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 by 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 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 in terms of 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 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 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 opposite to the angle A 1 H 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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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 | sin 2 α + cos 2 α = 1 | |
a | 1 | |
b | 1 | |
a | 1 1 | |
sinα |
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)
Trigonometric identities are equalities that establish a relationship between the sine, cosine, tangent and cotangent of one angle, which allows you to find any of these functions, provided that any other is known.
tg \alpha = \frac(\sin \alpha)(\cos \alpha), \enspace ctg \alpha = \frac(\cos \alpha)(\sin \alpha)
tg \alpha \cdot ctg \alpha = 1
This identity says that the sum of the square of the sine of one angle and the square of the cosine of one angle is equal to one, which in practice makes it possible to calculate the sine of one angle when its cosine is known and vice versa.
When converting trigonometric expressions, this identity is very often used, which allows you to replace the sum of the squares of the cosine and sine of one angle with one and also perform the replacement operation in reverse order.
Finding tangent and cotangent through sine and cosine
tg \alpha = \frac(\sin \alpha)(\cos \alpha),\enspace
These identities are formed from the definitions of sine, cosine, tangent and cotangent. After all, if you look, then by definition, the ordinate of y is the sine, and the abscissa of x is the cosine. Then the tangent will be equal to the ratio \frac(y)(x)=\frac(\sin \alpha)(\cos \alpha), and the ratio \frac(x)(y)=\frac(\cos \alpha)(\sin \alpha)- will be a cotangent.
We add that only for such angles \alpha for which the trigonometric functions included in them make sense, the identities will take place, ctg \alpha=\frac(\cos \alpha)(\sin \alpha).
For example: tg \alpha = \frac(\sin \alpha)(\cos \alpha) is valid for \alpha angles that are different from \frac(\pi)(2)+\pi z, a ctg \alpha=\frac(\cos \alpha)(\sin \alpha)- for an angle \alpha other than \pi z , z is an integer.
Relationship between tangent and cotangent
tg \alpha \cdot ctg \alpha=1
This identity is valid only for angles \alpha that are different from \frac(\pi)(2) z. Otherwise, either cotangent or tangent will not be determined.
Based on the points above, we get that tg \alpha = \frac(y)(x), a ctg\alpha=\frac(x)(y). Hence it follows that tg \alpha \cdot ctg \alpha = \frac(y)(x) \cdot \frac(x)(y)=1. Thus, the tangent and cotangent of one angle at which they make sense are mutually reciprocal numbers.
Relationships between tangent and cosine, cotangent and sine
tg^(2) \alpha + 1=\frac(1)(\cos^(2) \alpha)- the sum of the square of the tangent of the angle \alpha and 1 is equal to the inverse square of the cosine of this angle. This identity is valid for all \alpha other than \frac(\pi)(2)+ \pi z.
1+ctg^(2) \alpha=\frac(1)(\sin^(2)\alpha)- the sum of 1 and the square of the cotangent of the angle \alpha , equals the inverse square of the sine of the given angle. This identity is valid for any \alpha other than \pi z .
Examples with solutions to problems using trigonometric identities
Example 1
Find \sin \alpha and tg \alpha if \cos \alpha=-\frac12 and \frac(\pi)(2)< \alpha < \pi ;
Show Solution
Solution
The functions \sin \alpha and \cos \alpha are linked by the formula \sin^(2)\alpha + \cos^(2) \alpha = 1. Substituting into this formula \cos \alpha = -\frac12, we get:
\sin^(2)\alpha + \left (-\frac12 \right)^2 = 1
This equation has 2 solutions:
\sin \alpha = \pm \sqrt(1-\frac14) = \pm \frac(\sqrt 3)(2)
By condition \frac(\pi)(2)< \alpha < \pi . In the second quarter, the sine is positive, so \sin \alpha = \frac(\sqrt 3)(2).
To find tg \alpha , we use the formula tg \alpha = \frac(\sin \alpha)(\cos \alpha)
tg \alpha = \frac(\sqrt 3)(2) : \frac12 = \sqrt 3
Example 2
Find \cos \alpha and ctg \alpha if and \frac(\pi)(2)< \alpha < \pi .
Show Solution
Solution
Substituting into the formula \sin^(2)\alpha + \cos^(2) \alpha = 1 conditional number \sin \alpha=\frac(\sqrt3)(2), we get \left (\frac(\sqrt3)(2)\right)^(2) + \cos^(2) \alpha = 1. This equation has two solutions \cos \alpha = \pm \sqrt(1-\frac34)=\pm\sqrt\frac14.
By condition \frac(\pi)(2)< \alpha < \pi . In the second quarter, the cosine is negative, so \cos \alpha = -\sqrt\frac14=-\frac12.
In order to find ctg \alpha , we use the formula ctg \alpha = \frac(\cos \alpha)(\sin \alpha). We know the corresponding values.
ctg \alpha = -\frac12: \frac(\sqrt3)(2) = -\frac(1)(\sqrt 3).
The concepts of sine, cosine, tangent and cotangent are the main categories of trigonometry - a branch of mathematics, and are inextricably linked with the definition of an angle. Possession of this mathematical science requires memorization and understanding of formulas and theorems, as well as developed spatial thinking. That is why trigonometric calculations often cause difficulties for schoolchildren and students. To overcome them, you should become more familiar with trigonometric functions and formulas.
Concepts in trigonometry
To understand the basic concepts of trigonometry, you must first decide what a right triangle and an angle in a circle are, and why all basic trigonometric calculations are associated with them. A triangle in which one of the angles is 90 degrees is a right triangle. Historically, this figure was often used by people in architecture, navigation, art, astronomy. Accordingly, studying and analyzing the properties of this figure, people came to the calculation of the corresponding ratios of its parameters.
The main categories associated with right triangles are the hypotenuse and the legs. The hypotenuse is the side of a triangle that is opposite the right angle. The legs, respectively, are the other two sides. The sum of the angles of any triangle is always 180 degrees.
Spherical trigonometry is a section of trigonometry that is not studied at school, but in applied sciences such as astronomy and geodesy, scientists use it. A feature of a triangle in spherical trigonometry is that it always has a sum of angles greater than 180 degrees.
Angles of a triangle
In a right triangle, the sine of an angle is the ratio of the leg opposite the desired angle to the hypotenuse of the triangle. Accordingly, the cosine is the ratio of the adjacent leg and the hypotenuse. Both of these values always have a value less than one, since the hypotenuse is always longer than the leg.
The tangent of an angle is a value equal to the ratio of the opposite leg to the adjacent leg of the desired angle, or sine to cosine. The cotangent, in turn, is the ratio of the adjacent leg of the desired angle to the opposite cactet. The cotangent of an angle can also be obtained by dividing the unit by the value of the tangent.
unit circle
A unit circle in geometry is a circle whose radius is equal to one. Such a circle is constructed in the Cartesian coordinate system, with the center of the circle coinciding with the point of origin, and the initial position of the radius vector is determined by the positive direction of the X axis (abscissa axis). Each point of the circle has two coordinates: XX and YY, that is, the coordinates of the abscissa and ordinate. Selecting any point on the circle in the XX plane, and dropping the perpendicular from it to the abscissa axis, we get a right triangle formed by a radius to the selected point (let us denote it by the letter C), a perpendicular drawn to the X axis (the intersection point is denoted by the letter G), and a segment the abscissa axis between the origin (the point is denoted by the letter A) and the intersection point G. The resulting triangle ACG is a right triangle inscribed in a circle, where AG is the hypotenuse, and AC and GC are the legs. The angle between the radius of the circle AC and the segment of the abscissa axis with the designation AG, we define as α (alpha). So, cos α = AG/AC. Given that AC is the radius of the unit circle, and it is equal to one, it turns out that cos α=AG. Similarly, sin α=CG.
In addition, knowing these data, it is possible to determine the coordinate of point C on the circle, since cos α=AG, and sin α=CG, which means that point C has the given coordinates (cos α; sin α). Knowing that the tangent is equal to the ratio of the sine to the cosine, we can determine that tg α \u003d y / x, and ctg α \u003d x / y. Considering angles in a negative coordinate system, one can calculate that the sine and cosine values of some angles can be negative.
Calculations and basic formulas
Values of trigonometric functions
Having considered the essence of trigonometric functions through the unit circle, we can derive the values of these functions for some angles. The values are listed in the table below.
The simplest trigonometric identities
Equations in which there is an unknown value under the sign of the trigonometric function are called trigonometric. Identities with the value sin x = α, k is any integer:
- sin x = 0, x = πk.
- 2. sin x \u003d 1, x \u003d π / 2 + 2πk.
- sin x \u003d -1, x \u003d -π / 2 + 2πk.
- sin x = a, |a| > 1, no solutions.
- sin x = a, |a| ≦ 1, x = (-1)^k * arcsin α + πk.
Identities with the value cos x = a, where k is any integer:
- cos x = 0, x = π/2 + πk.
- cos x = 1, x = 2πk.
- cos x \u003d -1, x \u003d π + 2πk.
- cos x = a, |a| > 1, no solutions.
- cos x = a, |a| ≦ 1, х = ±arccos α + 2πk.
Identities with the value tg x = a, where k is any integer:
- tg x = 0, x = π/2 + πk.
- tg x \u003d a, x \u003d arctg α + πk.
Identities with value ctg x = a, where k is any integer:
- ctg x = 0, x = π/2 + πk.
- ctg x \u003d a, x \u003d arcctg α + πk.
Cast formulas
This category of constant formulas denotes methods by which you can go from trigonometric functions of the form to functions of the argument, that is, convert the sine, cosine, tangent and cotangent of an angle of any value to the corresponding indicators of the angle of the interval from 0 to 90 degrees for greater convenience of calculations.
The formulas for reducing functions for the sine of an angle look like this:
- sin(900 - α) = α;
- sin(900 + α) = cos α;
- sin(1800 - α) = sin α;
- sin(1800 + α) = -sin α;
- sin(2700 - α) = -cos α;
- sin(2700 + α) = -cos α;
- sin(3600 - α) = -sin α;
- sin(3600 + α) = sin α.
For the cosine of an angle:
- cos(900 - α) = sin α;
- cos(900 + α) = -sin α;
- cos(1800 - α) = -cos α;
- cos(1800 + α) = -cos α;
- cos(2700 - α) = -sin α;
- cos(2700 + α) = sin α;
- cos(3600 - α) = cos α;
- cos(3600 + α) = cos α.
The use of the above formulas is possible subject to two rules. First, if the angle can be represented as a value (π/2 ± a) or (3π/2 ± a), the value of the function changes:
- from sin to cos;
- from cos to sin;
- from tg to ctg;
- from ctg to tg.
The value of the function remains unchanged if the angle can be represented as (π ± a) or (2π ± a).
Secondly, the sign of the reduced function does not change: if it was initially positive, it remains so. The same is true for negative functions.
Addition Formulas
These formulas express the values of the sine, cosine, tangent, and cotangent of the sum and difference of two rotation angles in terms of their trigonometric functions. Angles are usually denoted as α and β.
The formulas look like this:
- sin(α ± β) = sin α * cos β ± cos α * sin.
- cos(α ± β) = cos α * cos β ∓ sin α * sin.
- tan(α ± β) = (tan α ± tan β) / (1 ∓ tan α * tan β).
- ctg(α ± β) = (-1 ± ctg α * ctg β) / (ctg α ± ctg β).
These formulas are valid for any angles α and β.
Double and triple angle formulas
The trigonometric formulas of a double and triple angle are formulas that relate the functions of the angles 2α and 3α, respectively, to the trigonometric functions of the angle α. Derived from addition formulas:
- sin2α = 2sinα*cosα.
- cos2α = 1 - 2sin^2α.
- tg2α = 2tgα / (1 - tg^2 α).
- sin3α = 3sinα - 4sin^3α.
- cos3α = 4cos^3α - 3cosα.
- tg3α = (3tgα - tg^3 α) / (1-tg^2 α).
Transition from sum to product
Considering that 2sinx*cosy = sin(x+y) + sin(x-y), simplifying this formula, we obtain the identity sinα + sinβ = 2sin(α + β)/2 * cos(α − β)/2. Similarly, sinα - sinβ = 2sin(α - β)/2 * cos(α + β)/2; cosα + cosβ = 2cos(α + β)/2 * cos(α − β)/2; cosα - cosβ = 2sin(α + β)/2 * sin(α − β)/2; tgα + tgβ = sin(α + β) / cosα * cosβ; tgα - tgβ = sin(α - β) / cosα * cosβ; cosα + sinα = √2sin(π/4 ∓ α) = √2cos(π/4 ± α).
Transition from product to sum
These formulas follow from the identities for the transition of the sum to the product:
- sinα * sinβ = 1/2*;
- cosα * cosβ = 1/2*;
- sinα * cosβ = 1/2*.
Reduction Formulas
In these identities, the square and cubic powers of the sine and cosine can be expressed in terms of the sine and cosine of the first power of a multiple angle:
- sin^2 α = (1 - cos2α)/2;
- cos^2α = (1 + cos2α)/2;
- sin^3 α = (3 * sinα - sin3α)/4;
- cos^3 α = (3 * cosα + cos3α)/4;
- sin^4 α = (3 - 4cos2α + cos4α)/8;
- cos^4 α = (3 + 4cos2α + cos4α)/8.
Universal substitution
The universal trigonometric substitution formulas express trigonometric functions in terms of the tangent of a half angle.
- sin x \u003d (2tgx / 2) * (1 + tg ^ 2 x / 2), while x \u003d π + 2πn;
- cos x = (1 - tg^2 x/2) / (1 + tg^2 x/2), where x = π + 2πn;
- tg x \u003d (2tgx / 2) / (1 - tg ^ 2 x / 2), where x \u003d π + 2πn;
- ctg x \u003d (1 - tg ^ 2 x / 2) / (2tgx / 2), while x \u003d π + 2πn.
Special cases
Particular cases of the simplest trigonometric equations are given below (k is any integer).
Private for sine:
| sin x value | x value |
|---|---|
| 0 | pk |
| 1 | π/2 + 2πk |
| -1 | -π/2 + 2πk |
| 1/2 | π/6 + 2πk or 5π/6 + 2πk |
| -1/2 | -π/6 + 2πk or -5π/6 + 2πk |
| √2/2 | π/4 + 2πk or 3π/4 + 2πk |
| -√2/2 | -π/4 + 2πk or -3π/4 + 2πk |
| √3/2 | π/3 + 2πk or 2π/3 + 2πk |
| -√3/2 | -π/3 + 2πk or -2π/3 + 2πk |
Cosine quotients:
| cos x value | x value |
|---|---|
| 0 | π/2 + 2πk |
| 1 | 2πk |
| -1 | 2 + 2πk |
| 1/2 | ±π/3 + 2πk |
| -1/2 | ±2π/3 + 2πk |
| √2/2 | ±π/4 + 2πk |
| -√2/2 | ±3π/4 + 2πk |
| √3/2 | ±π/6 + 2πk |
| -√3/2 | ±5π/6 + 2πk |
Private for tangent:
| tg x value | x value |
|---|---|
| 0 | pk |
| 1 | π/4 + πk |
| -1 | -π/4 + πk |
| √3/3 | π/6 + πk |
| -√3/3 | -π/6 + πk |
| √3 | π/3 + πk |
| -√3 | -π/3 + πk |
Cotangent quotients:
| ctg x value | x value |
|---|---|
| 0 | π/2 + πk |
| 1 | π/4 + πk |
| -1 | -π/4 + πk |
| √3 | π/6 + πk |
| -√3 | -π/3 + πk |
| √3/3 | π/3 + πk |
| -√3/3 | -π/3 + πk |
Theorems
Sine theorem
There are two versions of the theorem - simple and extended. Simple sine theorem: a/sin α = b/sin β = c/sin γ. In this case, a, b, c are the sides of the triangle, and α, β, γ are the opposite angles, respectively.
Extended sine theorem for an arbitrary triangle: a/sin α = b/sin β = c/sin γ = 2R. In this identity, R denotes the radius of the circle in which the given triangle is inscribed.
Cosine theorem
The identity is displayed in this way: a^2 = b^2 + c^2 - 2*b*c*cos α. In the formula, a, b, c are the sides of the triangle, and α is the angle opposite side a.
Tangent theorem
The formula expresses the relationship between the tangents of two angles, and the length of the sides opposite them. The sides are labeled a, b, c, and the corresponding opposite angles are α, β, γ. The formula of the tangent theorem: (a - b) / (a+b) = tg((α - β)/2) / tg((α + β)/2).
Cotangent theorem
Associates the radius of a circle inscribed in a triangle with the length of its sides. If a, b, c are the sides of a triangle, and A, B, C, respectively, are their opposite angles, r is the radius of the inscribed circle, and p is the half-perimeter of the triangle, the following identities hold:
- ctg A/2 = (p-a)/r;
- ctg B/2 = (p-b)/r;
- ctg C/2 = (p-c)/r.
Applications
Trigonometry is not only a theoretical science associated with mathematical formulas. Its properties, theorems and rules are used in practice by various branches of human activity - astronomy, air and sea navigation, music theory, geodesy, chemistry, acoustics, optics, electronics, architecture, economics, mechanical engineering, measuring work, computer graphics, cartography, oceanography, and many others.
Sine, cosine, tangent and cotangent are the basic concepts of trigonometry, with which you can mathematically express the relationship between angles and lengths of sides in a triangle, and find the desired quantities through identities, theorems and rules.
Lecture: Sine, cosine, tangent, cotangent of an arbitrary angle
Sine, cosine of an arbitrary angle
To understand what trigonometric functions are, let's turn to a circle with a unit radius. This circle is centered at the origin on the coordinate plane. To determine the given functions, we will use the radius vector OR, which starts at the center of the circle, and the point R is a point on the circle. This radius vector forms an angle alpha with the axis OH. Since the circle has a radius equal to one, then OR = R = 1.
If from the point R drop a perpendicular on the axis OH, then we get a right triangle with hypotenuse equal to one.
If the radius vector moves clockwise, then this direction is called negative, but if it moves counter-clockwise - positive.
The sine of an angle OR, is the ordinate of the point R vectors on a circle.
That is, to obtain the value of the sine of a given angle alpha, it is necessary to determine the coordinate At on surface.
How was this value obtained? Since we know that the sine of an arbitrary angle in a right triangle is the ratio of the opposite leg to the hypotenuse, we get that
And since R=1, then sin(α) = y 0 .
In the unit circle, the ordinate value cannot be less than -1 and greater than 1, which means that
The sine is positive in the first and second quarters of the unit circle, and negative in the third and fourth.
Cosine of an angle given circle formed by the radius vector OR, is the abscissa of the point R vectors on a circle.
That is, to obtain the value of the cosine of a given angle alpha, it is necessary to determine the coordinate X on surface.
The cosine of an arbitrary angle in a right triangle is the ratio of the adjacent leg to the hypotenuse, we get that
And since R=1, then cos(α) = x 0 .
In the unit circle, the value of the abscissa cannot be less than -1 and greater than 1, which means that
The cosine is positive in the first and fourth quadrants of the unit circle, and negative in the second and third.
tangentarbitrary angle the ratio of sine to cosine is calculated.
If we consider a right triangle, then this is the ratio of the opposite leg to the adjacent one. If we are talking about a unit circle, then this is the ratio of the ordinate to the abscissa.
Judging by these relationships, it can be understood that the tangent cannot exist if the value of the abscissa is zero, that is, at an angle of 90 degrees. The tangent can take all other values.
The tangent is positive in the first and third quarters of the unit circle, and negative in the second and fourth.
