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Trigonometric formulas and their names. Basic trigonometric identities

Trigonometric identities- these are equalities that establish a relationship between 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 one to replace the sum of the squares of the cosine and sine of one angle by one and also perform the replacement operation in the reverse order.

Finding tangent and cotangent using 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 at it, then by definition the ordinate y is a sine, and the abscissa x is a 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.

Let us add that only for such angles \alpha at which the trigonometric functions included in them make sense, the identities will hold, ctg \alpha=\frac(\cos \alpha)(\sin \alpha).

For example: tg \alpha = \frac(\sin \alpha)(\cos \alpha) is valid for angles \alpha 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 above points, we obtain that tg \alpha = \frac(y)(x), A ctg \alpha=\frac(x)(y). It follows that tg \alpha \cdot ctg \alpha = \frac(y)(x) \cdot \frac(x)(y)=1. Thus, the tangent and cotangent of the same angle at which they make sense are mutually inverse 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 is equal to the inverse square of the sine of the given angle. This identity is valid for any \alpha different from \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 related 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).

In order to find tan \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 given 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).

Trigonometry, as a science, originated in the Ancient East. The first trigonometric ratios were derived by astronomers to create an accurate calendar and orientation by the stars. These calculations related to spherical trigonometry, while in school course study the ratios of sides and angles of a plane triangle.

Trigonometry is a branch of mathematics that deals with the properties of trigonometric functions and the relationship between the sides and angles of triangles.

During the heyday of culture and science in the 1st millennium AD, knowledge spread from the Ancient East to Greece. But the main discoveries of trigonometry are the merit of the men of the Arab Caliphate. In particular, the Turkmen scientist al-Marazwi introduced functions such as tangent and cotangent, and compiled the first tables of values for sines, tangents and cotangents. The concepts of sine and cosine were introduced by Indian scientists. Trigonometry received a lot of attention in the works of such great figures of antiquity as Euclid, Archimedes and Eratosthenes.

Basic quantities of trigonometry

The basic trigonometric functions of a numeric argument are sine, cosine, tangent, and cotangent. Each of them has its own graph: sine, cosine, tangent and cotangent.

The formulas for calculating the values of these quantities are based on the Pythagorean theorem. It is better known to schoolchildren in the formulation: “Pythagorean pants are equal in all directions,” since the proof is given using the example of an isosceles right triangle.

Sine, cosine and other relationships establish the relationship between the acute angles and sides of any right triangle. Let us present formulas for calculating these quantities for angle A and trace the relationships between trigonometric functions:

As you can see, tg and ctg are inverse functions. If we imagine leg a as the product of sin A and hypotenuse c, and leg b as cos A * c, we get following formulas for tangent and cotangent:

Trigonometric circle

Graphically, the relationship between the mentioned quantities can be represented as follows:

Circumference, in in this case, represents all possible values of the angle α - from 0° to 360°. As can be seen from the figure, each function takes a negative or positive value depending on the size of the angle. For example, sin α will have a “+” sign if α belongs to the 1st and 2nd quarters of the circle, that is, it is in the range from 0° to 180°. For α from 180° to 360° (III and IV quarters), sin α can only be a negative value.

Let's try to build trigonometric tables for specific angles and find out the value of the quantities.

Values of α equal to 30°, 45°, 60°, 90°, 180° and so on are called special cases. The values of trigonometric functions for them are calculated and presented in the form of special tables.

These angles were not chosen at random. The designation π in the tables is for radians. Rad is the angle at which the length of a circle's arc corresponds to its radius. This value was introduced in order to establish a universal dependence; when calculating in radians, the actual length of the radius in cm does not matter.

Angles in tables for trigonometric functions correspond to radian values:

So, it is not difficult to guess that 2π is a complete circle or 360°.

Properties of trigonometric functions: sine and cosine

In order to consider and compare the basic properties of sine and cosine, tangent and cotangent, it is necessary to draw their functions. This can be done in the form of a curve located in a two-dimensional coordinate system.

Consider the comparative table of properties for sine and cosine:

Sine wave	Cosine
y = sinx	y = cos x
ODZ [-1; 1]	ODZ [-1; 1]
sin x = 0, for x = πk, where k ϵ Z	cos x = 0, for x = π/2 + πk, where k ϵ Z
sin x = 1, for x = π/2 + 2πk, where k ϵ Z	cos x = 1, at x = 2πk, where k ϵ Z
sin x = - 1, at x = 3π/2 + 2πk, where k ϵ Z	cos x = - 1, for x = π + 2πk, where k ϵ Z
sin (-x) = - sin x, i.e. the function is odd	cos (-x) = cos x, i.e. the function is even
	the function is periodic, the smallest period is 2π
sin x › 0, with x belonging to the 1st and 2nd quarters or from 0° to 180° (2πk, π + 2πk)	cos x › 0, with x belonging to the I and IV quarters or from 270° to 90° (- π/2 + 2πk, π/2 + 2πk)
sin x ‹ 0, with x belonging to the third and fourth quarters or from 180° to 360° (π + 2πk, 2π + 2πk)	cos x ‹ 0, with x belonging to the 2nd and 3rd quarters or from 90° to 270° (π/2 + 2πk, 3π/2 + 2πk)
increases in the interval [- π/2 + 2πk, π/2 + 2πk]	increases on the interval [-π + 2πk, 2πk]
decreases on intervals [π/2 + 2πk, 3π/2 + 2πk]	decreases on intervals
derivative (sin x)’ = cos x	derivative (cos x)’ = - sin x

Determining whether a function is even or not is very simple. It is enough to imagine a trigonometric circle with the signs of trigonometric quantities and mentally “fold” the graph relative to the OX axis. If the signs coincide, the function is even, otherwise it is odd.

The introduction of radians and the listing of the basic properties of sine and cosine waves allow us to present the following pattern:

It is very easy to verify that the formula is correct. For example, for x = π/2, the sine is 1, as is the cosine of x = 0. The check can be done by consulting tables or by tracing function curves for given values.

Properties of tangentsoids and cotangentsoids

The graphs of the tangent and cotangent functions differ significantly from the sine and cosine functions. The values tg and ctg are reciprocals of each other.

Y = tan x.
The tangent tends to the values of y at x = π/2 + πk, but never reaches them.
The smallest positive period of the tangentoid is π.
Tg (- x) = - tg x, i.e. the function is odd.
Tg x = 0, for x = πk.
The function is increasing.
Tg x › 0, for x ϵ (πk, π/2 + πk).
Tg x ‹ 0, for x ϵ (— π/2 + πk, πk).
Derivative (tg x)’ = 1/cos 2 ⁡x.

Consider the graphic image of the cotangentoid below in the text.

Main properties of cotangentoids:

Y = cot x.
Unlike the sine and cosine functions, in the tangentoid Y can take on the values of the set of all real numbers.
The cotangentoid tends to the values of y at x = πk, but never reaches them.
The smallest positive period of a cotangentoid is π.
Ctg (- x) = - ctg x, i.e. the function is odd.
Ctg x = 0, for x = π/2 + πk.
The function is decreasing.
Ctg x › 0, for x ϵ (πk, π/2 + πk).
Ctg x ‹ 0, for x ϵ (π/2 + πk, πk).
Derivative (ctg x)’ = - 1/sin 2 ⁡x Correct

Reference data for tangent (tg x) and cotangent (ctg x). Geometric definition, properties, graphs, formulas. Table of tangents and cotangents, derivatives, integrals, series expansions. Expressions through complex variables. Connection with hyperbolic functions.

Geometric definition

|BD| - length of the arc of a circle with center at point A.
α is the angle expressed in radians.

Tangent ( tan α) is a trigonometric function depending on the angle α between the hypotenuse and the leg of a right triangle, equal to the ratio of the length of the opposite leg |BC| to the length of the adjacent leg |AB| .

Cotangent ( ctg α) is a trigonometric function depending on the angle α between the hypotenuse and the leg of a right triangle, equal to the ratio of the length of the adjacent leg |AB| to the length of the opposite leg |BC| .

Tangent

Where n- whole.

In Western literature, tangent is denoted as follows:
.
;
;
.

Graph of the tangent function, y = tan x

Cotangent

Where n- whole.

In Western literature, cotangent is denoted as follows:
.
The following notations are also accepted:
;
;
.

Graph of the cotangent function, y = ctg x

Properties of tangent and cotangent

Periodicity

Functions y = tg x and y = ctg x are periodic with period π.

Parity

The tangent and cotangent functions are odd.

Areas of definition and values, increasing, decreasing

The tangent and cotangent functions are continuous in their domain of definition (see proof of continuity). The main properties of tangent and cotangent are presented in the table ( n- whole).

	y= tg x	y= ctg x
Scope and continuity
Range of values	-∞ < y < +∞	-∞ < y < +∞
Increasing		-
Descending	-
Extremes	-	-
Zeros, y = 0
Intercept points with the ordinate axis, x = 0	y= 0	-

Formulas

Expressions using sine and cosine

; ;
; ;
;

Formulas for tangent and cotangent from sum and difference

The remaining formulas are easy to obtain, for example

Product of tangents

Formula for the sum and difference of tangents

This table presents the values of tangents and cotangents for certain values of the argument.

Expressions using complex numbers

Expressions through hyperbolic functions

;
;

Derivatives

; .

.
Derivative of the nth order with respect to the variable x of the function:
.
Deriving formulas for tangent > > > ; for cotangent > > >

Integrals

Series expansions

To obtain the expansion of the tangent in powers of x, you need to take several terms of the expansion in power series for functions sin x And cos x and divide these polynomials by each other, . This produces the following formulas.

At .

at .
Where Bn- Bernoulli numbers. They are determined either from the recurrence relation:
;
;
Where .
Or according to Laplace's formula:

Inverse functions

The inverse functions of tangent and cotangent are arctangent and arccotangent, respectively.

Arctangent, arctg

, Where n- whole.

Arccotangent, arcctg

, Where n- whole.

References:
I.N. Bronstein, K.A. Semendyaev, Handbook of mathematics for engineers and college students, “Lan”, 2009.
G. Korn, Handbook of Mathematics for Scientists and Engineers, 2012.

– there will certainly be tasks on trigonometry. Trigonometry is often disliked for the need to cram a huge number of difficult formulas, teeming with sines, cosines, tangents and cotangents. The site already once gave advice on how to remember a forgotten formula, using the example of the Euler and Peel formulas.

And in this article we will try to show that it is enough to firmly know only five simplest trigonometric formulas, and have a general idea about the rest and deduce them along the way. It’s like with DNA: the molecule does not store the complete blueprints of a finished living being. Rather, it contains instructions for assembling it from available amino acids. So in trigonometry, knowing some general principles, we will get all the necessary formulas from a small set of those that must be kept in mind.

We will rely on the following formulas:

From the formulas for sine and cosine sums, knowing about the parity of the cosine function and the oddness of the sine function, substituting -b instead of b, we obtain formulas for differences:

Sine of the difference: sin(a-b) = sinacos(-b)+cosasin(-b) = sinacosb-cosasinb
Cosine of the difference: cos(a-b) = cosacos(-b)-sinasin(-b) = cosacosb+sinasinb

Putting a = b into the same formulas, we obtain the formulas for sine and cosine of double angles:

Sine of double angle: sin2a = sin(a+a) = sinacosa+cosasina = 2sinacosa
Cosine of double angle: cos2a = cos(a+a) = cosacosa-sinasina = cos2 a-sin2 a

The formulas for other multiple angles are obtained similarly:

Sine of a triple angle: sin3a = sin(2a+a) = sin2acosa+cos2asina = (2sinacosa)cosa+(cos2 a-sin2 a)sina = 2sinacos2 a+sinacos2 a-sin 3 a = 3 sinacos2 a-sin 3 a = 3 sina(1-sin2 a)-sin 3 a = 3 sina-4sin 3a
Cosine of triple angle: cos3a = cos(2a+a) = cos2acosa-sin2asina = (cos2 a-sin2 a)cosa-(2sinacosa)sina = cos 3 a- sin2 acosa-2sin2 acosa = cos 3 a-3 sin2 acosa = cos 3 a-3(1- cos2 a)cosa = 4cos 3 a-3 cosa

Before we move on, let's look at one problem.
Given: the angle is acute.
Find its cosine if
Solution given by one student:
Because , That sina= 3,a cosa = 4.
(From math humor)

So, the definition of tangent relates this function to both sine and cosine. But you can get a formula that relates the tangent only to the cosine. To derive it, we take the main trigonometric identity: sin 2 a+cos 2 a= 1 and divide it by cos 2 a. We get:

So the solution to this problem would be:

(Since the angle is acute, when extracting the root, the + sign is taken)

The formula for the tangent of a sum is another one that is difficult to remember. Let's output it like this:

Immediately displayed and

From the cosine formula for a double angle, you can obtain the sine and cosine formulas for half angles. To do this, to the left side of the double angle cosine formula:
cos2 a = cos 2 a-sin 2 a
we add one, and to the right - a trigonometric unit, i.e. the sum of the squares of sine and cosine.
cos2a+1 = cos2 a-sin2 a+cos2 a+sin2 a
2cos 2 a = cos2 a+1
Expressing cosa through cos2 a and performing a change of variables, we get:

The sign is taken depending on the quadrant.

Similarly, subtracting one from the left side of the equality and the sum of the squares of the sine and cosine from the right, we get:
cos2a-1 = cos2 a-sin2 a-cos2 a-sin2 a
2sin 2 a = 1-cos2 a

And finally, to convert the sum of trigonometric functions into a product, we use the following technique. Let's say we need to represent the sum of sines as a product sina+sinb. Let's introduce variables x and y such that a = x+y, b+x-y. Then
sina+sinb = sin(x+y)+ sin(x-y) = sin x cos y+ cos x sin y+ sin x cos y- cos x sin y=2 sin x cos y. Let us now express x and y in terms of a and b.

Since a = x+y, b = x-y, then . That's why

You can withdraw immediately

Formula for partitioning products of sine and cosine V amount: sinacosb = 0.5(sin(a+b)+sin(a-b))

We recommend that you practice and derive formulas on your own for converting the difference of sines and the sum and difference of cosines into the product, as well as for dividing the products of sines and cosines into the sum. Having completed these exercises, you will thoroughly master the skill of deriving trigonometric formulas and will not get lost even in the most difficult test, olympiad or testing.

The concepts of sine (), cosine (), tangent (), cotangent () are inextricably linked with the concept of angle. In order to have a good understanding of these, at first glance, complex concepts (which cause a state of horror in many schoolchildren), and to make sure that “the devil is not as terrible as he is painted,” let’s start from the very beginning and understand the concept of an angle.

Angle concept: radian, degree

Let's look at the picture. The vector has “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, of course, angle units!

Angle, in both geometry and trigonometry, can be measured in degrees and radians.

Angle (one degree) is the central angle in a circle subtended by a circular arc equal to 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 equal to, that is, this angle rests on a circular arc the size of the circumference.

An angle in radians is the central angle in a circle subtended by a circular arc whose length is equal to the radius of the circle. Well, did you figure it out? If not, then let's figure it out from the drawing.

So, the figure shows an angle equal to a radian, that is, this angle rests on a circular arc, the length of which is equal to the radius of the circle (the length is equal to the length or radius equal to length arcs). Thus, the arc length is calculated by the formula:

Where is the central angle in radians.

Well, knowing this, can you answer how many radians are contained in the angle described by the circle? Yes, for this you need to remember the formula for circumference. Here she is:

Well, now let’s correlate these two formulas and find that the angle described by the circle is equal. That is, by 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 there? That's right!

Got it? Then go ahead and fix it:

Having difficulties? Then look answers:

Right triangle: sine, cosine, tangent, cotangent of angle

So, we figured out the concept of an angle. But what is sine, cosine, tangent, and cotangent of an angle? Let's figure it out. For this it will help us right triangle.

What are the sides of a right triangle called? That's right, hypotenuse and legs: the hypotenuse is the side that lies opposite the right angle (in our example this is the side); legs are the two remaining sides and (those adjacent to right angle), and, if we consider the legs relative to the angle, then the leg is the adjacent leg, and the leg is the opposite. So, now let’s answer the question: what are sine, cosine, tangent and cotangent of an angle?

Sine of angle- this is the ratio of the opposite (distant) leg to the hypotenuse.

In our triangle.

Cosine of angle- this is the ratio of the adjacent (close) leg to the hypotenuse.

In our triangle.

Tangent of the angle- this is the ratio of the opposite (distant) side to the adjacent (close).

In our triangle.

Cotangent of 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 into 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, you need to remember that sine, cosine, tangent and cotangent as the ratios of the sides of a triangle do not depend on the lengths of these sides (at the same angle). Do not believe? 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 consolidate 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 angle.

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 will be very useful when studying trigonometry. Therefore, let's look at it in a little more detail.

As you can see, this circle is constructed in the Cartesian coordinate system. The radius of the circle is equal to one, while the center of the circle lies at the origin of coordinates, 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 on the circle corresponds to two numbers: the axis coordinate and the axis coordinate. What are these coordinate numbers? And in general, what do they have to do with the topic at hand? To do this, we need to remember about the considered right 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 the triangle equal to? That's right. In addition, we know that is the radius of the unit circle, which means . Let's substitute this value into our formula for cosine. Here's what happens:

What is the triangle equal to? Well, of course, ! Substitute the radius value into this formula and get:

So, can you tell what coordinates a point belonging to a circle has? Well, no way? What if you realize that and are just numbers? Which coordinate does it correspond to? Well, of course, the coordinates! And what coordinate does it correspond to? That's right, coordinates! Thus, period.

What then are and equal to? That's right, let's use the corresponding definitions of tangent and cotangent and get that, a.

What if the angle is larger? For example, like in this picture:

What has changed in in this example? Let's figure it out. To do this, let's turn again to a right triangle. Consider a right triangle: angle (as adjacent to an angle). What are the values of sine, cosine, tangent and cotangent for an angle? That's right, we adhere to the corresponding definitions of 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 apply to any rotation 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 value, 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 a circle is or. Is it possible to rotate the radius vector to or to? Well, of course you can! In the first case, therefore, the radius vector will make one full revolution and stop at position or.

In the second case, that is, the radius vector will make three full 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, etc. This list can be continued indefinitely. All these angles can be written by 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 are:

Here's a unit circle to help you:

Having difficulties? Then let's figure it out. So we know that:

From here, we determine the coordinates of the points corresponding to certain angle measures. Well, let's start in order: the angle 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, and then check the answers.

Answers:

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 of the trigonometric functions of angles in and, given in the table below, must be remembered:

Don't be scared, now we'll show you one example quite simple to remember the corresponding values:

To use this method, it is vital to remember the values of the sine for all three measures of angle (), as well as the value of the tangent of the angle. Knowing these values, it is quite simple 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 indicated in the figure. If you understand this and remember the diagram with the arrows, then it will be enough to remember all the values 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 get it out general formula to find the coordinates of a point.

For example, here is a circle in front of us:

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 a 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. The length of a segment can be expressed using the definition of cosine:

Then we have that for the point coordinate.

Using the same logic, we find the y coordinate value for the point. Thus,

So, in general view coordinates of points are determined by the formulas:

Coordinates of the center of the circle,

Circle radius,

The rotation angle of the vector radius.

As you can see, for the unit circle we are considering, these formulas are significantly reduced, since the coordinates of the center are equal to zero and the radius is equal to one:

Well, let's try out these formulas by practicing finding points on a circle?

1. Find the coordinates of a point on the unit circle obtained by rotating the point on.

2. Find the coordinates of a point on the unit circle obtained by rotating the point on.

3. Find the coordinates of a point on the unit circle obtained by rotating the point on.

4. 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 initial radius vector by.

5. 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 initial radius vector by.

Having trouble finding the coordinates of a point on a circle?

Solve these five examples (or get good at solving them) and you will learn to find them!

You can notice that. But we know what corresponds to a full revolution of the starting point. Thus, the desired point will be in the same position as when turning to. Knowing this, we find the required coordinates of the point:

2. The unit circle is centered at a point, which means we can use simplified formulas:

You can notice that. We know what corresponds to two full revolutions of the starting point. Thus, the desired point will be in the same position as when turning to. Knowing this, we find the required coordinates of the point:

Sine and cosine are table values. We recall their meanings and get:

Thus, the desired point has coordinates.

3. The unit circle is centered at a point, which means we can use simplified formulas:

You can notice that. Let's depict the example in question in the figure:

The radius makes angles equal to and with the axis. Knowing that the table values of cosine and sine are equal, and having determined that the cosine here takes negative meaning, and the sine is positive, we have:

Such examples are discussed in more detail when studying the formulas for reducing trigonometric functions in the topic.

Thus, the desired point has coordinates.

Angle of rotation of the radius of the vector (by condition)

To determine the corresponding signs of sine and cosine, we construct a unit circle and 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

Coordinates of the center of the circle (in our example,

Circle radius (by condition)

Angle of rotation of the radius of the vector (by condition).

Let's substitute all the values into the formula and get:

and - table values. Let’s remember and substitute them into the formula:

Thus, the desired point has coordinates.