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 dependencies establish the relationship between the acute angles and sides of any right triangle. Let us give 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 obtain the 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 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 comparison table 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 I and II 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. Enough to imagine trigonometric circle with the signs of trigonometric quantities and mentally “add” 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
Trigonometric identities- these are equalities that establish a connection 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 you to replace the sum of the squares of the cosine and sine of one angle with one and also perform the replacement operation in 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).
We will begin our study of trigonometry with the right triangle. Let's define what sine and cosine are, as well as tangent and cotangent acute angle. This is the basics of trigonometry.
Let us remind you that right angle is an angle equal to 90 degrees. In other words, half a turned angle.
Acute angle- less than 90 degrees.
Obtuse angle- greater than 90 degrees. In relation to such an angle, “obtuse” is not an insult, but a mathematical term :-)
Let's draw right triangle. A right angle is usually denoted by . Please note that the side opposite the corner is indicated by the same letter, only small. Thus, the side opposite angle A is designated .
The angle is denoted by the corresponding Greek letter.
Hypotenuse of a right triangle is the side opposite the right angle.
Legs- sides lying opposite acute angles.
The leg lying opposite the angle is called opposite(relative to angle). The other leg, which lies on one of the sides of the angle, is called adjacent.
Sinus The acute angle in a right triangle is the ratio of the opposite side to the hypotenuse:
Cosine acute angle in a right triangle - the ratio of the adjacent leg to the hypotenuse:
Tangent acute angle in a right triangle - the ratio of the opposite side to the adjacent:
Another (equivalent) definition: the tangent of an acute angle is the ratio of the sine of the angle to its cosine:
Cotangent acute angle in a right triangle - the ratio of the adjacent side to the opposite (or, which is the same, the ratio of cosine to sine):
Note the basic relationships for sine, cosine, tangent, and cotangent below. They will be useful to us when solving problems.
Let's prove some of them.
Okay, we have given definitions and written down formulas. But why do we still need sine, cosine, tangent and cotangent?
We know that the sum of the angles of any triangle is equal to.
We know the relationship between parties right triangle. This is the Pythagorean theorem: .
It turns out that knowing two angles in a triangle, you can find the third. Knowing the two sides of a right triangle, you can find the third. This means that the angles have their own ratio, and the sides have their own. But what should you do if in a right triangle you know one angle (except the right angle) and one side, but you need to find the other sides?
This is what people in the past encountered when making maps of the area and the starry sky. After all, it is not always possible to directly measure all sides of a triangle.
Sine, cosine and tangent - they are also called trigonometric angle functions- give relationships between parties And corners triangle. Knowing the angle, you can find all its trigonometric functions using special tables. And knowing the sines, cosines and tangents of the angles of a triangle and one of its sides, you can find the rest.
We will also draw a table of the values of sine, cosine, tangent and cotangent for “good” angles from to.
Please note the two red dashes in the table. At appropriate angle values, tangent and cotangent do not exist.
Let's look at several trigonometry problems from the FIPI Task Bank.
1. In a triangle, the angle is , . Find .
The problem is solved in four seconds.
Since , .
2. In a triangle, the angle is , , . Find .
Let's find it using the Pythagorean theorem.
The problem is solved.
Often in problems there are triangles with angles and or with angles and. Remember the basic ratios for them by heart!
For a triangle with angles and the leg opposite the angle at is equal to half of the hypotenuse.
A triangle with angles and is isosceles. In it, the hypotenuse is times larger than the leg.
We looked at problems solving right triangles - that is, finding unknown sides or angles. But that's not all! IN Unified State Exam options in mathematics there are many problems where the sine, cosine, tangent or cotangent of the external angle of a triangle appears. More on this in the next article.
First, consider a circle with radius 1 and center at (0;0). For any αЄR, the radius 0A can be drawn so that the radian measure of the angle between 0A and the 0x axis is equal to α. The counterclockwise direction is considered positive. Let the end of radius A have coordinates (a,b).
Definition of sine
Definition: The number b, equal to the ordinate of the unit radius constructed in the described way, is denoted by sinα and is called the sine of angle α.
Example: sin 3π cos3π/2 = 0 0 = 0
Definition of cosine
Definition: The number a, equal to the abscissa of the end of the unit radius constructed in the described way, is denoted by cosα and is called the cosine of the angle α.
Example: cos0 cos3π + cos3.5π = 1 (-1) + 0 = 2
These examples use the definition of the sine and cosine of an angle in terms of the coordinates of the end of the unit radius and the unit circle. For a more visual representation, you need to draw unit circle and plot the corresponding points on it, and then count their abscissas to calculate the cosine and the ordinate to calculate the sine.
Tangent definition
Definition: The function tgx=sinx/cosx for x≠π/2+πk, kЄZ, is called the cotangent of the angle x. The domain of definition of the function tgx is all real numbers, except x=π/2+πn, nЄZ.
Example: tg0 tgπ = 0 0 = 0
This example is similar to the previous one. To calculate the tangent of an angle, you need to divide the ordinate of a point by its abscissa.
Definition of cotangent
Definition: The function ctgx=cosx/sinx for x≠πk, kЄZ is called the cotangent of the angle x. The domain of definition of the function ctgx = is all real numbers except points x=πk, kЄZ.
Let's look at an example using a regular right triangle
To make it clearer what cosine, sine, tangent and cotangent are. Let's consider an example on a regular right triangle with angle y and sides a,b,c. Hypotenuse c, legs a and b respectively. The angle between the hypotenuse c and the leg b y.
Definition: The sine of the angle y is the ratio of the opposite side to the hypotenuse: siny = a/c
Definition: The cosine of the angle y is the ratio of the adjacent leg to the hypotenuse: cosy= in/c
Definition: The tangent of the angle y is the ratio of the opposite side to the adjacent side: tgy = a/b
Definition: The cotangent of the angle y is the ratio of the adjacent side to the opposite side: ctgy= in/a
Sine, cosine, tangent and cotangent are also called trigonometric functions. Each angle has its own sine and cosine. And almost everyone has their own tangent and cotangent.
It is believed that if we are given an angle, then its sine, cosine, tangent and cotangent are known to us! And vice versa. Given a sine, or any other trigonometric function, respectively, we know the angle. Even special tables have been created where trigonometric functions are written for each angle.
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.
An angle of (one degree) is called central angle in a circle, based on 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 it 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. To do this, a right triangle will help us.
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). Don't believe me? 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:
Doesn't 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:
Doesn't exist
Doesn't exist
Doesn't exist
Doesn't 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 for finding 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!
1.
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 value, 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.
4.
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.
SUMMARY AND BASIC FORMULAS
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) side to the adjacent (close) side.
The cotangent of an angle is the ratio of the adjacent (close) side to the opposite (far) side.