The leg is equal to the product of the hypotenuse and the cosine. Sine, cosine, tangent and cotangent: definitions in trigonometry, examples, formulas

In this article, we will show how definitions of sine, cosine, tangent and cotangent of angle and number in trigonometry. Here we will talk about notation, give examples of records, give graphic illustrations. In conclusion, we draw a parallel between the definitions of sine, cosine, tangent and cotangent in trigonometry and geometry.

Page navigation.

Definition of sine, cosine, tangent and cotangent

Let's follow how the concept of sine, cosine, tangent and cotangent is formed in the school mathematics course. In geometry lessons, the definition of sine, cosine, tangent and cotangent is given acute angle in a right triangle. And later trigonometry is studied, which refers to the sine, cosine, tangent and cotangent of the angle of rotation and the number. We give all these definitions, give examples and give the necessary comments.

Acute angle in a right triangle

From the course of geometry, the definitions of the sine, cosine, tangent and cotangent of an acute angle in a right triangle are known. They are given as the ratio of the sides of a right triangle. We present their formulations.

Definition.

Sine of an acute angle in a right triangle is the ratio of the opposite leg to the hypotenuse.

Definition.

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

Definition.

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

Definition.

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

The notation of sine, cosine, tangent and cotangent is also introduced there - sin, cos, tg and ctg, respectively.

For example, if ABC is right triangle with a right angle C, then the sine of the acute angle A is equal to the ratio of the opposite leg BC to the hypotenuse AB, that is, sin∠A=BC/AB.

These definitions allow you to calculate the values ​​of the sine, cosine, tangent and cotangent of an acute angle from the known lengths of the sides of a right triangle, as well as from the known values ​​of the sine, cosine, tangent, cotangent and the length of one of the sides, find the lengths of the other sides. For example, if we knew that in a right triangle the leg AC is 3 and the hypotenuse AB is 7 , then we could calculate the cosine of the acute angle A by definition: cos∠A=AC/AB=3/7 .

Angle of rotation

In trigonometry, they begin to look at the angle more widely - they introduce the concept of angle of rotation. The angle of rotation, unlike an acute angle, is not limited to frames from 0 to 90 degrees, the angle of rotation in degrees (and in radians) can be expressed by any real number from −∞ to +∞.

In this light, the definitions of the sine, cosine, tangent and cotangent are no longer an acute angle, but an angle of arbitrary magnitude - the angle of rotation. They are given through the x and y coordinates of the point A 1 , into which the so-called initial point A(1, 0) passes after it rotates through an angle α around the point O - the beginning of a rectangular Cartesian coordinate system and the center of the unit circle.

Definition.

Sine of rotation angleα is the ordinate of the point A 1 , that is, sinα=y .

Definition.

cosine of the angle of rotationα is called the abscissa of the point A 1 , that is, cosα=x .

Definition.

Tangent of rotation angleα is the ratio of the ordinate of point A 1 to its abscissa, that is, tgα=y/x .

Definition.

The cotangent of the angle of rotationα is the ratio of the abscissa of the point A 1 to its ordinate, that is, ctgα=x/y .

The sine and cosine are defined for any angle α , since we can always determine the abscissa and ordinate of a point, which is obtained by rotating the starting point through the angle α . And tangent and cotangent are not defined for any angle. The tangent is not defined for such angles α at which the initial point goes to a point with zero abscissa (0, 1) or (0, −1) , and this takes place at angles 90°+180° k , k∈Z (π /2+π k rad). Indeed, at such angles of rotation, the expression tgα=y/x does not make sense, since it contains division by zero. As for the cotangent, it is not defined for such angles α at which the starting point goes to a point with zero ordinate (1, 0) or (−1, 0) , and this is the case for angles 180° k , k ∈Z (π k rad).

So, the sine and cosine are defined for any rotation angles, the tangent is defined for all angles except 90°+180° k , k∈Z (π/2+π k rad), and the cotangent is for all angles except 180° ·k , k∈Z (π·k rad).

The notations already known to us appear in the definitions sin, cos, tg and ctg, they are also used to denote the sine, cosine, tangent and cotangent of the angle of rotation (sometimes you can find the notation tan and cotcorresponding to tangent and cotangent). So the sine of the rotation angle of 30 degrees can be written as sin30°, the records tg(−24°17′) and ctgα correspond to the tangent of the rotation angle −24 degrees 17 minutes and the cotangent of the rotation angle α . Recall that when writing the radian measure of an angle, the notation "rad" is often omitted. For example, the cosine of a rotation angle of three pi rads is usually denoted cos3 π .

In conclusion of this paragraph, it is worth noting that in talking about the sine, cosine, tangent and cotangent of the angle of rotation, the phrase “angle of rotation” or the word “rotation” is often omitted. That is, instead of the phrase "sine of the angle of rotation alpha", the phrase "sine of the angle of alpha" is usually used, or even shorter - "sine of alpha". The same applies to cosine, and tangent, and cotangent.

Let's also say that the definitions of the sine, cosine, tangent, and cotangent of an acute angle in a right triangle are consistent with the definitions just given for the sine, cosine, tangent, and cotangent of a rotation angle ranging from 0 to 90 degrees. We will substantiate this.

Numbers

Definition.

Sine, cosine, tangent and cotangent of a number t is a number equal to the sine, cosine, tangent and cotangent of the angle of rotation in t radians, respectively.

For example, the cosine of 8 π is, by definition, a number equal to the cosine of an angle of 8 π rad. And the cosine of the angle in 8 π rad is equal to one, therefore, the cosine of the number 8 π is equal to 1.

There is another approach to the definition of the sine, cosine, tangent and cotangent of a number. It consists in the fact that each real number t is assigned a point of the unit circle centered at the origin of the rectangular coordinate system, and the sine, cosine, tangent and cotangent are determined through the coordinates of this point. Let's dwell on this in more detail.

Let us show how the correspondence between real numbers and points of the circle is established:

• the number 0 is assigned the starting point A(1, 0) ;
• positive number t corresponds to the point of the unit circle, which we will get to if we move along the circle from the starting point in the counterclockwise direction and go through a path of length t;
• negative number t corresponds to the point of the unit circle, which we will get to if we move around the circle from the starting point in a clockwise direction and go through a path of length |t| .

Now let's move on to the definitions of sine, cosine, tangent and cotangent of the number t. Let us assume that the number t corresponds to a point of the circle A 1 (x, y) (for example, the number &pi/2; corresponds to the point A 1 (0, 1) ).

Definition.

The sine of a number t is the ordinate of the unit circle point corresponding to the number t , that is, sint=y .

Definition.

The cosine of a number t is called the abscissa of the point of the unit circle corresponding to the number t , that is, cost=x .

Definition.

Tangent of a number t is the ratio of the ordinate to the abscissa of the point of the unit circle corresponding to the number t, that is, tgt=y/x. In another equivalent formulation, the tangent of the number t is the ratio of the sine of this number to the cosine, that is, tgt=sint/cost .

Definition.

Cotangent of a number t is the ratio of the abscissa to the ordinate of the point of the unit circle corresponding to the number t, that is, ctgt=x/y. Another formulation is as follows: the tangent of the number t is the ratio of the cosine of the number t to the sine of the number t : ctgt=cost/sint .

Here we note that the definitions just given agree with the definition given at the beginning of this subsection. Indeed, the point of the unit circle corresponding to the number t coincides with the point obtained by rotating the starting point through an angle of t radians.

It is also worth clarifying this point. Let's say we have a sin3 entry. How to understand whether the sine of the number 3 or the sine of the rotation angle of 3 radians is in question? This is usually clear from the context, otherwise it probably doesn't matter.

Trigonometric functions of angular and numerical argument

According to the definitions given in the previous paragraph, each rotation angle α corresponds to a well-defined value sin α , as well as the value cos α . In addition, all rotation angles other than 90°+180° k , k∈Z (π/2+π k rad) correspond to the values ​​tgα , and other than 180° k , k∈Z (π k rad ) are the values ​​of ctgα . Therefore sinα, cosα, tgα and ctgα are functions of the angle α. In other words, these are functions of the angular argument.

Similarly, we can talk about the functions sine, cosine, tangent and cotangent of a numerical argument. Indeed, each real number t corresponds to a well-defined value of sint , as well as cost . In addition, all numbers other than π/2+π·k , k∈Z correspond to the values ​​tgt , and the numbers π·k , k∈Z correspond to the values ​​ctgt .

The functions sine, cosine, tangent and cotangent are called basic trigonometric functions.

It is usually clear from the context that we are dealing with trigonometric functions of an angular argument or a numerical argument. Otherwise, we can consider the independent variable as both a measure of the angle (the angle argument) and a numeric argument.

However, the school mainly studies numeric functions, that is, functions whose arguments, as well as their corresponding function values, are numbers. Therefore, if we are talking specifically about functions, it is expedient to consider trigonometric functions as functions of numerical arguments.

Connection of definitions from geometry and trigonometry

If we consider the angle of rotation α from 0 to 90 degrees, then the data in the context of trigonometry of the definition of the sine, cosine, tangent and cotangent of the angle of rotation are fully consistent with the definitions of the sine, cosine, tangent and cotangent of an acute angle in a right triangle, which are given in the geometry course. Let's substantiate this.

Draw a unit circle in the rectangular Cartesian coordinate system Oxy. Note the starting point A(1, 0) . Let's rotate it by an angle α ranging from 0 to 90 degrees, we get the point A 1 (x, y) . Let's drop the perpendicular A 1 H from the point A 1 to the Ox axis.

It is easy to see that in a right triangle the angle A 1 OH is equal to the angle of rotation α, the length of the leg OH adjacent to this angle is equal to the abscissa of the point A 1, that is, |OH|=x, the length of the leg A 1 H opposite to the angle is equal to the ordinate of the point A 1 , that is, |A 1 H|=y , and the length of the hypotenuse OA 1 is equal to one, since it is the radius of the unit circle. Then, by definition from geometry, the sine of an acute angle α in a right triangle A 1 OH is equal to the ratio of the opposite leg to the hypotenuse, that is, sinα=|A 1 H|/|OA 1 |=y/1=y . And by definition from trigonometry, the sine of the angle of rotation α is equal to the ordinate of the point A 1, that is, sinα=y. This shows that the definition of the sine of an acute angle in a right triangle is equivalent to the definition of the sine of the angle of rotation α for α from 0 to 90 degrees.

Similarly, it can be shown that the definitions of the cosine, tangent, and cotangent of an acute angle α are consistent with the definitions of the cosine, tangent, and cotangent of the angle of rotation α.

Bibliography.

1. Geometry. 7-9 grades: studies. for general education institutions / [L. S. Atanasyan, V. F. Butuzov, S. B. Kadomtsev and others]. - 20th ed. M.: Education, 2010. - 384 p.: ill. - ISBN 978-5-09-023915-8.
2. Pogorelov A.V. Geometry: Proc. for 7-9 cells. general education institutions / A. V. Pogorelov. - 2nd ed. - M.: Enlightenment, 2001. - 224 p.: ill. - ISBN 5-09-010803-X.
3. Algebra and elementary functions : Tutorial for students of the 9th grade of secondary school / E. S. Kochetkov, E. S. Kochetkova; Edited by Doctor of Physical and Mathematical Sciences O. N. Golovin. - 4th ed. Moscow: Education, 1969.
4. Algebra: Proc. for 9 cells. avg. school / Yu. N. Makarychev, N. G. Mindyuk, K. I. Neshkov, S. B. Suvorova; Ed. S. A. Telyakovsky.- M.: Enlightenment, 1990.- 272 p.: Ill.- ISBN 5-09-002727-7
5. Algebra and the beginning of the analysis: Proc. for 10-11 cells. general education institutions / A. N. Kolmogorov, A. M. Abramov, Yu. P. Dudnitsyn and others; Ed. A. N. Kolmogorova.- 14th ed.- M.: Enlightenment, 2004.- 384 p.: ill.- ISBN 5-09-013651-3.
6. Mordkovich A. G. Algebra and the beginnings of analysis. Grade 10. At 2 p. Ch. 1: a tutorial for educational institutions(profile level) / A. G. Mordkovich, P. V. Semenov. - 4th ed., add. - M.: Mnemosyne, 2007. - 424 p.: ill. ISBN 978-5-346-00792-0.
7. Algebra and the beginning of mathematical analysis. Grade 10: textbook. for general education institutions: basic and profile. levels /[Yu. M. Kolyagin, M. V. Tkacheva, N. E. Fedorova, M. I. Shabunin]; ed. A. B. Zhizhchenko. - 3rd ed. - I .: Education, 2010. - 368 p.: Ill. - ISBN 978-5-09-022771-1.
8. Bashmakov M.I. Algebra and the beginning of analysis: Proc. for 10-11 cells. avg. school - 3rd ed. - M.: Enlightenment, 1993. - 351 p.: ill. - ISBN 5-09-004617-4.
9. Gusev V. A., Mordkovich A. G. Mathematics (a manual for applicants to technical schools): Proc. allowance.- M.; Higher school, 1984.-351 p., ill.

Sinus acute angle α of a right triangle is the ratio opposite catheter to the hypotenuse.
It is denoted as follows: sin α.

Cosine acute angle α of a right triangle is the ratio of the adjacent leg to the hypotenuse.
It is denoted as follows: cos α.

Tangent acute angle α is the ratio of the opposite leg to the adjacent leg.
It is denoted as follows: tg α.

Cotangent acute angle α is the ratio of the adjacent leg to the opposite one.
It is designated as follows: ctg α.

The sine, cosine, tangent and cotangent of an angle depend only on the magnitude of the angle.

Rules:

Basic trigonometric identities in a right triangle:

(α - acute angle opposite the leg b and adjacent to the leg a . Side With - hypotenuse. β - the second acute angle).

b sinα = - c	sin 2 α + cos 2 α = 1
a cosα = - c	1 1 + tg 2 α = -- cos 2 α
b tgα = - a	1 1 + ctg 2 α = -- sin2α
a ctgα = - b	1 1 1 + -- = -- tg 2 α sin 2 α
sinα tgα = -- cosα

As the acute angle increasessinα andtg α increase, andcos α decreases.

For any acute angle α:

sin (90° - α) = cos α

cos (90° - α) = sin α

Explanatory example:

Let in a right triangle ABC
AB = 6,
BC = 3,
angle A = 30º.

Find the sine of angle A and the cosine of angle B.

Solution .

1) First, we find the value of angle B. Everything is simple here: since in a right triangle the sum of acute angles is 90º, then angle B \u003d 60º:

B \u003d 90º - 30º \u003d 60º.

2) Calculate sin A. We know that the sine is equal to the ratio of the opposite leg to the hypotenuse. For angle A, the opposite leg is side BC. So:

BC 3 1
sin A = -- = - = -
AB 6 2

3) Now we calculate cos B. We know that the cosine is equal to the ratio of the adjacent leg to the hypotenuse. For angle B, the adjacent leg is the same side BC. This means that we again need to divide BC into AB - that is, perform the same actions as when calculating the sine of angle A:

BC 3 1
cos B = -- = - = -
AB 6 2

The result is:
sin A = cos B = 1/2.

sin 30º = cos 60º = 1/2.

From this it follows that in a right triangle the sine of one acute angle is equal to the cosine of another acute angle - and vice versa. This is exactly what our two formulas mean:
sin (90° - α) = cos α
cos (90° - α) = sin α

Let's check it out again:

1) Let α = 60º. Substituting the value of α into the sine formula, we get:
sin (90º - 60º) = cos 60º.
sin 30º = cos 60º.

2) Let α = 30º. Substituting the value of α into the cosine formula, we get:
cos (90° - 30º) = sin 30º.
cos 60° = sin 30º.

(For more on trigonometry, see the Algebra section)

The ratio of the opposite leg to the hypotenuse is called sine of an acute angle right triangle.

\sin \alpha = \frac(a)(c)

Cosine of an acute angle of a right triangle

The ratio of the nearest leg to the hypotenuse is called cosine of an acute angle right triangle.

\cos \alpha = \frac(b)(c)

Tangent of an acute angle of a right triangle

The ratio of the opposite leg to the adjacent leg is called acute angle tangent right triangle.

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

Cotangent of an acute angle of a right triangle

The ratio of the adjacent leg to the opposite leg is called cotangent of an acute angle right triangle.

ctg \alpha = \frac(b)(a)

Sine of an arbitrary angle

The ordinate of the point on the unit circle to which the angle \alpha corresponds is called sine of an arbitrary angle rotation \alpha .

\sin \alpha=y

Cosine of an arbitrary angle

The abscissa of a point on the unit circle to which the angle \alpha corresponds is called cosine of an arbitrary angle rotation \alpha .

\cos \alpha=x

Tangent of an arbitrary angle

The ratio of the sine of an arbitrary rotation angle \alpha to its cosine is called tangent of an arbitrary angle rotation \alpha .

tg \alpha = y_(A)

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

Cotangent of an arbitrary angle

The ratio of the cosine of an arbitrary rotation angle \alpha to its sine is called cotangent of an arbitrary angle rotation \alpha .

ctg \alpha =x_(A)

ctg \alpha = \frac(\cos \alpha)(\sin \alpha)

An example of finding an arbitrary angle

If \alpha is some angle AOM , where M is a point on the unit circle, then

\sin \alpha=y_(M) , \cos \alpha=x_(M) , tg \alpha=\frac(y_(M))(x_(M)), ctg \alpha=\frac(x_(M))(y_(M)).

For example, if \angle AOM = -\frac(\pi)(4), then: the ordinate of the point M is -\frac(\sqrt(2))(2), the abscissa is \frac(\sqrt(2))(2) and that's why

\sin \left (-\frac(\pi)(4) \right)=-\frac(\sqrt(2))(2);

\cos \left (\frac(\pi)(4) \right)=\frac(\sqrt(2))(2);

tg;

ctg \left (-\frac(\pi)(4) \right)=-1.

Table of values ​​of sines of cosines of tangents of cotangents

The values ​​of the main frequently encountered angles are given in the table:

	0^(\circ) (0)	30^(\circ)\left(\frac(\pi)(6)\right)	45^(\circ)\left(\frac(\pi)(4)\right)	60^(\circ)\left(\frac(\pi)(3)\right)	90^(\circ)\left(\frac(\pi)(2)\right)	180^(\circ)\left(\pi\right)	270^(\circ)\left(\frac(3\pi)(2)\right)	360^(\circ)\left(2\pi\right)
\sin\alpha	0	\frac12	\frac(\sqrt 2)(2)	\frac(\sqrt 3)(2)	1	0	−1	0
\cos\alpha	1	\frac(\sqrt 3)(2)	\frac(\sqrt 2)(2)	\frac12	0	−1	0	1
tg\alpha	0	\frac(\sqrt 3)(3)	1	\sqrt3	—	0	—	0
ctg\alpha	—	\sqrt3	1	\frac(\sqrt 3)(3)	0	—	0	—

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 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 \(P\) . In this way,

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

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:

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

Javascript is disabled in your browser.
ActiveX controls must be enabled in order to make calculations!