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 angle. Ownership of this mathematical science requires memorization and understanding of formulas and theorems, as well as developed spatial thinking. That is why schoolchildren and students trigonometric calculations often cause difficulties. To overcome them, you should become more familiar with trigonometric functions and formulas.
Concepts in trigonometry
To understand basic concepts trigonometry, you must first decide what a right triangle and an angle in a circle are, and why all the basic trigonometric calculations are associated with them. A triangle in which one of the angles measures 90 degrees is rectangular. Historically, this figure was often used by people in architecture, navigation, art, and astronomy. Accordingly, by studying and analyzing the properties of this figure, people came to calculate the corresponding ratios of its parameters.
The main categories associated with right triangles are the hypotenuse and the legs. Hypotenuse - the side of a triangle opposite right angle. The legs, respectively, are the remaining two sides. The sum of the angles of any triangles is always 180 degrees.
Spherical trigonometry is a section of trigonometry that is not studied in school, but in applied sciences such as astronomy and geodesy, scientists use it. 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, cosine is the ratio adjacent leg and hypotenuse. Both of these values always have a magnitude less than one, since the hypotenuse is always longer than the leg.
Tangent of an angle is a value equal to the ratio opposite leg to the adjacent side of the desired angle, or sine to cosine. Cotangent, in turn, is the ratio of the adjacent side of the desired angle to the opposite side. The cotangent of an angle can also be obtained by dividing one by the tangent value.
Unit circle
A unit circle in geometry is a circle whose radius equal to one. Such a circle is constructed in Cartesian system coordinates, while the center of the circle coincides with the origin point, and starting position The radius vector is determined by the positive direction of the X axis (abscissa axis). Each point on the circle has two coordinates: XX and YY, that is, the coordinates of the abscissa and ordinate. By selecting any point on the circle in the XX plane and dropping a perpendicular from it to the abscissa axis, we obtain a right triangle formed by the radius to the selected point (denoted by the letter C), the perpendicular drawn to the X axis (the intersection point is denoted by the letter G), and the segment the abscissa axis is between the origin of coordinates (the point is designated 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 is defined as α (alpha). So, cos α = AG/AC. Considering that AC is the radius unit circle, and it is equal to one, it turns out that cos α=AG. Likewise, sin α=CG.
In addition, knowing this data, you can determine the coordinate of point C on the circle, since cos α=AG, and sin α=CG, which means point C has given coordinates(cos α;sin α). Knowing that the tangent equal to the ratio sine to cosine, we can determine that tan α = y/x, and cot α = x/y. By considering angles in a negative coordinate system, you can calculate that the sine and cosine values of some angles can be negative.
Calculations and basic formulas
Trigonometric function values
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 the sign of the trigonometric function contains unknown value, are called trigonometric. Identities with sin value x = α, k — any integer:
- sin x = 0, x = πk.
- 2. sin x = 1, x = π/2 + 2πk.
- sin x = -1, x = -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 = -1, x = π + 2πk.
- cos x = a, |a| > 1, no solutions.
- cos x = a, |a| ≦ 1, x = ±arccos α + 2πk.
Identities with the value tg x = a, where k is any integer:
- tan x = 0, x = π/2 + πk.
- tan x = a, x = arctan α + πk.
Identities with the value ctg x = a, where k is any integer:
- cot x = 0, x = π/2 + πk.
- ctg x = a, x = arcctg α + πk.
Reduction formulas
This category constant formulas denotes methods by which you can move from trigonometric functions of the form to functions of an argument, that is, reduce 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.
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 cosine of 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. Same with negative functions.
Addition formulas
These formulas express the values of sine, cosine, tangent and cotangent of the sum and difference of two rotation angles through their trigonometric functions. Typically the angles are 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) = (tg 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 double and triple angle trigonometric formulas are formulas that relate the functions of the angles 2α and 3α, respectively, to the trigonometric functions of the angle α. Derived from addition formulas:
- sin2α = 2sinα*cosα.
- cos2α = 1 - 2sin^2 α.
- tan2α = 2tgα / (1 - tan^2 α).
- sin3α = 3sinα - 4sin^3 α.
- cos3α = 4cos^3 α - 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 get 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; tanα + tanβ = 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 of the transition of a sum to a product:
- sinα * sinβ = 1/2*;
- cosα * cosβ = 1/2*;
- sinα * cosβ = 1/2*.
Degree reduction formulas
In these identities, square and cubic degree sine and cosine can be expressed through the sine and cosine of the first degree of a multiple angle:
- sin^2 α = (1 - cos2α)/2;
- cos^2 α = (1 + cos2α)/2;
- sin^3 α = (3 * sinα - sin3α)/4;
- cos^3 α = (3 * cosα + cos3α)/4;
- sin^4 α = (3 - 4cos2α + cos4α)/8;
- cos^4 α = (3 + 4cos2α + cos4α)/8.
Universal substitution
Formulas for universal trigonometric substitution express trigonometric functions in terms of the tangent of a half angle.
- sin x = (2tgx/2) * (1 + tan^2 x/2), with x = π + 2πn;
- cos x = (1 - tan^2 x/2) / (1 + tan^2 x/2), where x = π + 2πn;
- tg x = (2tgx/2) / (1 - tg^2 x/2), where x = π + 2πn;
- cot x = (1 - tg^2 x/2) / (2tgx/2), with x = π + 2πn.
Special cases
Special cases of protozoa trigonometric equations are given below (k is any integer).
Quotients for sine:
| Sin x value | x value |
|---|---|
| 0 | πk |
| 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 |
Quotients for cosine:
| 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 |
Quotients for tangent:
| tg x value | x value |
|---|---|
| 0 | πk |
| 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 |
Quotients for cotangent:
| 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
Theorem of sines
There are two versions of the theorem - simple and extended. Simple theorem sines: a/sin α = b/sin β = c/sin γ. In this case, a, b, c are the sides of the triangle, and α, β, γ are the opposite angles, respectively.
Extended sine theorem for arbitrary triangle: a/sin α = b/sin β = c/sin γ = 2R. In this identity, R denotes the radius of the circle in which the given triangle is inscribed.
Cosine theorem
The identity is displayed as follows: a^2 = b^2 + c^2 - 2*b*c*cos α. In the formula, a, b, c are the sides of the triangle, and α is the angle opposite to side a.
Tangent theorem
The formula expresses the relationship between the tangents of two angles and the length of the sides opposite them. The sides are labeled a, b, c, and the corresponding opposite angles are α, β, γ. Formula of the tangent theorem: (a - b) / (a+b) = tan((α - β)/2) / tan((α + β)/2).
Cotangent theorem
Connects the radius of a circle inscribed in a triangle with the length of its sides. If a, b, c are the sides of the triangle, and A, B, C, respectively, are the angles opposite them, r is the radius of the inscribed circle, and p is the semi-perimeter of the triangle, the following identities are valid:
- cot A/2 = (p-a)/r;
- cot B/2 = (p-b)/r;
- cot C/2 = (p-c)/r.
Application
Trigonometry - not only theoretical science related to mathematical formulas. Its properties, theorems and rules are used in practice by various industries. human activity- astronomy, aerial and sea navigation, music theory, geodesy, chemistry, acoustics, optics, electronics, architecture, economics, mechanical engineering, measuring work, computer graphics, cartography, oceanography, and many others.
Sine, cosine, tangent and cotangent are the basic concepts of trigonometry, with the help of which one can mathematically express the relationships between the angles and lengths of the sides in a triangle, and find the required quantities through identities, theorems and rules.
I won't try to convince you not to write cheat sheets. Write! Including cheat sheets on trigonometry. Later I plan to explain why cheat sheets are needed and why cheat sheets are useful. And here is information on how not to learn, but remember some trigonometric formulas. So - trigonometry without a cheat sheet! We use associations for memorization.
1. Addition formulas:
Cosines always “come in pairs”: cosine-cosine, sine-sine.
And one more thing: cosines are “inadequate”. “Everything is not right” for them, so they change the signs: “-” to “+”, and vice versa.
Sinuses - “mix”: sine-cosine, cosine-sine.
2. Sum and difference formulas:
cosines always “come in pairs”. By adding two cosines - “koloboks”, we get a pair of cosines - “koloboks”. And by subtracting, we definitely won’t get any koloboks. We get a couple of sines. Also with a minus ahead.
Sinuses - “mix” :
3. Formulas for converting a product into a sum and difference.
When do we get a cosine pair? When we add cosines. That's why
When do we get a couple of sines? When subtracting cosines. From here:
“Mixing” is obtained both when adding and subtracting sines. What's more fun: adding or subtracting? That's right, fold. And for the formula they take addition:
In the first and third formulas, the sum is in parentheses. Rearranging the places of the terms does not change the sum. The order is important only for the second formula. But, in order not to get confused, for ease of remembering, in all three formulas in the first brackets we take the difference
and secondly - the amount
Cheat sheets in your pocket give you peace of mind: if you forget the formula, you can copy it. And they give you confidence: if you fail to use the cheat sheet, you can easily remember the formulas.
Sine and cosine originally arose from the need to calculate quantities in right triangles. It was noticed that if the degree measure of the angles in a right triangle is not changed, then the aspect ratio, no matter how much these sides change in length, always remains the same.
This is how the concepts of sine and cosine were introduced. Sinus acute angle in a right triangle is the ratio of the opposite side to the hypotenuse, and the cosine is the ratio of the side adjacent to the hypotenuse.
Theorems of cosines and sines
But cosines and sines can be used for more than just right triangles. To find the value of an obtuse or acute angle or side of any triangle, it is enough to apply the theorem of cosines and sines.
The cosine theorem is quite simple: “The square of the side of a triangle equal to the sum the squares of the other two sides minus twice the product of these sides by the cosine of the angle between them.”
There are two interpretations of the sine theorem: small and extended. According to the small one: “In a triangle, the angles are proportional opposing parties». This theorem often expanded due to the property of the circumscribed circle of a triangle: “In a triangle, the angles are proportional to the opposite sides, and their ratio is equal to the diameter of the circumscribed circle.”
Derivatives
The derivative is a mathematical tool that shows how quickly a function changes relative to a change in its argument. Derivatives are used in geometry, and in a number of technical disciplines.
When solving problems, you need to know the tabular values of the derivatives of trigonometric functions: sine and cosine. The derivative of a sine is a cosine, and a cosine is a sine, but with a minus sign.
Application in mathematics
Sines and cosines are especially often used when solving right triangles and tasks associated with them.
The convenience of sines and cosines is also reflected in technology. It was easy to evaluate angles and sides using the theorems of cosines and sines, breaking down complex figures and objects into “simple” triangles. Engineers often deal with aspect ratio calculations and degree measures, spent a lot of time and effort to calculate the cosines and sines of non-tabular angles.
Then Bradis tables came to the rescue, containing thousands of values of sines, cosines, tangents and cotangents different angles. IN Soviet time some teachers forced their students to memorize pages of Bradis tables.
Radian - angular magnitude arcs, length equal to the radius or 57.295779513° degrees.
Degree (in geometry) - 1/360th part of a circle or 1/90th part of a right angle.
π = 3.141592653589793238462… ( approximate value Pi numbers).
Cosine table for angles: 0°, 30°, 45°, 60°, 90°, 120°, 135°, 150°, 180°, 210°, 225°, 240°, 270°, 300°, 315°, 330°, 360°.
| Angle x (in degrees) | 0° | 30° | 45° | 60° | 90° | 120° | 135° | 150° | 180° | 210° | 225° | 240° | 270° | 300° | 315° | 330° | 360° |
|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
| Angle x (in radians) | 0 | π/6 | π/4 | π/3 | π/2 | 2 x π/3 | 3 x π/4 | 5 x π/6 | π | 7 x π/6 | 5 x π/4 | 4 x π/3 | 3 x π/2 | 5 x π/3 | 7 x π/4 | 11 x π/6 | 2 x π |
| cos x | 1 | √3/2 (0,8660) | √2/2 (0,7071) | 1/2 (0,5) | 0 | -1/2 (-0,5) | -√2/2 (-0,7071) | -√3/2 (-0,8660) | -1 | -√3/2 (-0,8660) | -√2/2 (-0,7071) | -1/2 (-0,5) | 0 | 1/2 (0,5) | √2/2 (0,7071) | √3/2 (0,8660) | 1 |
Solving simple trigonometric equations.
Solving trigonometric equations of any level of complexity ultimately comes down to solving the simplest trigonometric equations. And in this best helper again it turns out to be a trigonometric circle.
Let's recall the definitions of cosine and sine.
The cosine of an angle is the abscissa (that is, the coordinate along the axis) of a point on the unit circle corresponding to a rotation through a given angle.
The sine of an angle is the ordinate (that is, the coordinate along the axis) of a point on the unit circle corresponding to a rotation through a given angle.
The positive direction of movement on the trigonometric circle is counterclockwise. A rotation of 0 degrees (or 0 radians) corresponds to a point with coordinates (1;0)
We use these definitions to solve simple trigonometric equations.
1. Solve the equation
This equation is satisfied by all values of the rotation angle that correspond to points on the circle whose ordinate is equal to .
Let's mark a point with ordinate on the ordinate axis:
Let's carry out horizontal line parallel to the x-axis until it intersects with the circle. We get two points lying on the circle and having an ordinate. These points correspond to rotation angles in and radians:
If we, leaving the point corresponding to the angle of rotation by radians, go around full circle, then we will arrive at a point corresponding to the rotation angle per radian and having the same ordinate. That is, this rotation angle also satisfies our equation. We can make as many “idle” revolutions as we like, returning to the same point, and all these angle values will satisfy our equation. The number of “idle” revolutions will be denoted by the letter (or). Since we can make these revolutions in both positive and negative directions, (or) can take on any integer values.
That is, the first series of solutions to the original equation has the form:
, , - set of integers (1)
Similarly, the second series of solutions has the form:
, Where , . (2)
As you might have guessed, this series of solutions is based on the point on the circle corresponding to the angle of rotation by .
These two series of solutions can be combined into one entry:
If we're in this let's take the notes(that is, even), then we get the first series of solutions.
If we take (that is, odd) in this entry, then we get the second series of solutions.
2. Now let's solve the equation
Since this is the abscissa of a point on the unit circle obtained by rotating through an angle, we mark the point with the abscissa on the axis:
Let's carry out vertical line parallel to the axis until it intersects with the circle. We will get two points lying on the circle and having an abscissa. These points correspond to rotation angles in and radians. Recall that when moving clockwise we get a negative rotation angle:
Let us write down two series of solutions:
,
,
(We get to the desired point by going from the main full circle, that is.
Let's combine these two series into one entry:
3. Solve the equation
The tangent line passes through the point with coordinates (1,0) of the unit circle parallel to the OY axis
Let's mark a point on it with an ordinate equal to 1 (we are looking for the tangent of which angles is equal to 1):
Let's connect this point to the origin of coordinates with a straight line and mark the points of intersection of the line with the unit circle. The intersection points of the straight line and the circle correspond to the angles of rotation on and :
Since the points corresponding to the rotation angles that satisfy our equation lie at a distance of radians from each other, we can write the solution this way:
4. Solve the equation
The line of cotangents passes through the point with the coordinates of the unit circle parallel to the axis.
Let's mark a point with abscissa -1 on the line of cotangents:
Let's connect this point to the origin of the straight line and continue it until it intersects with the circle. This straight line will intersect the circle at points corresponding to the angles of rotation in and radians:
Since these points are separated from each other by a distance equal to , then common decision We can write this equation like this:
In the given examples illustrating the solution of the simplest trigonometric equations, tabular values of trigonometric functions were used.
However, if the right side of the equation contains a non-tabular value, then we substitute the value into the general solution of the equation:
SPECIAL SOLUTIONS:
Let us mark the points on the circle whose ordinate is 0:
Let us mark a single point on the circle whose ordinate is 1:
Let us mark a single point on the circle whose ordinate is equal to -1:
Since it is customary to indicate values closest to zero, we write the solution as follows:
Let us mark the points on the circle whose abscissa is equal to 0:
5.
Let us mark a single point on the circle whose abscissa is equal to 1:
Let us mark a single point on the circle whose abscissa is equal to -1:
And slightly more complex examples:
1.
The sine is equal to one if the argument is equal to
The argument of our sine is equal, so we get:
Let's divide both sides of the equality by 3:
Answer:
2.
Cosine equal to zero, if the cosine argument is equal to
The argument of our cosine is equal to , so we get:
Let's express , to do this we first move to the right with the opposite sign:
Let's simplify the right side:
Divide both sides by -2:
Note that the sign in front of the term does not change, since k can take any integer value.
Answer:
And finally, watch the video tutorial “Selecting roots in a trigonometric equation using trigonometric circle"
This concludes our conversation about solving simple trigonometric equations. Next time we will talk about how to decide.
In this article we will take a comprehensive look. Basic trigonometric identities are equalities that establish a connection between the sine, cosine, tangent and cotangent of one angle, and allow one to find any of these trigonometric functions through a known other.
Let us immediately list the main trigonometric identities that we will analyze in this article. Let's write them down in a table, and below we'll give the output of these formulas and provide the necessary explanations.
Page navigation.
Relationship between sine and cosine of one angle
Sometimes they do not talk 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 main 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'll talk about this in more detail in the following paragraphs.
That is, special interest represents precisely the equality, which was given the name of the main trigonometric identity.
Before proving the main 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 when transformation 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: 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 tangent and cotangent with sine and cosine of one angle of view and 
follow immediately from the definitions of sine, cosine, tangent and cotangent. Indeed, by definition, sine is the ordinate of y, cosine is the abscissa of x, 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, 
.
Thanks to such obviousness of the identities and Tangent and cotangent are often defined not through the ratio of abscissa and ordinate, but through the ratio of 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.
In conclusion of this paragraph, it should be noted that the identities and take place for all angles at which the trigonometric functions included in them make sense. So the formula is valid for any , other than (otherwise the denominator will have 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
Even more obvious trigonometric identity than the previous two, 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 , in otherwise either tangent or cotangent is not defined.
Proof of the formula very simple. By definition and from where 
. The proof could have been carried out a little differently. Since , That 
.
So, the tangent and cotangent of the same angle at which they make sense are .