What is the tangent of a rectangular angle? What are sine and cosine in trigonometry? Formulas for tangent and cotangent from sum and difference
Trigonometry is a branch of mathematical science that studies trigonometric functions and their use in geometry. The development of trigonometry began in ancient Greece. During the Middle Ages, scientists from the Middle East and India made important contributions to the development of this science.
This article is devoted to the basic concepts and definitions of trigonometry. It discusses the definitions of the basic trigonometric functions: sine, cosine, tangent and cotangent. Their meaning is explained and illustrated in the context of geometry.
Initially, the definitions of trigonometric functions whose argument is an angle were expressed in terms of the ratio of the sides of a right triangle.
Definitions of trigonometric functions
The sine of an angle (sin α) is the ratio of the leg opposite this angle to the hypotenuse.
Cosine of the angle (cos α) - the ratio of the adjacent leg to the hypotenuse.
Angle tangent (t g α) - the ratio of the opposite side to the adjacent side.
Angle cotangent (c t g α) - the ratio of the adjacent side to the opposite side.
These definitions are given for the acute angle of a right triangle!
Let's give an illustration.
In triangle ABC with right angle C, the sine of angle A is equal to the ratio of leg BC to hypotenuse AB.
The definitions of sine, cosine, tangent and cotangent allow you to calculate the values of these functions from the known lengths of the sides of the triangle.
Important to remember!
The range of values of sine and cosine is from -1 to 1. In other words, sine and cosine take values from -1 to 1. The range of values of tangent and cotangent is the entire number line, that is, these functions can take on any values.
The definitions given above apply to acute angles. In trigonometry, the concept of a rotation angle is introduced, the value of which, unlike an acute angle, is not limited to 0 to 90 degrees. The rotation angle in degrees or radians is expressed by any real number from - ∞ to + ∞.
In this context, we can define sine, cosine, tangent and cotangent of an angle of arbitrary magnitude. Let us imagine a unit circle with its center at the origin of the Cartesian coordinate system.
The initial point A with coordinates (1, 0) rotates around the center of the unit circle through a certain angle α and goes to point A 1. The definition is given in terms of the coordinates of point A 1 (x, y).
Sine (sin) of the rotation angle
The sine of the rotation angle α is the ordinate of point A 1 (x, y). sin α = y
Cosine (cos) of the rotation angle
The cosine of the rotation angle α is the abscissa of point A 1 (x, y). cos α = x
Tangent (tg) of the rotation angle
The tangent of the angle of rotation α is the ratio of the ordinate of point A 1 (x, y) to its abscissa. t g α = y x
Cotangent (ctg) of the rotation angle
The cotangent of the rotation angle α is the ratio of the abscissa of point A 1 (x, y) to its ordinate. c t g α = x y
Sine and cosine are defined for any rotation angle. This is logical, because the abscissa and ordinate of a point after rotation can be determined at any angle. The situation is different with tangent and cotangent. The tangent is undefined when a point after rotation goes to a point with zero abscissa (0, 1) and (0, - 1). In such cases, the expression for tangent t g α = y x simply does not make sense, since it contains division by zero. The situation is similar with cotangent. The difference is that the cotangent is not defined in cases where the ordinate of a point goes to zero.
Important to remember!
Sine and cosine are defined for any angles α.
Tangent is defined for all angles except α = 90° + 180° k, k ∈ Z (α = π 2 + π k, k ∈ Z)
Cotangent is defined for all angles except α = 180° k, k ∈ Z (α = π k, k ∈ Z)
When solving practical examples, do not say “sine of the angle of rotation α”. The words “angle of rotation” are simply omitted, implying that it is already clear from the context what is being discussed.
Numbers
What about the definition of sine, cosine, tangent and cotangent of a number, and not the angle of rotation?
Sine, cosine, tangent, cotangent of a number
Sine, cosine, tangent and cotangent of a number t is a number that is respectively equal to sine, cosine, tangent and cotangent in t radian.
For example, the sine of the number 10 π is equal to the sine of the rotation angle of 10 π rad.
There is another approach to determining the sine, cosine, tangent and cotangent of a number. Let's take a closer look at it.
Any real number t a point on the unit circle is associated with the center at the origin of the rectangular Cartesian coordinate system. Sine, cosine, tangent and cotangent are determined through the coordinates of this point.
The starting point on the circle is point A with coordinates (1, 0).
Positive number t
Negative number t corresponds to the point to which the starting point will go if it moves around the circle counterclockwise and passes the path t.
Now that the connection between a number and a point on a circle has been established, we move on to the definition of sine, cosine, tangent and cotangent.
Sine (sin) of t
Sine of a number t- ordinate of a point on the unit circle corresponding to the number t. sin t = y
Cosine (cos) of t
Cosine of a number t- abscissa of the point of the unit circle corresponding to the number t. cos t = x
Tangent (tg) of t
Tangent of a number t- the ratio of the ordinate to the abscissa of a point on the unit circle corresponding to the number t. t g t = y x = sin t cos t
The latest definitions are in accordance with and do not contradict the definition given at the beginning of this paragraph. Point on the circle corresponding to the number t, coincides with the point to which the starting point goes after turning by an angle t radian.
Trigonometric functions of angular and numeric argument
Each value of the angle α corresponds to a certain value of the sine and cosine of this angle. Just like all angles α other than α = 90 ° + 180 ° k, k ∈ Z (α = π 2 + π k, k ∈ Z) correspond to a certain tangent value. Cotangent, as stated above, is defined for all α except α = 180° k, k ∈ Z (α = π k, k ∈ Z).
We can say that sin α, cos α, t g α, c t g α are functions of the angle alpha, or functions of the angular argument.
Similarly, we can talk about sine, cosine, tangent and cotangent as functions of a numerical argument. Every real number t corresponds to a certain value of the sine or cosine of a number t. All numbers other than π 2 + π · k, k ∈ Z, correspond to a tangent value. Cotangent, similarly, is defined for all numbers except π · k, k ∈ Z.
Basic functions of trigonometry
Sine, cosine, tangent and cotangent are the basic trigonometric functions.
It is usually clear from the context which argument of the trigonometric function (angular argument or numeric argument) we are dealing with.
Let's return to the definitions given at the very beginning and the alpha angle, which lies in the range from 0 to 90 degrees. The trigonometric definitions of sine, cosine, tangent, and cotangent are entirely consistent with the geometric definitions given by the aspect ratios of a right triangle. Let's show it.
Let's take a unit circle with a center in a rectangular Cartesian coordinate system. Let's rotate the starting point A (1, 0) by an angle of up to 90 degrees and draw a perpendicular to the abscissa axis from the resulting point A 1 (x, y). In the resulting right triangle, the angle A 1 O H is equal to the angle of rotation α, the length of the leg O H is equal to the abscissa of the point A 1 (x, y). The length of the leg opposite the angle is equal to the ordinate of the point A 1 (x, y), and the length of the hypotenuse is equal to one, since it is the radius of the unit circle.
In accordance with the definition from geometry, the sine of angle α is equal to the ratio of the opposite side to the hypotenuse.
sin α = A 1 H O A 1 = y 1 = y
This means that determining the sine of an acute angle in a right triangle through the aspect ratio is equivalent to determining the sine of the rotation angle α, with alpha lying in the range from 0 to 90 degrees.
Similarly, the correspondence of definitions can be shown for cosine, tangent and cotangent.
If you notice an error in the text, please highlight it and press Ctrl+Enter
Lecture: Sine, cosine, tangent, cotangent of an arbitrary angle
Sine, cosine of an arbitrary angle
To understand what trigonometric functions are, let's look at a circle with unit radius. This circle has a center at the origin on the coordinate plane. To determine the given functions we will use the radius vector OR, which starts at the center of the circle, and the point R is a point on the circle. This radius vector forms an angle alpha with the axis OH. Since the circle has a radius equal to one, then OR = R = 1.
If from the point R lower the perpendicular to the axis OH, then we get a right triangle with a hypotenuse equal to one.
If the radius vector moves clockwise, then this direction is called negative, if it moves counterclockwise - positive.
Sine of the angle OR, is the ordinate of the point R vector on a circle.
That is, to obtain the value of the sine of a given angle alpha, it is necessary to determine the coordinate U on surface.
How was this value obtained? Since we know that the sine of an arbitrary angle in a right triangle is the ratio of the opposite leg to the hypotenuse, we get that
And since R=1, That sin(α) = y 0 .
In a unit circle, the ordinate value cannot be less than -1 and greater than 1, which means
The sine takes a positive value in the first and second quarters of the unit circle, and negative in the third and fourth.
Cosine of the angle given circle formed by the radius vector OR, is the abscissa of the point R vector on a circle.
That is, to obtain the cosine value of a given angle alpha, it is necessary to determine the coordinate X on surface.
The cosine of an arbitrary angle in a right triangle is the ratio of the adjacent leg to the hypotenuse, we get that
And since R=1, That cos(α) = x 0 .
In the unit circle, the abscissa value cannot be less than -1 and greater than 1, which means
The cosine takes a positive value in the first and fourth quarters of the unit circle, and negative in the second and third.
Tangentarbitrary angle The ratio of sine to cosine is calculated.
If we consider a right triangle, then this is the ratio of the opposite side to the adjacent side. If we are talking about the unit circle, then this is the ratio of the ordinate to the abscissa.
Judging by these relationships, it can be understood that the tangent cannot exist if the abscissa value is zero, that is, at an angle of 90 degrees. The tangent can take all other values.
The tangent is positive in the first and third quarters of the unit circle, and negative in the second and fourth.
|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 a power series for the 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.
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 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
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 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, the 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 a negative value and the sine takes a positive value, 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.
Sine is one of the basic trigonometric functions, the use of which is not limited to geometry alone. Tables for calculating trigonometric functions, like engineering calculators, are not always at hand, and calculating the sine is sometimes needed to solve various problems. In general, calculating the sine will help consolidate drawing skills and knowledge of trigonometric identities.
Games with ruler and pencil
A simple task: how to find the sine of an angle drawn on paper? To solve, you will need a regular ruler, a triangle (or compass) and a pencil. The simplest way to calculate the sine of an angle is by dividing the far leg of a triangle with a right angle by the long side - the hypotenuse. Thus, you first need to complete the acute angle to the shape of a right triangle by drawing a line perpendicular to one of the rays at an arbitrary distance from the vertex of the angle. We will need to maintain an angle of exactly 90°, for which we need a clerical triangle.
Using a compass is a little more accurate, but will take more time. On one of the rays you need to mark 2 points at a certain distance, set a radius on the compass approximately equal to the distance between the points, and draw semicircles with centers at these points until the intersections of these lines are obtained. By connecting the intersection points of our circles with each other, we get a strict perpendicular to the ray of our angle; all that remains is to extend the line until it intersects with another ray.
In the resulting triangle, you need to use a ruler to measure the side opposite the corner and the long side on one of the rays. The ratio of the first dimension to the second will be the desired value of the sine of the acute angle.
Find the sine for an angle greater than 90°
For an obtuse angle the task is not much more difficult. We need to draw a ray from the vertex in the opposite direction using a ruler to form a straight line with one of the rays of the angle we are interested in. The resulting acute angle should be treated as described above; the sines of adjacent angles that together form a reverse angle of 180° are equal.
Calculating sine using other trigonometric functions
Also, calculating the sine is possible if the values of other trigonometric functions of the angle or at least the lengths of the sides of the triangle are known. Trigonometric identities will help us with this. Let's look at common examples.
How to find the sine with a known cosine of an angle? The first trigonometric identity, based on the Pythagorean theorem, states that the sum of the squares of the sine and cosine of the same angle is equal to one.
How to find the sine with a known tangent of an angle? The tangent is obtained by dividing the far side by the near side or dividing the sine by the cosine. Thus, the sine will be the product of the cosine and the tangent, and the square of the sine will be the square of this product. We replace the squared cosine with the difference between unity and the square sine according to the first trigonometric identity and, through simple manipulations, we reduce the equation to the calculation of the square sine through the tangent; accordingly, to calculate the sine, you will have to extract the root of the result obtained.
How to find the sine with a known cotangent of an angle? The value of the cotangent can be calculated by dividing the length of the leg closest to the angle by the length of the far one, as well as dividing the cosine by the sine, that is, the cotangent is a function inverse to the tangent relative to the number 1. To calculate the sine, you can calculate the tangent using the formula tg α = 1 / ctg α and use the formula in the second option. You can also derive a direct formula by analogy with tangent, which will look like this.
How to find the sine of three sides of a triangle
There is a formula for finding the length of the unknown side of any triangle, not just a right triangle, from two known sides using the trigonometric function of the cosine of the opposite angle. She looks like this.
