, containing thousands of values of sines, cosines, tangents and cotangents of different angles. IN Soviet time some teachers forced their students to memorize pages of Bradis tables.

Radian is the angular value of an arc whose length is equal to the radius or 57.295779513° degrees.

Degree (in geometry) - 1/360th part of a circle or 1/90th part of a right angle.

π = 3.141592653589793238462… (approximate value of Pi).

Cosine table for angles: 0°, 30°, 45°, 60°, 90°, 120°, 135°, 150°, 180°, 210°, 225°, 240°, 270°, 300°, 315°, 330°, 360°.

Angle x (in degrees)	0°	30°	45°	60°	90°	120°	135°	150°	180°	210°	225°	240°	270°	300°	315°	330°	360°
Angle x (in radians)	0	π/6	π/4	π/3	π/2	2 x π/3	3 x π/4	5 x π/6	π	7 x π/6	5 x π/4	4 x π/3	3 x π/2	5 x π/3	7 x π/4	11 x π/6	2 x π
cos x	1	√3/2 (0,8660)	√2/2 (0,7071)	1/2 (0,5)	0	-1/2 (-0,5)	-√2/2 (-0,7071)	-√3/2 (-0,8660)	-1	-√3/2 (-0,8660)	-√2/2 (-0,7071)	-1/2 (-0,5)	0	1/2 (0,5)	√2/2 (0,7071)	√3/2 (0,8660)	1

How to find sine?

Studying geometry helps develop thinking. This subject is necessarily included in school training. In everyday life, knowledge of this subject can be useful - for example, when planning an apartment.

From the history

The geometry course also includes trigonometry, which studies trigonometric functions. In trigonometry we study sines, cosines, tangents and cotangents of angles.

But on this moment Let's start with the simplest thing - sine. Let's take a closer look at the very first concept - the sine of an angle in geometry. What is sine and how to find it?

The concept of “sine angle” and sinusoids

The sine of an angle is the ratio of the values of the opposite side and the hypotenuse of a right triangle. This is a direct trigonometric function, which is written as “sin (x)”, where (x) is the angle of the triangle.

On the graph, the sine of an angle is indicated by a sine wave with its own characteristics. A sine wave looks like a continuous wavy line that lies within certain limits on the coordinate plane. The function is odd, therefore it is symmetrical about 0 on the coordinate plane (it comes out from the origin of the coordinates).

The domain of definition of this function lies in the range from -1 to +1 on the Cartesian coordinate system. The period of the sine angle function is 2 Pi. This means that every 2 Pi the pattern repeats and the sine wave goes through a full cycle.

Sine wave equation

sin x = a/c
where a is the leg opposite to the angle of the triangle
c - hypotenuse of a right triangle

Properties of the sine of an angle

sin(x) = - sin(x). This feature demonstrates that the function is symmetrical, and if the values x and (-x) are plotted on the coordinate system in both directions, then the ordinates of these points will be opposite. They will be at an equal distance from each other.
Another feature of this function is that the graph of the function increases on the segment [- P/2 + 2 Pn]; [P/2 + 2Pn], where n is any integer. A decrease in the graph of the sine of the angle will be observed on the segment: [P/2 + 2Pn]; [3P/2 + 2Pn].
sin(x) > 0 when x is in the range (2Пn, П + 2Пn)
(x)< 0, когда х находится в диапазоне (-П+2Пn, 2Пn)

The values of the sines of the angle are determined using special tables. Such tables have been created to facilitate the process of calculating complex formulas and equations. It is easy to use and contains not only the values of the sin(x) function, but also the values of other functions.

Moreover, a table of standard values of these functions is included in the compulsory memory study, like a multiplication table. This is especially true for classes with a physical and mathematical bias. In the table you can see the values of the main angles used in trigonometry: 0, 15, 30, 45, 60, 75, 90, 120, 135, 150, 180, 270 and 360 degrees.

There is also a table defining the values of trigonometric functions of non-standard angles. Taking advantage different tables, you can easily calculate the sine, cosine, tangent and cotangent of some angles.

Equations are made with trigonometric functions. Solving these equations is easy if you know the simple ones trigonometric identities and reductions of functions, for example, such as sin (P/2 + x) = cos (x) and others. A separate table has also been compiled for such reductions.

How to find the sine of an angle

When the task is to find the sine of an angle, and according to the condition we only have the cosine, tangent, or cotangent of the angle, we can easily calculate what we need using trigonometric identities.

sin 2 x + cos 2 x = 1

From this equation, we can find both sine and cosine, depending on which value is unknown. We get a trigonometric equation with one unknown:

sin 2 x = 1 - cos 2 x
sin x = ± √ 1 - cos 2 x
cot 2 x + 1 = 1 / sin 2 x

From this equation you can find the value of the sine, knowing the value of the cotangent of the angle. To simplify, replace sin 2 x = y and you have a simple equation. For example, the cotangent value is 1, then:

1 + 1 = 1/y
2 = 1/y
2у = 1
y = 1/2

Now we perform the reverse replacement of the player:

sin 2 x = ½
sin x = 1 / √2

Since we took the cotangent value for the standard angle (45 0), the obtained values can be checked in the table.

If you have a tangent value and need to find the sine, another trigonometric identity will help:

tg x * ctg x = 1

It follows that:

cot x = 1 / tan x

In order to find the sine of a non-standard angle, for example, 240 0, you need to use angle reduction formulas. We know that π corresponds to 180 0. Thus, we express our equality using standard angles by expansion.

240 0 = 180 0 + 60 0

We need to find the following: sin (180 0 + 60 0). In trigonometry there are reduction formulas that in this case will come in handy. This is the formula:

sin (π + x) = - sin (x)

Thus, the sine of an angle of 240 degrees is equal to:

sin (180 0 + 60 0) = - sin (60 0) = - √3/2

In our case, x = 60, and P, respectively, 180 degrees. We found the value (-√3/2) from the table of values of functions of standard angles.

In this way, non-standard angles can be expanded, for example: 210 = 180 + 30.

Trigonometric identities- these are equalities that establish a relationship between sine, cosine, tangent and cotangent of one angle, which allows you to find any of these functions, provided that any other is known.

tg \alpha = \frac(\sin \alpha)(\cos \alpha), \enspace ctg \alpha = \frac(\cos \alpha)(\sin \alpha)

tg \alpha \cdot ctg \alpha = 1

This identity says that the sum of the square of the sine of one angle and the square of the cosine of one angle is equal to one, which in practice makes it possible to calculate the sine of one angle when its cosine is known and vice versa.

When converting trigonometric expressions, this identity is very often used, which allows you to replace the sum of the squares of the cosine and sine of one angle with one and also perform the replacement operation in the reverse order.

Finding tangent and cotangent using sine and cosine

tg \alpha = \frac(\sin \alpha)(\cos \alpha),\enspace

These identities are formed from the definitions of sine, cosine, tangent and cotangent. After all, if you look at it, then by definition the ordinate y is a sine, and the abscissa x is a cosine. Then the tangent will be equal to the ratio \frac(y)(x)=\frac(\sin \alpha)(\cos \alpha), and the ratio \frac(x)(y)=\frac(\cos \alpha)(\sin \alpha)- will be a cotangent.

Let us add that only for such angles \alpha at which the trigonometric functions included in them make sense, the identities will hold, ctg \alpha=\frac(\cos \alpha)(\sin \alpha).

For example: tg \alpha = \frac(\sin \alpha)(\cos \alpha) is valid for angles \alpha that are different from \frac(\pi)(2)+\pi z, A ctg \alpha=\frac(\cos \alpha)(\sin \alpha)- for an angle \alpha other than \pi z, z is an integer.

Relationship between tangent and cotangent

tg \alpha \cdot ctg \alpha=1

This identity is valid only for angles \alpha that are different from \frac(\pi)(2) z. Otherwise, either cotangent or tangent will not be determined.

Based on the above points, we obtain that tg \alpha = \frac(y)(x), A ctg \alpha=\frac(x)(y). It follows that tg \alpha \cdot ctg \alpha = \frac(y)(x) \cdot \frac(x)(y)=1. Thus, the tangent and cotangent of the same angle at which they make sense are mutually inverse numbers.

Relationships between tangent and cosine, cotangent and sine

tg^(2) \alpha + 1=\frac(1)(\cos^(2) \alpha)- the sum of the square of the tangent of the angle \alpha and 1 is equal to the inverse square of the cosine of this angle. This identity is valid for all \alpha other than \frac(\pi)(2)+ \pi z.

1+ctg^(2) \alpha=\frac(1)(\sin^(2)\alpha)- the sum of 1 and the square of the cotangent of the angle \alpha is equal to the inverse square of the sine of the given angle. This identity is valid for any \alpha different from \pi z.

Examples with solutions to problems using trigonometric identities

Example 1

Find \sin \alpha and tg \alpha if \cos \alpha=-\frac12 And \frac(\pi)(2)< \alpha < \pi ;

Show solution

Solution

The functions \sin \alpha and \cos \alpha are related by the formula \sin^(2)\alpha + \cos^(2) \alpha = 1. Substituting into this formula \cos \alpha = -\frac12, we get:

\sin^(2)\alpha + \left (-\frac12 \right)^2 = 1

This equation has 2 solutions:

\sin \alpha = \pm \sqrt(1-\frac14) = \pm \frac(\sqrt 3)(2)

By condition \frac(\pi)(2)< \alpha < \pi . In the second quarter the sine is positive, so \sin \alpha = \frac(\sqrt 3)(2).

In order to find tan \alpha, we use the formula tg \alpha = \frac(\sin \alpha)(\cos \alpha)

tg \alpha = \frac(\sqrt 3)(2) : \frac12 = \sqrt 3

Example 2

Find \cos \alpha and ctg \alpha if and \frac(\pi)(2)< \alpha < \pi .

Show solution

Solution

Substituting into the formula \sin^(2)\alpha + \cos^(2) \alpha = 1 given number \sin \alpha=\frac(\sqrt3)(2), we get \left (\frac(\sqrt3)(2)\right)^(2) + \cos^(2) \alpha = 1. This equation has two solutions \cos \alpha = \pm \sqrt(1-\frac34)=\pm\sqrt\frac14.

By condition \frac(\pi)(2)< \alpha < \pi . In the second quarter the cosine is negative, so \cos \alpha = -\sqrt\frac14=-\frac12.

In order to find ctg \alpha , we use the formula ctg \alpha = \frac(\cos \alpha)(\sin \alpha). We know the corresponding values.

ctg \alpha = -\frac12: \frac(\sqrt3)(2) = -\frac(1)(\sqrt 3).

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) 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 ( 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 a power series for the 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

The inverse functions of tangent and cotangent are arctangent and arccotangent, respectively.

Arctangent, arctg

, Where n- whole.

Arccotangent, arcctg

, Where n- whole.

References:
I.N. Bronstein, K.A. Semendyaev, Handbook of mathematics for engineers and college students, “Lan”, 2009.
G. Korn, Handbook of Mathematics for scientific workers and engineers, 2012.

We will begin our study of trigonometry with the right triangle. Let's define what sine and cosine are, as well as tangent and cotangent of an acute angle. This is the basics of trigonometry.

Let us remind you that right angle is an angle equal to 90 degrees. In other words, half a turned angle.

Sharp corner- less than 90 degrees.

Obtuse angle- greater than 90 degrees. In relation to such an angle, “obtuse” is not an insult, but a mathematical term :-)

Let's draw a right triangle. A right angle is usually denoted by . Please note that the side opposite the corner is indicated by the same letter, only small. Thus, the side opposite angle A is designated .

The angle is indicated by the corresponding Greek letter.

Hypotenuse of a right triangle is the side opposite the right angle.

Legs- sides lying opposite acute angles.

The leg lying opposite the angle is called opposite(relative to angle). The other leg, which lies on one of the sides of the angle, is called adjacent.

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

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

Tangent acute angle in a right triangle - the ratio of the opposite side to the adjacent:

Another (equivalent) definition: the tangent of an acute angle is the ratio of the sine of the angle to its cosine:

Cotangent acute angle in a right triangle - the ratio of the adjacent side to the opposite (or, which is the same, the ratio of cosine to sine):

Note the basic relationships for sine, cosine, tangent, and cotangent below. They will be useful to us when solving problems.

Let's prove some of them.

Okay, we have given definitions and written down formulas. But why do we still need sine, cosine, tangent and cotangent?

We know that the sum of the angles of any triangle is equal to.

We know the relationship between parties right triangle. This is the Pythagorean theorem: .

It turns out that knowing two angles in a triangle, you can find the third. Knowing the two sides of a right triangle, you can find the third. This means that the angles have their own ratio, and the sides have their own. But what should you do if in a right triangle you know one angle (except the right angle) and one side, but you need to find the other sides?

This is what people in the past encountered when making maps of the area and the starry sky. After all, it is not always possible to directly measure all sides of a triangle.

Sine, cosine and tangent - they are also called trigonometric angle functions- give relationships between parties And corners triangle. Knowing the angle, you can find all its trigonometric functions using special tables. And knowing the sines, cosines and tangents of the angles of a triangle and one of its sides, you can find the rest.

We will also draw a table of the values of sine, cosine, tangent and cotangent for “good” angles from to.

Please note the two red dashes in the table. At appropriate angle values, tangent and cotangent do not exist.

Let's look at several trigonometry problems from the FIPI Task Bank.

1. In a triangle, the angle is , . Find .

The problem is solved in four seconds.

Because the , .

2. In a triangle, the angle is , , . Find .

Let's find it using the Pythagorean theorem.

The problem is solved.

Often in problems there are triangles with angles and or with angles and. Remember the basic ratios for them by heart!

For a triangle with angles and the leg opposite the angle at is equal to half of the hypotenuse.

A triangle with angles and is isosceles. In it, the hypotenuse is times larger than the leg.

We looked at problems solving right triangles - that is, finding unknown sides or angles. But that's not all! IN Unified State Exam options in mathematics there are many problems where the sine, cosine, tangent or cotangent of the external angle of a triangle appears. More on this in the next article.