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 denoted 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! There are many problems in the Unified State Examination in mathematics that involve sine, cosine, tangent or cotangent of an external angle of a triangle. More on this in the next article.
Back forward
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Lesson objectives:
- introduce the concepts of sine, cosine and tangent of an acute angle of a right triangle;
- show how sine, cosine and tangent are used in solving problems;
- development of skills to observe, compare, analyze and draw conclusions.
During the classes
Updating knowledge (identifying the main problem of the lesson)
Conducted in the form of a frontal survey.
Teacher. On the board you see a summary of 6 problems< Рисунок 1>. Remember which of these problems you already know how to solve? Solve these problems. Formulate the corresponding theorems.
Picture 1
Students:
Task 1. Answer: 5. In a right triangle, the leg opposite the 30° angle is equal to half the hypotenuse.
Task 2. Answer: 41°. The sum of the interior angles of a triangle is 180°.
Task 3. Answer: 10. The square of the hypotenuse of a right triangle is equal to the sum of the squares of the legs.
Problems 4-6 we can't decide.
Teacher. Why can’t you solve problems 4-6? What question arises?
Students. We don't know what tgB, sinA, cosB are.
Teacher. sinA, cosB, tanB is read: “sine of angle A”, “cosine of angle B” and “tangent of angle B”. Today we will learn what each of these expressions means and learn how to solve problems like 4-6.
Introduction of new material
Conducted in the form of a heuristic conversation.
Teacher. Draw right triangles with legs 3 and 4, 6 and 8. Label them ABC and A 1 B 1 C 1 so that B and B 1 are angles opposite to legs 4 and 8, and right angles are C, C 1. Are angles B and B1 equal? Why?
Students. Equal because the triangles are similar. AC: BC = A 1 C 1: B 1 C 1 (3: 4 = 6: 8) and the angles between them are right.<Рисунок 2>
Teacher. Equalities of what other relations follow from the similarity of triangles ABC and A 1 B 1 C 1?
Students. BC: AB = B 1 C 1: A 1 B 1, AC: AB = A 1 C 1: A 1 B 1.
Teacher. AC: AB = A 1 C 1: A 1 B 1 = sinB = sinB 1.
BC: AB = B 1 C 1: A 1 B 1 = cosB = cosB 1. AC: BC = A 1 C 1: B 1 C 1 = tgB = tgB 1. Leg AC is opposite to angle B, and leg BC is adjacent to this angle. State the definitions of sine, cosine and tangent.
Students. The sine of an acute angle of a right triangle is the ratio of the opposite side to the hypotenuse.
The cosine of an acute angle of a right triangle is the ratio of the adjacent leg to the hypotenuse.
The tangent of an acute angle of a right triangle is the ratio of the opposite side to the adjacent side.
Teacher. Write down the sine, cosine and tangent of angle A yourself (slide 1). The resulting formulas (1), (2), (3):
(1)
So, we learned what sine, cosine and tangent of an acute angle of a right triangle are. In general, the concepts of sine, cosine and tangent have a long history. By studying the relationship between the sides and angles of a triangle, ancient scientists found ways to calculate the various elements of a triangle. This knowledge was mainly used to solve problems of practical astronomy, to determine inaccessible distances.
Consolidation
Teacher. Let's solve problem No. 591 (a, b).
The task is displayed on the screen (slide 2). Task “a” is solved on the board with a full explanation; “b” – independently, followed by checking each other.
Find the sine, cosine and tangent of angles A and B of triangle ABC with right angle C, if: a) BC = 8, AB = 17; b) BC = 21, AC = 20.
Solution. a) = . = , using the Pythagorean theorem we find AC = 15,
= ; b), using the Pythagorean theorem we find AB = 29, . . .Teacher. Now let's return to problems 4–6<Рисунок 1>. Let's discuss what is known in problems 4–6 and what needs to be found?
Task 4. What is known? What do you need to find?
Students. BC = 7 and tan B = 3.5 are known. We need to find the AC.
Teacher. What is tg B?
Students. .
Teacher. We work with the formula. The formula consists of three components. Name them. What components are known? Which component is unknown? Can you find it? Find it.
Students. AC = BC * tg B = 7 * 3.5 = 24.5
Teacher. Using this example, solve problems 5 and 6<Рисунок 1>. 1 student works on a closed board
Teacher.
1. Tell me, did you manage to find the required unknowns?
2. What was the order of your actions?
3. Maybe there are other solutions?
Students.1. Yes. Easily. Following the example. Problem 5. Answer: 10. Problem 6. Answer: 2.5
2. First, we replace the sine and cosine of the corresponding angles by definition with the corresponding ratios, then we put the known data in the resulting proportions, after which we find the unknown unknowns.
Teacher. What general conclusion can be drawn after solving problems 4–6? What new problems have we learned to solve in a right triangle? Think and formulate your conclusion.
Students. If in a right triangle you know one side and the ratio of that side to one of the other sides, or one side and the ratio of one of the other sides to a known side (either sine, cosine, or tangent), then you can find this second side.
Problem solving.
Now try to solve these problems 7–9<Рисунок 3>.
Figure 3
Students. We don't know how to solve them.
Teacher. Let's return to problem 1<Рисунок 1>. Let's change the problem condition. Let NK = 5, NM = 10. Find the angle M.
Students. Angle M is equal to 30°, since the leg opposite the angle M is equal to half the hypotenuse.
Teacher. That is, it turns out that if the sine of the angle is 0.5, then the angle is 30°. Now let’s solve problems No. 592 (a, c, d)
No. 592. Construct an angle a, if: a) c) d) .
Solution.
a) On the sides of the right angle we will place segments of length 1 and 2 and connect the ends of the segments. In the resulting triangle, the angle opposite leg 1 is the desired angle a;
c) 0.2 = . On one side of the right angle from its vertex we lay off a segment of length 1. Construct a circle of radius 5 with the center at the end of the laid off segment. The point of intersection of the circle with the second side of the right angle is connected to the end of the segment laid out on the first side of the angle. In the resulting triangle, the angle adjacent to the leg of length 1 is the angle a; (slide 4)
e) On one side of the right angle from its vertex we lay off a segment of length 1. Construct a circle of radius 2 with the center at the end of the laid off segment. The point of intersection of the circle with the second side of the right angle is connected to the end of the segment laid out on the first side of the angle. In the resulting triangle, the angle opposite to the leg of length 1 is the desired angle a.(slide 5)
You have built the angles, which means you have found the angles. They can be measured and presented in table form.
Similarly, you can solve problems 7-9<Рисунок 3>
Summarizing
Teacher. Answer the questions:
1. What are the sine, cosine and tangent of a right angle in a right triangle?
2. There are 6 elements in a right triangle. What new problems have you learned to solve today? What is your order of action? Test your ability to perform these actions correctly (Individual cards are distributed).
Approximate contents of the cards: 1. In triangle ABC, angle C is a right angle, BC = 2, Find AB. 2. In triangle ABC, angle C is a straight line, AC = 8, . Find AB. 3. In triangle ABC, angle C is 90°, AC = 6, . Find the sun.
Students compare their work with ready-made solutions on the corresponding cards.
Homework assignments: question 15 on page 159; No. 591(c,d),592(b,d,f) (slide 6)
References
- Geometry. Grades 7–9: textbook. for educational organizations / [L.S. Atanasyan, V.F. Butuzov, S.B. Kadomtsev and others]. – 2nd ed. – M.: Education, 2014.
Trigonometry is a branch of mathematical science that studies trigonometric functions and their use in geometry. The development of trigonometry began in ancient Greece. During the Middle Ages, scientists from the Middle East and India made important contributions to the development of this science.
This article is devoted to the basic concepts and definitions of trigonometry. It discusses the definitions of the basic trigonometric functions: sine, cosine, tangent and cotangent. Their meaning is explained and illustrated in the context of geometry.
Yandex.RTB R-A-339285-1
Initially, the definitions of trigonometric functions whose argument is an angle were expressed in terms of the ratio of the sides of a right triangle.
Definitions of trigonometric functions
The sine of an angle (sin α) is the ratio of the leg opposite this angle to the hypotenuse.
Cosine of the angle (cos α) - the ratio of the adjacent leg to the hypotenuse.
Angle tangent (t g α) - the ratio of the opposite side to the adjacent side.
Angle cotangent (c t g α) - the ratio of the adjacent side to the opposite side.
These definitions are given for the acute angle of a right triangle!
Let's give an illustration.
In triangle ABC with 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 is from -1 to 1. In other words, sine and cosine take values from -1 to 1. The range of values of tangent and cotangent is the entire number line, that is, these functions can take on any values.
The definitions given above apply to acute angles. In trigonometry, the concept of a rotation angle is introduced, the value of which, unlike an acute angle, is not limited to 0 to 90 degrees. The rotation angle in degrees or radians is expressed by any real number from - ∞ to + ∞.
In this context, we can define sine, cosine, tangent and cotangent of an angle of arbitrary magnitude. Let us imagine a unit circle with its center at the origin of the Cartesian coordinate system.
The initial point A with coordinates (1, 0) rotates around the center of the unit circle through a certain angle α and goes to point A 1. The definition is given in terms of the coordinates of point A 1 (x, y).
Sine (sin) of the rotation angle
The sine of the rotation angle α is the ordinate of point A 1 (x, y). sin α = y
Cosine (cos) of the rotation angle
The cosine of the rotation angle α is the abscissa of point A 1 (x, y). cos α = x
Tangent (tg) of the rotation angle
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 rotation angle
The cotangent of the rotation angle 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 rotation angle. This is logical, because the abscissa and ordinate of a point after rotation can be determined at any angle. The situation is different with tangent and cotangent. The tangent is undefined when a point after rotation goes to a point with zero abscissa (0, 1) and (0, - 1). In such cases, the expression for tangent t g α = y x simply does not make sense, since it contains division by zero. The situation is similar with cotangent. The difference is that the cotangent is not defined in cases where the ordinate of a point goes to zero.
Important to remember!
Sine and cosine are defined for any angles α.
Tangent is defined for all angles except α = 90° + 180° k, k ∈ Z (α = π 2 + π k, k ∈ Z)
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 already clear from the context what is being discussed.
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 the number 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 take a closer look at it.
Any real number t a point on the unit circle is associated with the center at the origin of the rectangular Cartesian coordinate system. Sine, cosine, tangent and cotangent are determined through the coordinates of this point.
The starting point on the circle is point A with coordinates (1, 0).
Positive number t
Negative number t corresponds to the point to which the starting point will go if it moves around the circle counterclockwise and passes the path t.
Now that the connection between a number and a point on a circle has been established, we move on to the definition of sine, cosine, tangent and cotangent.
Sine (sin) of t
Sine of a number t- ordinate of a point on the unit circle corresponding to the number t. sin t = y
Cosine (cos) of t
Cosine of a number t- abscissa of the point of the unit circle corresponding to the number t. cos t = x
Tangent (tg) of t
Tangent of a number t- the ratio of the ordinate to the abscissa of a point on the unit circle corresponding to the number t. t g t = y x = sin t cos t
The latest definitions are in accordance with and do not contradict the definition given at the beginning of this paragraph. Point on the circle corresponding to the number t, coincides with the point to which the starting point goes after turning 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. Just like all angles α other than α = 90 ° + 180 ° k, k ∈ Z (α = π 2 + π k, k ∈ Z) correspond to a certain tangent value. Cotangent, as stated above, is defined for all α except 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, we can talk about sine, cosine, tangent and cotangent as functions of a numerical argument. Every real number t corresponds to a certain value of the sine or cosine of a number t. All numbers other than π 2 + π · k, k ∈ Z, correspond to a tangent value. Cotangent, similarly, is defined for all numbers except π · k, k ∈ Z.
Basic functions of trigonometry
Sine, cosine, tangent and cotangent are the basic trigonometric functions.
It is usually clear from the context which argument of the trigonometric function (angular argument or numeric argument) we are dealing with.
Let's return to the definitions given at the very beginning and the alpha angle, which lies in the range from 0 to 90 degrees. The trigonometric definitions of sine, cosine, tangent, and cotangent are entirely consistent with the geometric definitions given by the aspect ratios of a right triangle. Let's show it.
Let's take a unit circle with a center in a rectangular Cartesian coordinate system. Let's rotate the starting point A (1, 0) by an angle of up to 90 degrees and draw a perpendicular to the abscissa axis from the resulting point A 1 (x, y). In the resulting right 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 the point A 1 (x, y). The length of the leg opposite the angle is equal to the ordinate of the point A 1 (x, y), and the length of the hypotenuse is equal to one, since it is the radius of the unit circle.
In accordance with the definition from geometry, the sine of angle α is equal to the ratio of the opposite side 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 triangle through the aspect ratio is equivalent to determining the sine of the rotation angle 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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In this article we will show how to give definitions of sine, cosine, tangent and cotangent of an angle and number in trigonometry. Here we will talk about notations, give examples of entries, and give graphic illustrations. In conclusion, let us 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 see how the idea of sine, cosine, tangent and cotangent is formed in a 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 talks about sine, cosine, tangent and cotangent of the angle of rotation and number. Let us present all these definitions, give examples and give the necessary comments.
Acute angle in a right triangle
From the geometry course we know the definitions of sine, cosine, tangent and cotangent of an acute angle in a right triangle. They are given as the ratio of the sides of a right triangle. Let us give their formulations.
Definition.
Sine of an acute angle in a right triangle is the ratio of the opposite side 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– this is the ratio of the opposite side to the adjacent side.
Definition.
Cotangent of an acute angle in a right triangle- this is the ratio of the adjacent side to the opposite side.
The designations for sine, cosine, tangent and cotangent are also introduced there - sin, cos, tg and ctg, respectively.
For example, if ABC is a right triangle with right angle C, then the sine of the acute angle A is equal to the ratio of the opposite side BC to the hypotenuse AB, that is, sin∠A=BC/AB.
These definitions allow you to calculate the values of 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 sine, cosine, tangent, cotangent and the length of one of the sides to find the lengths of the other sides. For example, if we knew that in a right triangle the leg AC is equal to 3 and the hypotenuse AB is equal to 7, then we could calculate the value of the cosine of the acute angle A by definition: cos∠A=AC/AB=3/7.
Rotation angle
In trigonometry, they begin to look at the angle more broadly - they introduce the concept of angle of rotation. The magnitude of the rotation angle, unlike an acute angle, is not limited to 0 to 90 degrees; the rotation angle in degrees (and in radians) can be expressed by any real number from −∞ to +∞.
In this light, the definitions of sine, cosine, tangent and cotangent are given not of an acute angle, but of an angle of arbitrary size - the angle of rotation. They are given through the x and y coordinates of the point A 1, to which the so-called starting point A(1, 0) goes after its rotation by an angle α around the point O - the beginning of the rectangular Cartesian coordinate system and the center of the unit circle.
Definition.
Sine of rotation angleα is the ordinate of point A 1, that is, sinα=y.
Definition.
Cosine of the rotation angleα is called the abscissa of 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, tanα=y/x.
Definition.
Cotangent of the rotation angleα is the ratio of the abscissa of point A 1 to its ordinate, that is, ctgα=x/y.
Sine and cosine are defined for any angle α, since we can always determine the abscissa and ordinate of the point, which is obtained by rotating the starting point by angle α. But tangent and cotangent are not defined for any angle. The tangent is not defined for angles α at which the starting point goes to a point with zero abscissa (0, 1) or (0, −1), and this occurs 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 angles α at which the starting point goes to the point with the zero ordinate (1, 0) or (−1, 0), and this occurs for angles 180° k, k ∈Z (π·k rad).
So, sine and cosine are defined for any rotation angles, tangent is defined for all angles except 90°+180°k, k∈Z (π/2+πk rad), and cotangent is defined for all angles except 180° ·k , k∈Z (π·k rad).
The definitions include the designations already known to us sin, cos, tg and ctg, they are also used to designate sine, cosine, tangent and cotangent of the angle of rotation (sometimes you can find the designations tan and cotcorresponding to tangent and cotangent). So the sine of a rotation angle of 30 degrees can be written as sin30°, the entries 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 designation “rad” is often omitted. For example, the cosine of a rotation angle of three pi rad is usually denoted cos3·π.
In conclusion of this point, it is worth noting that when talking about 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 rotation angle alpha,” the phrase “sine of the alpha angle” or even shorter, “sine alpha,” is usually used. The same applies to cosine, tangent, and cotangent.
We will also say that the definitions of sine, cosine, tangent and cotangent of an acute angle in a right triangle are consistent with the definitions just given for sine, cosine, tangent and cotangent of an angle of rotation ranging from 0 to 90 degrees. We will justify 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 rotation angle in t radians, respectively.
For example, the cosine of the number 8·π by definition is a number equal to the cosine of the angle of 8·π rad. And the cosine of an angle of 8·π rad is equal to one, therefore, the cosine of the number 8·π is equal to 1.
There is another approach to determining the sine, cosine, tangent and cotangent of a number. It consists in the fact that each real number t is associated with a point on the unit circle with the center at the origin of the rectangular coordinate system, and sine, cosine, tangent and cotangent are determined through the coordinates of this point. Let's look at this in more detail.
Let us show how a correspondence is established between real numbers and points on a circle:
- the number 0 is assigned the starting point A(1, 0);
- the positive number t is associated with a point on the unit circle, which we will get to if we move along the circle from the starting point in a counterclockwise direction and walk a path of length t;
- the negative number t is associated with a point on the unit circle, which we will get to if we move along the circle from the starting point in a clockwise direction and walk a path of length |t| .
Now we 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 on the circle A 1 (x, y) (for example, the number &pi/2; corresponds to the point A 1 (0, 1) ).
Definition.
Sine of the number t is the ordinate of the point on the unit circle corresponding to the number t, that is, sint=y.
Definition.
Cosine of the 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 the number t is the ratio of the ordinate to the abscissa of a point on the unit circle corresponding to the number t, that is, tgt=y/x. In another equivalent formulation, the tangent of a number t is the ratio of the sine of this number to the cosine, that is, tgt=sint/cost.
Definition.
Cotangent of the number t is the ratio of the abscissa to the ordinate of a point on the unit circle corresponding to the number t, that is, ctgt=x/y. Another formulation is this: 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 are consistent with the definition given at the beginning of this paragraph. Indeed, the point on the unit circle corresponding to the number t coincides with the point obtained by rotating the starting point by an angle of t radians.
It is still worth clarifying this point. Let's say we have the entry sin3. How can we understand whether we are talking about the sine of the number 3 or the sine of the rotation angle of 3 radians? This is usually clear from the context, otherwise it is likely not of fundamental importance.
Trigonometric functions of angular and numeric argument
According to the definitions given in the previous paragraph, each angle of rotation α corresponds to a very specific value sinα, as well as the value cosα. In addition, all rotation angles other than 90°+180°k, k∈Z (π/2+πk rad) correspond to tgα values, and values other than 180°k, k∈Z (πk rad ) – values of ctgα . Therefore sinα, cosα, tanα and ctgα are functions of the angle α. In other words, these are functions of the angular argument.
We can speak similarly about the functions sine, cosine, tangent and cotangent of a numerical argument. Indeed, each real number t corresponds to a very specific value sint, as well as cost. In addition, all numbers other than π/2+π·k, k∈Z correspond to values tgt, and numbers π·k, k∈Z - values ctgt.
The functions sine, cosine, tangent and cotangent are called basic trigonometric functions.
It is usually clear from the context whether we are dealing with trigonometric functions of an angular argument or a numerical argument. Otherwise, we can think of the independent variable as both a measure of the angle (angular argument) and a numeric argument.
However, at school we mainly study numerical functions, that is, functions whose arguments, as well as their corresponding function values, are numbers. Therefore, if we are talking specifically about functions, then it is advisable to consider trigonometric functions as functions of numerical arguments.
Relationship between definitions from geometry and trigonometry
If we consider the rotation angle α ranging from 0 to 90 degrees, then the definitions of sine, cosine, tangent and cotangent of the rotation angle in the context of trigonometry are fully consistent with the definitions of sine, cosine, tangent and cotangent of an acute angle in a right triangle, which are given in the geometry course. Let's justify this.
Let us depict the unit circle in the rectangular Cartesian coordinate system Oxy. Let's mark the starting point A(1, 0) . Let's rotate it by an angle α ranging from 0 to 90 degrees, we get point A 1 (x, y). Let us drop the perpendicular A 1 H from 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 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 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 rotation angle α is equal to the ordinate of point A 1, that is, sinα=y. This shows that determining the sine of an acute angle in a right triangle is equivalent to determining the sine of the rotation angle α when α is from 0 to 90 degrees.
Similarly, it can be shown that the definitions of cosine, tangent and cotangent of an acute angle α are consistent with the definitions of cosine, tangent and cotangent of the rotation angle α.
Bibliography.
- Geometry. 7-9 grades: textbook for general education institutions / [L. S. Atanasyan, V. F. Butuzov, S. B. Kadomtsev, etc.]. - 20th ed. M.: Education, 2010. - 384 p.: ill. - ISBN 978-5-09-023915-8.
- Pogorelov A.V. Geometry: Textbook. for 7-9 grades. general education institutions / A. V. Pogorelov. - 2nd ed. - M.: Education, 2001. - 224 p.: ill. - ISBN 5-09-010803-X.
- Algebra and elementary functions: Textbook for students of 9th grade of secondary school / E. S. Kochetkov, E. S. Kochetkova; Edited by Doctor of Physical and Mathematical Sciences O. N. Golovin. - 4th ed. M.: Education, 1969.
- Algebra: Textbook for 9th grade. avg. school/Yu. N. Makarychev, N. G. Mindyuk, K. I. Neshkov, S. B. Suvorova; Ed. S. A. Telyakovsky. - M.: Education, 1990. - 272 pp.: ill. - ISBN 5-09-002727-7
- Algebra and the beginning of analysis: Proc. for 10-11 grades. general education institutions / A. N. Kolmogorov, A. M. Abramov, Yu. P. Dudnitsyn and others; Ed. A. N. Kolmogorov. - 14th ed. - M.: Education, 2004. - 384 pp.: ill. - ISBN 5-09-013651-3.
- Mordkovich A. G. Algebra and the beginnings of analysis. Grade 10. In 2 parts. Part 1: textbook for general education institutions (profile level) / A. G. Mordkovich, P. V. Semenov. - 4th ed., add. - M.: Mnemosyne, 2007. - 424 p.: ill. ISBN 978-5-346-00792-0.
- Algebra and the beginning of mathematical analysis. 10th grade: textbook. for general education institutions: basic and profile. levels /[Yu. M. Kolyagin, M. V. Tkacheva, N. E. Fedorova, M. I. Shabunin]; edited by A. B. Zhizhchenko. - 3rd ed. - I.: Education, 2010.- 368 p.: ill.- ISBN 978-5-09-022771-1.
- Bashmakov M. I. Algebra and the beginnings of analysis: Textbook. for 10-11 grades. avg. school - 3rd ed. - M.: Education, 1993. - 351 p.: ill. - ISBN 5-09-004617-4.
- Gusev V. A., Mordkovich A. G. Mathematics (a manual for those entering technical schools): Proc. allowance.- M.; Higher school, 1984.-351 p., ill.
One of the areas of mathematics that students struggle with the most is trigonometry. It is not surprising: in order to freely master this area of knowledge, you need spatial thinking, the ability to find sines, cosines, tangents, cotangents using formulas, simplify expressions, and be able to use the number pi in calculations. In addition, you need to be able to use trigonometry when proving theorems, and this requires either a developed mathematical memory or the ability to derive complex logical chains.
Origins of trigonometry
Getting acquainted with this science should begin with the definition of sine, cosine and tangent of an angle, but first you need to understand what trigonometry does in general.
Historically, the main object of study in this branch of mathematical science was right triangles. The presence of an angle of 90 degrees makes it possible to carry out various operations that allow one to determine the values of all parameters of the figure in question using two sides and one angle or two angles and one side. In the past, people noticed this pattern and began to actively use it in the construction of buildings, navigation, astronomy and even in art.
First stage
Initially, people talked about the relationship between angles and sides exclusively using the example of right triangles. Then special formulas were discovered that made it possible to expand the boundaries of use in everyday life of this branch of mathematics.
The study of trigonometry in school today begins with right triangles, after which students use the acquired knowledge in physics and solving abstract trigonometric equations, which begin in high school.
Spherical trigonometry
Later, when science reached the next level of development, formulas with sine, cosine, tangent, and cotangent began to be used in spherical geometry, where different rules apply, and the sum of the angles in a triangle is always more than 180 degrees. This section is not studied in school, but it is necessary to know about its existence, at least because the earth’s surface, and the surface of any other planet, is convex, which means that any surface marking will be “arc-shaped” in three-dimensional space.
Take the globe and the thread. Attach the thread to any two points on the globe so that it is taut. Please note - it has taken on the shape of an arc. Spherical geometry deals with such forms, which is used in geodesy, astronomy and other theoretical and applied fields.
Right triangle
Having learned a little about the ways of using trigonometry, let's return to basic trigonometry in order to further understand what sine, cosine, tangent are, what calculations can be performed with their help and what formulas to use.
The first step is to understand the concepts related to a right triangle. First, the hypotenuse is the side opposite the 90 degree angle. It is the longest. We remember that according to the Pythagorean theorem, its numerical value is equal to the root of the sum of the squares of the other two sides.
For example, if the two sides are 3 and 4 centimeters respectively, the length of the hypotenuse will be 5 centimeters. By the way, the ancient Egyptians knew about this about four and a half thousand years ago.
The two remaining sides, which form a right angle, are called legs. In addition, we must remember that the sum of the angles in a triangle in a rectangular coordinate system is equal to 180 degrees.
Definition
Finally, with a firm understanding of the geometric basis, one can turn to the definition of sine, cosine and tangent of an angle.
The sine of an angle is the ratio of the opposite leg (i.e., the side opposite the desired angle) to the hypotenuse. The cosine of an angle is the ratio of the adjacent side to the hypotenuse.
Remember that neither sine nor cosine can be greater than one! Why? Because the hypotenuse is by default the longest. No matter how long the leg is, it will be shorter than the hypotenuse, which means their ratio will always be less than one. Thus, if in your answer to a problem you get a sine or cosine with a value greater than 1, look for an error in the calculations or reasoning. This answer is clearly incorrect.
Finally, the tangent of an angle is the ratio of the opposite side to the adjacent side. Dividing the sine by the cosine will give the same result. Look: according to the formula, we divide the length of the side by the hypotenuse, then divide by the length of the second side and multiply by the hypotenuse. Thus, we get the same relationship as in the definition of tangent.
Cotangent, accordingly, is the ratio of the side adjacent to the corner to the opposite side. We get the same result by dividing one by the tangent.
So, we have looked at the definitions of what sine, cosine, tangent and cotangent are, and we can move on to formulas.
The simplest formulas
In trigonometry you cannot do without formulas - how to find sine, cosine, tangent, cotangent without them? But this is exactly what is required when solving problems.
The first formula you need to know when starting to study trigonometry says that the sum of the squares of the sine and cosine of an angle is equal to one. This formula is a direct consequence of the Pythagorean theorem, but it saves time if you need to know the size of the angle rather than the side.
Many students cannot remember the second formula, which is also very popular when solving school problems: the sum of one and the square of the tangent of an angle is equal to one divided by the square of the cosine of the angle. Take a closer look: this is the same statement as in the first formula, only both sides of the identity were divided by the square of the cosine. It turns out that a simple mathematical operation makes the trigonometric formula completely unrecognizable. Remember: knowing what sine, cosine, tangent and cotangent are, transformation rules and several basic formulas, you can at any time derive the required more complex formulas on a sheet of paper.
Formulas for double angles and addition of arguments
Two more formulas that you need to learn are related to the values of sine and cosine for the sum and difference of angles. They are presented in the figure below. Please note that in the first case, sine and cosine are multiplied both times, and in the second, the pairwise product of sine and cosine is added.
There are also formulas associated with double angle arguments. They are completely derived from the previous ones - as a practice, try to get them yourself by taking the alpha angle equal to the beta angle.
Finally, note that double angle formulas can be rearranged to reduce the power of sine, cosine, tangent alpha.
Theorems
The two main theorems in basic trigonometry are the sine theorem and the cosine theorem. With the help of these theorems, you can easily understand how to find the sine, cosine and tangent, and therefore the area of the figure, and the size of each side, etc.
The sine theorem states that dividing the length of each side of a triangle by the opposite angle results in the same number. Moreover, this number will be equal to two radii of the circumscribed circle, that is, the circle containing all the points of a given triangle.
The cosine theorem generalizes the Pythagorean theorem, projecting it onto any triangles. It turns out that from the sum of the squares of the two sides, subtract their product multiplied by the double cosine of the adjacent angle - the resulting value will be equal to the square of the third side. Thus, the Pythagorean theorem turns out to be a special case of the cosine theorem.
Careless mistakes
Even knowing what sine, cosine and tangent are, it is easy to make a mistake due to absent-mindedness or an error in the simplest calculations. To avoid such mistakes, let's take a look at the most popular ones.
First, you should not convert fractions to decimals until you get the final result - you can leave the answer as a fraction unless otherwise stated in the conditions. Such a transformation cannot be called a mistake, but it should be remembered that at each stage of the problem new roots may appear, which, according to the author’s idea, should be reduced. In this case, you will waste your time on unnecessary mathematical operations. This is especially true for values such as the root of three or the root of two, because they are found in problems at every step. The same goes for rounding “ugly” numbers.
Further, note that the cosine theorem applies to any triangle, but not the Pythagorean theorem! If you mistakenly forget to subtract twice the product of the sides multiplied by the cosine of the angle between them, you will not only get a completely wrong result, but you will also demonstrate a complete lack of understanding of the subject. This is worse than a careless mistake.
Thirdly, do not confuse the values for angles of 30 and 60 degrees for sines, cosines, tangents, cotangents. Remember these values, because the sine of 30 degrees is equal to the cosine of 60, and vice versa. It is easy to confuse them, as a result of which you will inevitably get an erroneous result.
Application
Many students are in no hurry to start studying trigonometry because they do not understand its practical meaning. What is sine, cosine, tangent for an engineer or astronomer? These are concepts with which you can calculate the distance to distant stars, predict the fall of a meteorite, or send a research probe to another planet. Without them, it is impossible to build a building, design a car, calculate the load on a surface or the trajectory of an object. And these are just the most obvious examples! After all, trigonometry in one form or another is used everywhere, from music to medicine.
Finally
So you're sine, cosine, tangent. You can use them in calculations and successfully solve school problems.
The whole point of trigonometry comes down to the fact that using the known parameters of a triangle you need to calculate the unknowns. There are six parameters in total: the length of three sides and the size of three angles. The only difference in the tasks lies in the fact that different input data are given.
You now know how to find sine, cosine, tangent based on the known lengths of the legs or hypotenuse. Since these terms mean nothing more than a ratio, and a ratio is a fraction, the main goal of a trigonometry problem is to find the roots of an ordinary equation or system of equations. And here regular school mathematics will help you.