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 get acquainted with trigonometric functions and formulas in more detail.

Concepts in trigonometry

To understand the basic concepts of trigonometry, you must first determine what a right-angled triangle and an angle in a circle are, and why all the basic trigonometric calculations are associated with them. A triangle in which one of the corners is 90 degrees is rectangular. 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-angled triangles are hypotenuse and legs. The hypotenuse is the side of the triangle opposite the right angle. The legs, respectively, are the other two sides. The sum of the angles of any triangles 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. The peculiarity of a triangle in spherical trigonometry is that it always has a sum of angles of more than 180 degrees.

Angles of a triangle

In a right-angled triangle, the sine of an angle is the ratio of the leg opposite to 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 ​​are always 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. Cotangent, in turn, is the ratio of the adjacent leg of the desired angle to the opposite leg. The cotangent of an angle can also be obtained by dividing one by the tangent value.

Unit circle

A unit circle in geometry is a circle whose radius is equal to one. Such a circle is constructed in a Cartesian coordinate system, while the center of the circle coincides with the origin point, and the initial position of the radius vector is determined along the positive direction of the X axis (abscissa). Each point of the circle has two coordinates: XX and YY, that is, the coordinates of the abscissas and ordinates. Selecting any point on the circle in the XX plane, and dropping the perpendicular from it to the abscissa axis, we obtain a right-angled triangle formed by the radius to the selected point (denote it by the letter C), by the 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 designated by the letter A) and the intersection point G. The resulting triangle ACG is a right-angled 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. Considering 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 sine to cosine, we can determine that tg α = y / x, and ctg α = x / y. Considering angles in a negative coordinate system, you can calculate that the values ​​of the sine and cosine of some angles may be negative.

Calculations and basic formulas

Values ​​of trigonometric functions

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

Simplest trigonometric identities

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

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

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

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

Casting formulas

This category of constant formulas denotes methods that can be used to switch from trigonometric functions of the form to functions of an argument, that is, to bring 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 converting 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. Likewise with negative functions.

Addition formulas

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

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 values ​​of the angles α and β.

Double and triple angle formulas

Double and triple angle trigonometric formulas 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 ^ 2 α.
3. tg2α = 2tgα / (1 - tg ^ 2 α).
4. sin3α = 3sinα - 4sin ^ 3 α.
5. cos3α = 4cos ^ 3 α - 3cosα.
6. tg3α = (3tgα - tan ^ 3 α) / (1-tan ^ 2 α).

The transition from sum to product

Taking into account that 2sinx * cozy = 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).

Moving from work to sum

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

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

Degree 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 the multiple angle:

• sin ^ 2 α = (1 - cos2α) / 2;
• cos ^ 2 α = (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

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

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

Special cases

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

Private for sinus:

The quotients for the cosine are:

Private for tangent:

Private for cotangent:

Theorems

Sine theorem

There are two versions of the theorem - simple and extended. Simple theorem of sines: 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, respectively, opposite angles.

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 as follows: 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 to side a.

Tangent theorem

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

Cotangent theorem

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

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

Applied application

Trigonometry is not only a theoretical science related to mathematical formulas. Its properties, theorems and rules are used in practice by different 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 the help of which you can mathematically express the relationship between the angles and the lengths of the sides in a triangle, and find the required quantities through identities, theorems and rules.

Trigonometric function values ​​table

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

see also useful materials:

For determining the value of the trigonometric function, find it at the intersection of the trigonometric function line. For example, sine 30 degrees - look for a column with the heading sin (sine) and find the intersection of this table column with the line "30 degrees", at their intersection we read the result - one second. Similarly, we find cosine 60 degrees, sinus 60 degrees (once again, at the intersection of the sin (sine) column and 60 degrees 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 given in radians... To do this, use the second column of angle values. Thanks to this, the value of popular angles can be converted from degrees to radians. For example, let's find an angle of 60 degrees in the first line and read its value in radians below it. 60 degrees is equal to π / 3 radians.

The number pi uniquely expresses the dependence of the circumference on the degree measure of the angle. So pi radians are equal to 180 degrees.

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

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

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

If a dash (tangent (tg) 90 degrees, cotangent (ctg) 180 degrees) is indicated in the table of values ​​of trigonometric functions instead of the function value, then the function has no definite meaning for this value of the degree measure of the angle. If there is no dash - the cell is empty, then we have not yet entered the required value. We are interested in what requests users come to us and supplement the table with new values, despite the fact that the current data on the values ​​of the cosines, sines and tangents of the most frequently encountered angle values ​​is quite 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 in Bradis tables")

Trigonometry is a branch of mathematics that studies trigonometric functions and their use in geometry. The development of trigonometry began in the days of ancient Greece. During the Middle Ages, scientists from the Middle East and India made an important contribution to the development of this science.

This article is devoted to the basic concepts and definitions of trigonometry. It discusses the definitions of the main trigonometric functions: sine, cosine, tangent and cotangent. Their meaning is explained and illustrated in the context of geometry.

Initially, the definitions of trigonometric functions, the argument of which is an angle, were expressed in terms of the ratios of the sides of a right-angled triangle.

Definitions of trigonometric functions

The sine of the angle (sin α) is the ratio of the leg opposite to this angle to the hypotenuse.

The cosine of the angle (cos α) is the ratio of the adjacent leg to the hypotenuse.

The tangent of the angle (t g α) is the ratio of the opposite leg to the adjacent one.

Angle cotangent (c t g α) - the ratio of the adjacent leg to the opposite one.

These definitions are given for an acute angle of a right triangle!

Here's an illustration.

In a triangle ABC with a right angle C, the sine of angle A is equal to the ratio of leg BC to hypotenuse AB.

The definitions of sine, cosine, tangent and cotangent allow you to calculate the values ​​of these functions from the known lengths of the sides of the triangle.

Important to remember!

The range of values ​​of sine and cosine: from -1 to 1. In other words, the sine and cosine take values ​​from -1 to 1. The range of values ​​of the tangent and cotangent is the whole number line, that is, these functions can take any values.

The definitions given above are for sharp corners. In trigonometry, the concept of a rotation angle is introduced, the value of which, unlike an acute angle, is not limited to a frame from 0 to 90 degrees. The angle of rotation in degrees or radians is expressed by any real number from - ∞ to + ∞.

In this context, you can give a definition of sine, cosine, tangent and cotangent of an angle of arbitrary magnitude. Imagine the unit circle centered at the origin of the Cartesian coordinate system.

The starting point A with coordinates (1, 0) rotates around the center of the unit circle by some angle α and goes to point A 1. The definition is given through the coordinates of the point A 1 (x, y).

Sine (sin) of the angle of rotation

The sine of the angle of rotation α is the ordinate of point A 1 (x, y). sin α = y

The cosine (cos) of the angle of rotation

The cosine of the angle of rotation α is the abscissa of point A 1 (x, y). cos α = x

Tangent (tg) angle of rotation

The tangent of the angle of rotation α is the ratio of the ordinate of point A 1 (x, y) to its abscissa. t g α = y x

Cotangent (ctg) of the angle of rotation

The cotangent of the angle of rotation α is the ratio of the abscissa of point A 1 (x, y) to its ordinate. c t g α = x y

Sine and cosine are defined for any angle of rotation. This is logical, because the abscissa and ordinate of a point after turning can be determined at any angle. The situation is different with tangent and cotangent. The tangent is not defined when the point after turning goes to the point with zero abscissa (0, 1) and (0, - 1). In such cases, the expression for the tangent t g α = y x simply does not make sense, since it contains division by zero. The situation is similar with the cotangent. The difference is that the cotangent is not defined when the ordinate of a point vanishes.

Important to remember!

Sine and cosine are defined for any angle α.

The tangent is defined for all angles except α = 90 ° + 180 ° k, k ∈ Z (α = π 2 + π k, k ∈ Z)

The cotangent is defined for all angles except α = 180 ° k, k ∈ Z (α = π k, k ∈ Z)

When solving practical examples, do not say "sine of the angle of rotation α". The words "angle of rotation" are simply omitted, implying that it is clear from the context what it is about.

Numbers

What about the definition of sine, cosine, tangent and cotangent of a number, and not the angle of rotation?

Sine, cosine, tangent, cotangent of a number

Sine, cosine, tangent and cotangent of a number t is a number that is, respectively, equal to sine, cosine, tangent and cotangent in t radian.

For example, the sine of 10 π is equal to the sine of the rotation angle of 10 π rad.

There is another approach to determining the sine, cosine, tangent and cotangent of a number. Let's consider it in more detail.

Any real number t a point on the unit circle with a center at the origin of a rectangular Cartesian coordinate system is assigned. Sine, cosine, tangent and cotangent are defined through the coordinates of this point.

The starting point on the circle is point A with coordinates (1, 0).

A positive number t

Negative number t corresponds to the point to which the starting point will go if it moves counterclockwise along the circle and traverses the path t.

Now that the connection between the number and the point on the circle is established, we proceed to the definition of sine, cosine, tangent and cotangent.

The sine (sin) of t

Sine of number t is the ordinate of the point of the unit circle corresponding to the number t. sin t = y

Cosine (cos) of number t

Cosine number t is the abscissa of the point of the unit circle corresponding to the number t. cos t = x

The tangent (tg) of the number t

Tangent of number t- the ratio of the ordinate to the abscissa of the point of the unit circle corresponding to the number t. t g t = y x = sin t cos t

The latter definitions are consistent with and do not contradict the definition given at the beginning of this clause. The point on the circle corresponding to the number t, coincides with the point to which the starting point goes after rotation by an angle t radian.

Trigonometric functions of angular and numeric argument

Each value of the angle α corresponds to a certain value of the sine and cosine of this angle. As well as all angles α other than α = 90 ° + 180 ° k, k ∈ Z (α = π 2 + π k, k ∈ Z) there corresponds a certain value of the tangent. The cotangent, as mentioned above, is defined for all α, except for α = 180 ° k, k ∈ Z (α = π k, k ∈ Z).

We can say that sin α, cos α, t g α, c t g α are functions of the angle alpha, or functions of the angular argument.

Similarly, you can talk about sine, cosine, tangent and cotangent as functions of a numeric argument. To every real number t corresponds to a specific value of the sine or cosine of a number t... All numbers other than π 2 + π · k, k ∈ Z, correspond to the value of the tangent. The cotangent is similarly defined for all numbers except π k, k ∈ Z.

Basic functions of trigonometry

Sine, cosine, tangent and cotangent are basic trigonometric functions.

It is usually clear from the context which argument of the trigonometric function (angle argument or numeric argument) we are dealing with.

Let's return to the data at the very beginning of the definitions and the angle alpha, lying in the range from 0 to 90 degrees. The trigonometric definitions of sine, cosine, tangent and cotangent are completely consistent with the geometric definitions given using the aspect ratios of a right-angled triangle. Let's show it.

Take the unit circle centered in a rectangular Cartesian coordinate system. Let's rotate the starting point A (1, 0) by an angle up to 90 degrees and draw a perpendicular to the abscissa axis from the resulting point A 1 (x, y). In the resulting right-angled triangle, the angle A 1 O H is equal to the angle of rotation α, the length of the leg O H is equal to the abscissa of point A 1 (x, y). The length of the leg opposite to the corner is equal to the ordinate of point A 1 (x, y), and the length of the hypotenuse is equal to one, since it is the radius of the unit circle.

According to the definition from geometry, the sine of the angle α is equal to the ratio of the opposite leg to the hypotenuse.

sin α = A 1 H O A 1 = y 1 = y

This means that determining the sine of an acute angle in a right-angled triangle through the aspect ratio is equivalent to determining the sine of the angle of rotation α, with alpha lying in the range from 0 to 90 degrees.

Similarly, the correspondence of definitions can be shown for cosine, tangent and cotangent.

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1. One of the legs of a right-angled triangle is 25 cm. Calculate the length of the second leg if the angle adjacent to the known leg is 36º.
   

   Solution:

   According to the definition, the tangent of an acute angle in a right-angled triangle is equal to the ratio of the opposite leg to the adjacent leg. Leg a = 25 cm is adjacent to the angle α = 36º, and the unknown leg b is opposite. Then:

   $$ tg (\ alpha) = \ frac (b) (a) $$, hence $$ b = a \ cdot tg (\ alpha) $$

   Let's make the substitution:

   $$ b = 25 \ cdot tg (36 ^ 0) = 25 \ cdot 0.727 = 18.175 cm $$

   Answer:

   $$ b = 18.175 cm $$

2. Calculate the value of the expression: $$ 2 + tg (12 ^ 0) - tg ^ 2 \ left (\ frac (\ pi) (5) \ right) $$
   

   Solution:

   When substituting, take into account that one of the angles is measured in degrees, the other in radians:

   $$ 2 + tg (12 ^ 0) - tg ^ 2 \ left (\ frac (\ pi) (5) \ right) = 2 + 0.213 - 0.727 ^ 2 \ approx 1.684 $$

   Answer:
3. To calculate the height of the Cheops pyramid, the scientist waited until the Sun from where he is, touches its top. Then he measured the angular height of the Sun above the horizon, it turned out to be 21º, and the distance to the pyramid was 362 m. What is its height?
   

   Solution:

   The height of the pyramid H and the distance L to it are the legs of a right-angled triangle, the hypotenuse of which is a sunbeam. Then the tangent of the angle at which the Sun is seen at the top of the pyramid is:

   $$ tg \ alpha = \ frac (H) (L) $$, we calculate the height by transforming the formula:

   $$ H = L \ cdot tg (\ alpha) = 362 \ cdot tg (21 ^ 0) = 138.96 $$

   Answer:

   $$ H = 138.96 $$

4. Find tg α if the opposite leg is 6 cm and the adjacent leg is 5 cm.
   

   Solution:

   A-priory

   $$ tg \ alpha = \ frac (b) (a) $$

   $$ tg \ alpha = \ frac (6) (5) = 1.2 $$

   So the angle is $$ \ alpha = 50 ^ (\ circ) $$.

   Answer:

   $$ tg \ alpha = 1.2 $$

5. Find tg α if the opposite leg is 8 cm and the hypotenuse is 10 cm.
   

   Solution:

   Using the Pythagorean formula, we find the adjacent leg of the triangle:

   $$ a = \ sqrt ((c ^ 2 - b ^ 2)) $$

   $$ a = \ sqrt ((10 ^ 2 - 8 ^ 2)) = \ sqrt (36) = 6 \ cm $$

   A-priory

   $$ tg \ \ alpha = \ frac (8) (6) = 1.333 $$

   So the angle is $$ \ alpha = 53 ^ (\ circ) $$.

   Answer:

   $$ tg \ alpha = 1.333 $$

6. Find tg α if the adjacent leg is 2 times larger than the opposite one, and the hypotenuse is 5√5 cm.
   

   Solution:

   Using the Pythagorean formula, we find the legs of the triangle:

   $$ c = \ sqrt ((b ^ 2 + 4b ^ 2)) = \ sqrt ((5b ^ 2)) = b \ sqrt (5) $$

   $$ b = \ frac (c) (\ sqrt (5)) = \ frac (5 \ sqrt (5)) (\ sqrt (5)) = 5 \ cm $$

   $$ a = 5 \ cdot 2 = 10 \ cm $$

   A-priory

   $$ tg \ \ alpha = \ frac (b) (a) $$

   $$ tg \ \ alpha = \ frac (5) (10) = 0.5 $$

   Hence, the angle $$ \ alpha = 27 ^ (\ circ) $$.

   Answer:

   $$ tg \ alpha = 0.5 $$

7. Find tan α if the hypotenuse is 12 cm and the angle β = 30 °.
   

   Solution:

   Let's find the leg adjacent to the desired angle. It is known that a leg lying opposite an angle of 30 ° is equal to half of the hypotenuse. Means,

   $$ a = 6 \ cm $$

   By the Pythagorean theorem, we find the leg opposite to the desired angle:

   $$ b = \ sqrt ((c ^ 2 + a ^ 2)) $$

   $$ b = \ sqrt ((144-36)) = \ sqrt (108) = 6 \ sqrt (3) $$

   A-priory

   $$ tg \ \ alpha = \ frac (b) (a) $$

   $$ tg \ \ alpha = \ frac (6 \ sqrt (3)) (6) = \ sqrt (3) = 1.732 $$

   So the angle is $$ \ alpha = 60 ^ (\ circ) $$.

   Answer:

   $$ tg \ alpha = 1.732 $$

8. Find tg α if the opposite and adjacent legs are equal and the hypotenuse is 6√2cm.
   

   Solution:

   A-priory

   $$ tg \ \ alpha = \ frac (b) (a) $$

   $$ tg \ \ alpha = 1 $$

   So the angle is $$ \ alpha = 45 ^ (\ circ) $$.

   Answer:

   Lecture: Sine, cosine, tangent, cotangent of an arbitrary angle

   Sine, cosine of an arbitrary angle

   
   

   To understand what trigonometric functions are, let's turn to a circle with a unit radius. This circle is centered at the origin on the coordinate plane. To determine the given functions, we will use the radius vector OR which starts at the center of the circle and point R is the point of the circle. This radius vector forms an angle alpha with the axis OH... Since the circle has a radius equal to one, then OP = R = 1.

   

   If from the point R lower the perpendicular to the axis OH, then we get a right-angled triangle with a hypotenuse equal to one.

   
   

   If the radius vector moves clockwise, then this direction is called negative, if it moves counterclockwise - positive.

   

   
   

   Sine angle OR, is the ordinate of the point R vectors on a circle.

   That is, to obtain the sine value of a given angle alpha, it is necessary to determine the coordinate Have on surface.

   How was this value obtained? Since we know that the sine of an arbitrary angle in a right-angled triangle is the ratio of the opposite leg to the hypotenuse, we get that

   

   And since R = 1, then sin (α) = y 0 .

   
   

   In the unit circle, the value of the ordinate cannot be less than -1 and more than 1, which means that

   

   

   The sine is positive in the first and second quarters of the unit circle, and negative in the third and fourth.

   Cosine angle the given circle formed by the radius vector OR, is the abscissa of the point R vectors on a circle.

   That is, to obtain the value of the cosine of a given angle alpha, it is necessary to determine the coordinate NS on surface.

   
   

   The cosine of an arbitrary angle in a right-angled triangle is the ratio of the adjacent leg to the hypotenuse, we get that

   
   

   And since R = 1, then cos (α) = x 0 .

   In the unit circle, the value of the abscissa cannot be less than -1 and more than 1, which means that

   

   

   The cosine is positive in the first and fourth quarters of the unit circle, and negative in the second and third.

   Tangentarbitrary angle the ratio of sine to cosine is considered.

   If we consider a right-angled triangle, then this is the ratio of the opposite leg to the adjacent one. If we are talking about the unit circle, then this is the ratio of the ordinate to the abscissa.

   

   Judging by these ratios, one can understand that the tangent cannot exist if the value of the abscissa is zero, that is, at an angle of 90 degrees. The tangent can take all other values.

   

   The tangent is positive in the first and third quarters of the unit circle, and negative in the second and fourth.