Sin is the ratio of the opposite leg to the hypotenuse. Sine, cosine, tangent and cotangent: definitions in trigonometry, examples, formulas
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 cotangent value can be calculated by dividing the length of the near leg from the leg angle by the length of the far one, as well as 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.
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.
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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 of an acute angle in a right triangle is given. 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 a 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) ;
- a positive number t is associated with a point on the unit circle, which we will get to if we move around the circle from the starting point in a counterclockwise direction and go through a path of length t;
- a negative number t is associated with a point on 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 of sinα, as well as the value of cosα. 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 numerical functions, that is, functions whose arguments, as well as the corresponding function values, are numbers. Therefore, if we are talking about functions, then it is advisable 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.
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- 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.
- Algebra and elementary functions: Textbook for students of grade 9 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.
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In this article, we will take a comprehensive look at . Basic trigonometric identities are equalities that establish a relationship between the sine, cosine, tangent and cotangent of one angle, and allow you to find any of these trigonometric functions through a known other.
We immediately list the main trigonometric identities, which we will analyze in this article. We write them down in a table, and below we give the derivation of these formulas and give the necessary explanations.
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Relationship between sine and cosine of one angle
Sometimes they talk not about the main trigonometric identities listed in the table above, but about one single basic trigonometric identity kind 
. The explanation for this fact is quite simple: the equalities are obtained from the basic trigonometric identity after dividing both of its parts by and respectively, and the equalities 
And 
follow from the definitions of sine, cosine, tangent, and cotangent. We will discuss this in more detail in the following paragraphs.
That is, it is the equality that is of particular interest, which was given the name of the main trigonometric identity.
Before proving the basic trigonometric identity, we give its formulation: the sum of the squares of the sine and cosine of one angle is identically equal to one. Now let's prove it.
The basic trigonometric identity is very often used in transformation of trigonometric expressions. It allows the sum of the squares of the sine and cosine of one angle to be replaced by one. No less often, the basic trigonometric identity is used in reverse order: the unit is replaced by the sum of the squares of the sine and cosine of any angle.
Tangent and cotangent through sine and cosine
Identities connecting the tangent and cotangent with the sine and cosine of one angle of the form and 
immediately follow from the definitions of sine, cosine, tangent and cotangent. Indeed, by definition, the sine is the ordinate of y, the cosine is the abscissa of x, the tangent is the ratio of the ordinate to the abscissa, that is, 
, and the cotangent is the ratio of the abscissa to the ordinate, that is, 
.
Due to this obviousness of the identities and often the definitions of tangent and cotangent are given not through the ratio of the abscissa and the ordinate, but through the ratio of the sine and cosine. So the tangent of an angle is the ratio of the sine to the cosine of this angle, and the cotangent is the ratio of the cosine to the sine.
To conclude this section, it should be noted that the identities and hold for all such angles for which the trigonometric functions in them make sense. So the formula is valid for any other than (otherwise the denominator will be zero, and we did not define division by zero), and the formula - for all , different from , where z is any .
Relationship between tangent and cotangent
An even more obvious trigonometric identity than the two previous ones is the identity connecting the tangent and cotangent of one angle of the form . It is clear that it takes place for any angles other than , otherwise either the tangent or the cotangent is not defined.
Proof of the formula very simple. By definition and from where 
. The proof could have been carried out in a slightly different way. Since and , then 
.
So, the tangent and cotangent of one angle, at which they make sense, is.
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 | — |
Instruction
The first option is classic, using paper, a protractor and a pencil (or pen). By definition, sine angle equal to the opposite leg to the hypotenuse of a right triangle. That is, to calculate the value, you need to use a protractor to build a right-angled triangle, one of the angles of which is equal to the one whose sine you are interested in. Then measure the length of the hypotenuse and the opposite leg and divide the second by the first with the desired accuracy.
The second option is school. From school, everyone remembers the “Bradis tables”, containing thousands of trigonometric values from different angles. You can search for both the paper edition and its electronic counterpart in pdf format - they are available online. Having found the tables, find the value sinus necessary angle won't be difficult.
The third option is the best. If you have access to, then you can use the standard Windows calculator. It should be switched to advanced mode. To do this, in the "View" section of the menu, select the item "Engineering". The view of the calculator will change - it will appear, in particular, buttons for calculating trigonometric functions. Now enter the value angle, whose sine you want to calculate. You can do this both from the keyboard and by clicking the desired calculator keys with the mouse cursor. Or you can just paste the value you need (CTRL + C and CTRL + V). After that, select the units in which it should be calculated - for trigonometric functions, these can be radians, degrees, or rads. This is done by selecting one of the three switch values located below the input field of the calculated value. Now, by pressing the button labeled "sin", get the answer to your question.
The fourth option is the most modern. In the era of the Internet, there are on the net offering almost every problem that arises. Online calculators of trigonometric functions with a user-friendly interface, more advanced functionality are not to be found at all. The best of them offer to calculate not only the values of a single function, but also rather complex expressions from several functions.
Functions sinus and co sinus belong to the area of mathematics, which is called trigonometry, and therefore the functions themselves are called trigonometric. According to the oldest of the definitions, they express the size of an acute angle in a right triangle in terms of the ratio of the lengths of its sides. Calculating values sinus and with the current level of development of electronic technology - a fairly simple task.
You will need
- Windows Calculator.
Instruction
Use to calculate sinus and the angle - the calculation of trigonometric functions is provided in most of them. Given the presence of a calculator in many mobile phones, some wrist and other mobile gadgets, not to mention computers, this is perhaps an affordable way to calculate sinus but. If you decide to use the computer's software calculator, then look for a link to launch it in the main menu of the OS. If it's Windows, press the Win button, select "All Programs" from the menu, go to the "Accessories" subsection and click on the "Calculator" line. To access the commands for calculating trigonometric functions in the launched application, press the key combination Alt + 2.
If in the initial value of the angle, sinus which you want to calculate is given in , make sure that next to the inscription “ ” in the calculator interface