The concepts of sine (), cosine (), tangent (), cotangent () are inextricably linked with the concept of angle. In order to understand well these, at first glance, complex concepts (which cause a state of horror in many schoolchildren), and make sure that “the devil is not as scary as he is painted”, let's start from the very beginning and understand the concept of an angle.
The concept of angle: radian, degree
Let's look at the picture. The vector "turned" relative to the point by a certain amount. So the measure of this rotation relative to the initial position will be corner.
What else do you need to know about the concept of angle? Well, units of angle, of course!
Angle, both in geometry and trigonometry, can be measured in degrees and radians.
The angle at (one degree) is the central angle in the circle, based on a circular arc equal to the part of the circle. Thus, the entire circle consists of "pieces" of circular arcs, or the angle described by the circle is equal.
That is, the figure above shows an angle that is equal, that is, this angle is based on a circular arc the size of the circumference.
An angle in radians is called the central angle in a circle, based on a circular arc, the length of which is equal to the radius of the circle. Well, did you understand? If not, then let's look at the picture.
So, the figure shows an angle equal to a radian, that is, this angle is based on a circular arc, the length of which is equal to the radius of the circle (the length is equal to the length or the radius is equal to the length of the arc). Thus, the length of the arc is calculated by the formula:
Where is the central angle in radians.
Well, knowing this, can you answer how many radians contains an angle described by a circle? Yes, for this you need to remember the formula for the circumference of a circle. There she is:
Well, now let's correlate these two formulas and get that the angle described by the circle is equal. That is, correlating the value in degrees and radians, we get that. Respectively, . As you can see, unlike "degrees", the word "radian" is omitted, since the unit of measurement is usually clear from the context.
How many radians are? That's right!
Got it? Then fasten forward:
Any difficulties? Then look answers:
Right triangle: sine, cosine, tangent, cotangent of an angle
So, with the concept of the angle figured out. But what is the sine, cosine, tangent, cotangent of an angle? Let's figure it out. For this, a right triangle will help us.
What are the sides of a right triangle called? That's right, the hypotenuse and legs: the hypotenuse is the side that lies opposite the right angle (in our example, this is the side); the legs are the two remaining sides and (those adjacent to right angle), moreover, if we consider the legs relative to the angle, then the leg is the adjacent leg, and the leg is the opposite one. So, now let's answer the question: what are the sine, cosine, tangent and cotangent of an angle?
Sine of an angle is the ratio of the opposite (far) leg to the hypotenuse.
in our triangle.
Cosine of an angle- this is the ratio of the adjacent (close) leg to the hypotenuse.
in our triangle.
Angle tangent- this is the ratio of the opposite (far) leg to the adjacent (close).
in our triangle.
Cotangent of an angle- this is the ratio of the adjacent (close) leg to the opposite (far).
in our triangle.
These definitions are necessary remember! To make it easier to remember which leg to divide by what, you need to clearly understand that in tangent and cotangent only the legs sit, and the hypotenuse appears only in sinus and cosine. And then you can come up with a chain of associations. For example, this one:
cosine→touch→touch→adjacent;
Cotangent→touch→touch→adjacent.
First of all, it is necessary to remember that the sine, cosine, tangent and cotangent as ratios of the sides of a triangle do not depend on the lengths of these sides (at one angle). Do not trust? Then make sure by looking at the picture:
Consider, for example, the cosine of an angle. By definition, from a triangle: , but we can calculate the cosine of an angle from a triangle: . You see, the lengths of the sides are different, but the value of the cosine of one angle is the same. Thus, the values of sine, cosine, tangent and cotangent depend solely on the magnitude of the angle.
If you understand the definitions, then go ahead and fix them!
For the triangle shown in the figure below, we find.
Well, did you get it? Then try it yourself: calculate the same for the corner.
Unit (trigonometric) circle
Understanding the concepts of degrees and radians, we considered a circle with a radius equal to. Such a circle is called single. It is very useful in the study of trigonometry. Therefore, we dwell on it in a little more detail.
As you can see, this circle is built in Cartesian system coordinates. The radius of the circle is equal to one, while the center of the circle lies at the origin, the initial position of the radius vector is fixed along the positive direction of the axis (in our example, this is the radius).
Each point of the circle corresponds to two numbers: the coordinate along the axis and the coordinate along the axis. What are these coordinate numbers? And in general, what do they have to do with the topic at hand? To do this, remember about the considered right-angled triangle. In the figure above, you can see two whole right triangles. Consider a triangle. It is rectangular because it is perpendicular to the axis.
What is equal to from a triangle? That's right. In addition, we know that is the radius of the unit circle, and therefore, . Substitute this value into our cosine formula. Here's what happens:
And what is equal to from a triangle? Well, of course, ! Substitute the value of the radius into this formula and get:
So, can you tell me what are the coordinates of a point that belongs to the circle? Well, no way? And if you realize that and are just numbers? What coordinate does it correspond to? Well, of course, the coordinate! What coordinate does it correspond to? That's right, coordinate! Thus, the point.
And what then are equal and? That's right, let's use the appropriate definitions of tangent and cotangent and get that, a.
What if the angle is larger? Here, for example, as in this picture:
What has changed in this example? Let's figure it out. For this, we turn again to right triangle. Consider a right triangle: an angle (as adjacent to an angle). What is the value of the sine, cosine, tangent and cotangent of an angle? That's right, we adhere to the relevant definitions trigonometric functions:
Well, as you can see, the value of the sine of the angle still corresponds to the coordinate; the value of the cosine of the angle - the coordinate; and the values of tangent and cotangent to the corresponding ratios. Thus, these relations are applicable to any rotations of the radius vector.
It has already been mentioned that the initial position of the radius vector is along the positive direction of the axis. So far we have rotated this vector counterclockwise, but what happens if we rotate it clockwise? Nothing extraordinary, you will also get an angle of a certain size, but only it will be negative. Thus, when rotating the radius vector counterclockwise, we get positive angles, and when rotating clockwise - negative.
So, we know that a whole revolution of the radius vector around the circle is or. Is it possible to rotate the radius vector by or by? Well, of course you can! In the first case, therefore, the radius vector will make one complete revolution and stop at position or.
In the second case, that is, the radius vector will make three complete revolutions and stop at position or.
Thus, from the above examples, we can conclude that angles that differ by or (where is any integer) correspond to the same position of the radius vector.
The figure below shows an angle. The same image corresponds to the corner, and so on. This list can be continued indefinitely. All these angles can be written with the general formula or (where is any integer)
Now, knowing the definitions of the basic trigonometric functions and using the unit circle, try to answer what the values \u200b\u200bare equal to:
Here's a unit circle to help you:
Any difficulties? Then let's figure it out. So we know that:
From here, we determine the coordinates of the points corresponding to certain measures of the angle. Well, let's start in order: the corner at corresponds to a point with coordinates, therefore:
Does not exist;
Further, adhering to the same logic, we find out that the corners in correspond to points with coordinates, respectively. Knowing this, it is easy to determine the values of trigonometric functions at the corresponding points. Try it yourself first, then check the answers.
Answers:
Does not exist
Does not exist
Does not exist
Does not exist
Thus, we can make the following table:
There is no need to remember all these values. It is enough to remember the correspondence between the coordinates of points on the unit circle and the values of trigonometric functions:
But the values \u200b\u200bof the trigonometric functions of the angles in and, given in the table below, must be remembered:
Do not be afraid, now we will show one of the examples rather simple memorization of the corresponding values:
To use this method, it is vital to remember the values of the sine for all three measures of the angle (), as well as the value of the tangent of the angle in. Knowing these values, it is quite easy to restore the entire table - the cosine values are transferred in accordance with the arrows, that is:
Knowing this, you can restore the values for. The numerator " " will match and the denominator " " will match. Cotangent values are transferred in accordance with the arrows shown in the figure. If you understand this and remember the diagram with arrows, then it will be enough to remember the entire value from the table.
Coordinates of a point on a circle
Is it possible to find a point (its coordinates) on a circle, knowing the coordinates of the center of the circle, its radius and angle of rotation?
Well, of course you can! Let's bring out general formula to find the coordinates of a point.
Here, for example, we have such a circle:
We are given that the point is the center of the circle. The radius of the circle is equal. It is necessary to find the coordinates of the point obtained by rotating the point by degrees.
As can be seen from the figure, the coordinate of the point corresponds to the length of the segment. The length of the segment corresponds to the coordinate of the center of the circle, that is, it is equal to. The length of a segment can be expressed using the definition of cosine:
Then we have that for the point the coordinate.
By the same logic, we find the value of the y coordinate for the point. In this way,
So in general view point coordinates are determined by the formulas:
Circle center coordinates,
circle radius,
Angle of rotation of the radius vector.
As you can see, for the unit circle we are considering, these formulas are significantly reduced, since the coordinates of the center are zero, and the radius is equal to one:
Well, let's try these formulas for a taste, practicing finding points on a circle?
1. Find the coordinates of a point on a unit circle obtained by turning a point on.
2. Find the coordinates of a point on a unit circle obtained by rotating a point on.
3. Find the coordinates of a point on a unit circle obtained by turning a point on.
4. Point - the center of the circle. The radius of the circle is equal. It is necessary to find the coordinates of the point obtained by rotating the initial radius vector by.
5. Point - the center of the circle. The radius of the circle is equal. It is necessary to find the coordinates of the point obtained by rotating the initial radius vector by.
Having trouble finding the coordinates of a point on a circle?
Solve these five examples (or understand the solution well) and you will learn how to find them!
1.
It can be seen that. And we know what corresponds to a full turn of the starting point. Thus, the desired point will be in the same position as when turning to. Knowing this, we find the desired coordinates of the point:
2. The circle is unit with a center at a point, which means that we can use simplified formulas:
It can be seen that. We know what corresponds to two complete rotations of the starting point. Thus, the desired point will be in the same position as when turning to. Knowing this, we find the desired coordinates of the point:
Sine and cosine are tabular values. We remember their values and get:
Thus, the desired point has coordinates.
3. The circle is unit with a center at a point, which means that we can use simplified formulas:
It can be seen that. Let's depict the considered example in the figure:
The radius makes angles with the axis equal to and. Knowing that the tabular values of the cosine and sine are equal, and having determined that the cosine here takes a negative value, and the sine is positive, we have:
Similar examples are analyzed in more detail when studying the formulas for reducing trigonometric functions in the topic.
Thus, the desired point has coordinates.
4.
Angle of rotation of the radius vector (by condition)
To determine the corresponding signs of sine and cosine, we construct a unit circle and an angle:
As you can see, the value, that is, is positive, and the value, that is, is negative. Knowing the tabular values of the corresponding trigonometric functions, we obtain that:
Let's substitute the obtained values into our formula and find the coordinates:
Thus, the desired point has coordinates.
5. To solve this problem, we use formulas in general form, where
The coordinates of the center of the circle (in our example,
Circle radius (by condition)
Angle of rotation of the radius vector (by condition).
Substitute all the values into the formula and get:
and - table values. We remember and substitute them into the formula:
Thus, the desired point has coordinates.
SUMMARY AND BASIC FORMULA
The sine of an angle is the ratio of the opposite (far) leg to the hypotenuse.
The cosine of an angle is the ratio of the adjacent (close) leg to the hypotenuse.
The tangent of an angle is the ratio of the opposite (far) leg to the adjacent (close).
The cotangent of an angle is the ratio of the adjacent (close) leg to the opposite (far).
In this article, we will take a comprehensive look at . Main trigonometric identities are equalities that establish a relationship between the sine, cosine, tangent and cotangent of one angle, and allow you to find any of these trigonometric functions through a known other.
We immediately list the main trigonometric identities, which we will analyze in this article. We write them down in a table, and below we give the derivation of these formulas and give the necessary explanations.
Page navigation.
Relationship between sine and cosine of one angle
Sometimes they talk not about the main trigonometric identities listed in the table above, but about one single basic trigonometric identity kind 
. The explanation for this fact is quite simple: the equalities are obtained from the basic trigonometric identity after dividing both of its parts by and respectively, and the equalities 
and 
follow from the definitions of sine, cosine, tangent, and cotangent. We will discuss this in more detail in the following paragraphs.
That is, it is the equality that is of particular interest, which was given the name of the main trigonometric identity.
Before proving the basic trigonometric identity, we give its formulation: the sum of the squares of the sine and cosine of one angle is identically equal to one. Now let's prove it.
The basic trigonometric identity is very often used in transformation trigonometric expressions . It allows the sum of the squares of the sine and cosine of one angle to be replaced by one. No less often, the basic trigonometric identity is used in reverse order: the unit is replaced by the sum of the squares of the sine and cosine of any angle.
Tangent and cotangent through sine and cosine
Identities connecting the tangent and cotangent with the sine and cosine of one angle of the form and 
immediately follow from the definitions of sine, cosine, tangent and cotangent. Indeed, by definition, the sine is the ordinate of y, the cosine is the abscissa of x, the tangent is the ratio of the ordinate to the abscissa, that is, 
, and the cotangent is the ratio of the abscissa to the ordinate, that is, 
.
Due to this obviousness of the identities and often the definitions of tangent and cotangent are given not through the ratio of the abscissa and the ordinate, but through the ratio of the sine and cosine. So the tangent of an angle is the ratio of the sine to the cosine of this angle, and the cotangent is the ratio of the cosine to the sine.
To conclude this section, it should be noted that the identities and hold for all such angles for which the trigonometric functions in them make sense. So the formula is valid for any other than (otherwise the denominator will be zero, and we did not define division by zero), and the formula - for all , different from , where z is any .
Relationship between tangent and cotangent
An even more obvious trigonometric identity than the two previous ones is the identity connecting the tangent and cotangent of one angle of the form . It is clear that it takes place for any angles other than , otherwise either the tangent or the cotangent is not defined.
Proof of the formula very simple. By definition and from where 
. The proof could have been carried out in a slightly different way. Since and , then 
.
So, the tangent and cotangent of one angle, at which they make sense, is.
One of the branches of mathematics with which schoolchildren cope with the greatest difficulties is trigonometry. No wonder: 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 apply trigonometry when proving theorems, and this requires either a developed mathematical memory or the ability to deduce complex logical chains.
Origins of trigonometry
Acquaintance with this science should begin with the definition of the sine, cosine and tangent of the angle, but first you need to figure out what trigonometry does in general.
Historically, right triangles have been the main object of study in this section of mathematical science. 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 under consideration 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 art.
First stage
Initially, people talked about the relationship of angles and sides exclusively on 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 section of mathematics.
The study of trigonometry at school today begins with right-angled triangles, after which the acquired knowledge is used by students in physics and solving abstract trigonometric equations, work with which begins in high school.
Spherical trigonometry
Later, when science reached the next level of development, formulas with sine, cosine, tangent, cotangent began to be used in spherical geometry, where other rules apply, and the sum of the angles in a triangle is always more than 180 degrees. This section is not studied at school, but it is necessary to know about its existence, at least because earth's surface, and the surface of any other planet is convex, which means that any marking of the surface will be "arc-shaped" in three-dimensional space.
Take the globe and thread. Attach the thread to any two points on the globe so that it is taut. Pay attention - it has acquired the shape of an arc. It is with such forms that spherical geometry, which is used in geodesy, astronomy, and other theoretical and applied fields, deals.
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. She 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 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 that 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 180 degrees.
Definition
Finally, with a solid understanding of the geometric base, we can turn to the definition of the 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 leg 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 that their ratio will always be less than one. Thus, if you get a sine or cosine with a value greater than 1 in the answer to the problem, look for an error in calculations or reasoning. This answer is clearly wrong.
Finally, the tangent of an angle is the ratio of the opposite side to the adjacent side. The same result will give the division of the sine by the cosine. Look: in accordance with the formula, we divide the length of the side by the hypotenuse, after which we divide by the length of the second side and multiply by the hypotenuse. Thus, we get the same ratio as in the definition of tangent.
The cotangent, respectively, is the ratio of the side adjacent to the corner to the opposite side. We get the same result by dividing the unit by the tangent.
So, we have considered the definitions of what sine, cosine, tangent and cotangent are, and we can deal with formulas.
The simplest formulas
In trigonometry, one cannot do without formulas - how to find sine, cosine, tangent, cotangent without them? And this is exactly what is required when solving problems.
The first formula that 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 want to know the value of the angle, not the side.
Many students cannot remember the second formula, which is also very popular in solving school tasks: 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: after all, 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, the conversion rules and a few basic formulas, you can at any time derive the required more complex formulas on a piece of paper.
Double angle formulas and addition of arguments
Two more formulas that you need to learn are related to the values \u200b\u200bof the sine and cosine for the sum and difference of the angles. They are shown in the figure below. Please note that in the first case, the sine and cosine are multiplied both times, and in the second, the pairwise product of the sine and cosine is added.
There are also formulas associated with arguments in the form double angle. They are completely derived from the previous ones - as a practice, try to get them yourself by taking the alpha angle equal to the angle beta.
Finally, note that the double angle formulas can be converted to lower the degree 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 \u200b\u200bthe figure, and the size of each side, etc.
The sine theorem states that as a result of dividing the length of each of the sides of the triangle by the value of the opposite angle, we get the same number. Moreover, this number will be equal to two radii of the circumscribed circle, that is, the circle containing all points of the 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 angle adjacent to them - 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.
Mistakes due to inattention
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 get acquainted with the most popular of them.
Firstly, you should not convert ordinary fractions to decimals until the final result is obtained - you can leave the answer in the form common fraction unless the condition states otherwise. Such a transformation cannot be called a mistake, but it should be remembered that at each stage of the task, new roots may appear, which, according to the author's idea, should be reduced. In this case, you will waste time on unnecessary mathematical operations. This is especially true for values such as the root of three or two, because they occur in tasks at every step. The same applies to 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 also demonstrate a complete misunderstanding 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 mix them up, 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 it applied sense. What is sine, cosine, tangent for an engineer or astronomer? These are concepts thanks to which you can calculate the distance to distant stars, predict the fall of a meteorite, send a research probe to another planet. Without them, it is impossible to build a building, design a car, calculate the load on the 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 are sine, cosine, tangent. You can use them in calculations and successfully solve school problems.
The whole essence of trigonometry boils down to the fact that unknown parameters must be calculated from the known parameters of the triangle. There are six parameters in total: the lengths of three sides and the magnitudes of three angles. The whole difference in the tasks lies in the fact that different input data are given.
How to find the sine, cosine, tangent based on the known lengths of the legs or the hypotenuse, you now know. Since these terms mean nothing more than ratio, and ratio is a fraction, main goal finding the roots of an ordinary equation or a system of equations becomes a trigonometric problem. And here you will be helped by ordinary school mathematics.
The formulas for the sum and difference of sines and cosines for two angles α and β allow you to go from the sum of the indicated angles to the product of the angles α + β 2 and α - β 2 . We note right away that you should not confuse the formulas for the sum and difference of sines and cosines with the formulas for sines and cosines of the sum and difference. Below we list these formulas, give their derivation and show examples of application for specific problems.
Yandex.RTB R-A-339285-1
Formulas for the sum and difference of sines and cosines
Let's write down how the sum and difference formulas for sines and cosines look like
Sum and difference formulas for sines
sin α + sin β = 2 sin α + β 2 cos α - β 2 sin α - sin β = 2 sin α - β 2 cos α + β 2
Sum and difference formulas for cosines
cos α + cos β = 2 cos α + β 2 cos α - β 2 cos α - cos β = - 2 sin α + β 2 cos α - β 2, cos α - cos β = 2 sin α + β 2 β -α 2
These formulas are valid for any angles α and β. The angles α + β 2 and α - β 2 are called, respectively, the half-sum and half-difference of the angles alpha and beta. We give a formulation for each formula.
Definitions of sum and difference formulas for sines and cosines
The sum of the sines of two angles is equal to twice the product of the sine of the half-sum of these angles and the cosine of the half-difference.
Difference of sines of two angles is equal to twice the product of the sine of the half-difference of these angles and the cosine of the half-sum.
The sum of the cosines of two angles is equal to twice the product of the cosine of the half-sum and the cosine of the half-difference of these angles.
Difference of cosines of two angles is equal to twice the product of the sine of the half-sum and the cosine of the half-difference of these angles, taken with a negative sign.
Derivation of formulas for the sum and difference of sines and cosines
To derive formulas for the sum and difference of the sine and cosine of two angles, addition formulas are used. We present them below
sin (α + β) = sin α cos β + cos α sin β sin (α - β) = sin α cos β - cos α sin β cos (α + β) = cos α cos β - sin α sin β cos (α - β) = cos α cos β + sin α sin β
We also represent the angles themselves as the sum of half-sums and half-differences.
α \u003d α + β 2 + α - β 2 \u003d α 2 + β 2 + α 2 - β 2 β \u003d α + β 2 - α - β 2 \u003d α 2 + β 2 - α 2 + β 2
We proceed directly to the derivation of the sum and difference formulas for sin and cos.
Derivation of the formula for the sum of sines
In the sum sin α + sin β, we replace α and β with the expressions for these angles given above. Get
sin α + sin β = sin α + β 2 + α - β 2 + sin α + β 2 - α - β 2
Now we apply the addition formula to the first expression, and the sine formula of the angle differences to the second one (see the formulas above)
sin α + β 2 + α - β 2 = sin α + β 2 cos α - β 2 + cos α + β 2 sin α - β 2 sin α + β 2 - α - β 2 = sin α + β 2 cos α - β 2 - cos α + β 2 sin α - β 2 sin α + β 2 + α - β 2 + sin α + β 2 - α - β 2 = sin α + β 2 cos α - β 2 + cos α + β 2 sin α - β 2 + sin α + β 2 cos α - β 2 - cos α + β 2 sin α - β 2
sin α + β 2 cos α - β 2 + cos α + β 2 sin α - β 2 + sin α + β 2 cos α - β 2 - cos α + β 2 sin α - β 2 = = 2 sin α + β 2 cos α - β 2
The steps for deriving the rest of the formulas are similar.
Derivation of the formula for the difference of sines
sin α - sin β = sin α + β 2 + α - β 2 - sin α + β 2 - α - β 2 sin α + β 2 + α - β 2 - sin α + β 2 - α - β 2 = sin α + β 2 cos α - β 2 + cos α + β 2 sin α - β 2 - sin α + β 2 cos α - β 2 - cos α + β 2 sin α - β 2 = = 2 sin α - β 2 cos α + β 2
Derivation of the formula for the sum of cosines
cos α + cos β = cos α + β 2 + α - β 2 + cos α + β 2 - α - β 2 cos α + β 2 + α - β 2 + cos α + β 2 - α - β 2 = cos α + β 2 cos α - β 2 - sin α + β 2 sin α - β 2 + cos α + β 2 cos α - β 2 + sin α + β 2 sin α - β 2 = = 2 cos α + β 2 cos α - β 2
Derivation of the cosine difference formula
cos α - cos β = cos α + β 2 + α - β 2 - cos α + β 2 - α - β 2 cos α + β 2 + α - β 2 - cos α + β 2 - α - β 2 = cos α + β 2 cos α - β 2 - sin α + β 2 sin α - β 2 - cos α + β 2 cos α - β 2 + sin α + β 2 sin α - β 2 = = - 2 sin α + β 2 sin α - β 2
Examples of solving practical problems
To begin with, we will check one of the formulas by substituting specific angle values into it. Let α = π 2 , β = π 6 . Let's calculate the value of the sum of the sines of these angles. First, we use the table of basic values of trigonometric functions, and then we apply the formula for the sum of sines.
Example 1. Checking the formula for the sum of the sines of two angles
α \u003d π 2, β \u003d π 6 sin π 2 + sin π 6 \u003d 1 + 1 2 \u003d 3 2 sin π 2 + sin π 6 \u003d 2 sin π 2 + π 6 2 cos π 2 - π 6 2 \u003d 2 sin π 3 cos π 6 \u003d 2 3 2 3 2 \u003d 3 2
Let us now consider the case when the values of the angles differ from the basic values presented in the table. Let α = 165°, β = 75°. Let us calculate the value of the difference between the sines of these angles.
Example 2. Applying the sine difference formula
α = 165 ° , β = 75 ° sin α - sin β = sin 165 ° - sin 75 ° sin 165 - sin 75 = 2 sin 165 ° - sin 75 ° 2 cos 165 ° + sin 75 ° 2 = = 2 sin 45 ° cos 120 ° = 2 2 2 - 1 2 = 2 2
Using the formulas for the sum and difference of sines and cosines, you can go from the sum or difference to the product of trigonometric functions. Often these formulas are called formulas for the transition from sum to product. The formulas for the sum and difference of sines and cosines are widely used in solving trigonometric equations and in converting trigonometric expressions.
If you notice a mistake in the text, please highlight it and press Ctrl+Enter
The ratios between the main trigonometric functions - sine, cosine, tangent and cotangent - are given trigonometric formulas. And since there are quite a lot of connections between trigonometric functions, this also explains the abundance of trigonometric formulas. Some formulas connect the trigonometric functions of the same angle, others - the functions of a multiple angle, others - allow you to lower the degree, the fourth - to express all functions through the tangent of a half angle, etc.
In this article, we will list in order all the main trigonometric formulas, which are sufficient to solve the vast majority of trigonometry problems. For ease of memorization and use, we will group them according to their purpose, and enter them into tables.
Page navigation.
Basic trigonometric identities
Basic trigonometric identities set the relationship between the sine, cosine, tangent and cotangent of one angle. They follow from the definition of sine, cosine, tangent and cotangent, as well as the concept of the unit circle. They allow you to express one trigonometric function through any other.
For a detailed description of these trigonometry formulas, their derivation and application examples, see the article.
Cast formulas
Cast formulas follow from the properties of sine, cosine, tangent and cotangent, that is, they reflect the property of periodicity of trigonometric functions, the property of symmetry, and also the property of shift by a given angle. These trigonometric formulas allow you to move from working with arbitrary angles to working with angles ranging from zero to 90 degrees.
The rationale for these formulas, a mnemonic rule for memorizing them, and examples of their application can be studied in the article.
Addition Formulas
Trigonometric addition formulas show how the trigonometric functions of the sum or difference of two angles are expressed in terms of the trigonometric functions of these angles. These formulas serve as the basis for the derivation of the following trigonometric formulas.
Formulas for double, triple, etc. corner
Formulas for double, triple, etc. angle (they are also called multiple angle formulas) show how the trigonometric functions of double, triple, etc. angles () are expressed in terms of trigonometric functions of a single angle. Their derivation is based on addition formulas.
More detailed information is collected in the article formulas for double, triple, etc. angle .
Half Angle Formulas
Half Angle Formulas show how the trigonometric functions of a half angle are expressed in terms of the cosine of an integer angle. These trigonometric formulas follow from the double angle formulas.
Their conclusion and examples of application can be found in the article.
Reduction formulas
Trigonometric formulas for decreasing degrees are designed to facilitate the transition from natural powers of trigonometric functions to sines and cosines in the first degree, but multiple angles. In other words, they allow one to reduce the powers of trigonometric functions to the first.
Formulas for the sum and difference of trigonometric functions
main destination sum and difference formulas for trigonometric functions consists in the transition to the product of functions, which is very useful when simplifying trigonometric expressions. These formulas are also widely used in solving trigonometric equations, as they allow factoring the sum and difference of sines and cosines.
Formulas for the product of sines, cosines and sine by cosine
The transition from the product of trigonometric functions to the sum or difference is carried out through the formulas for the product of sines, cosines and sine by cosine.
Copyright by clever students
All rights reserved.
Protected by copyright law. No part of www.site, including internal materials and external design, may be reproduced in any form or used without the prior written permission of the copyright holder.
