The cosine of an angle is equal to the ratio. Sine, cosine, tangent, cotangent of an acute angle

The concepts of sine, cosine, tangent and cotangent are the main categories of trigonometry - a branch of mathematics, and are inextricably linked with the definition of an angle. Possession of this mathematical science requires memorization and understanding of formulas and theorems, as well as developed spatial thinking. That is why trigonometric calculations often cause difficulties for schoolchildren and students. To overcome them, you should become more familiar with trigonometric functions and formulas.

Concepts in trigonometry

To understand the basic concepts of trigonometry, you must first decide what is right triangle and the angle in a circle, and why all the basic trigonometric calculations are associated with them. A triangle in which one of the angles is 90 degrees is a right triangle. Historically, this figure was often used by people in architecture, navigation, art, astronomy. Accordingly, studying and analyzing the properties of this figure, people came to the calculation of the corresponding ratios of its parameters.

The main categories associated with right triangles are the hypotenuse and the legs. The hypotenuse is the side of a triangle that is opposite the right angle. The legs, respectively, are the other two sides. The sum of the angles of any triangle is always 180 degrees.

Spherical trigonometry is a section of trigonometry that is not studied at school, but in applied sciences such as astronomy and geodesy, scientists use it. A feature of a triangle in spherical trigonometry is that it always has a sum of angles greater than 180 degrees.

Angles of a triangle

In a right triangle, the sine of an angle is the ratio of the leg opposite the desired angle to the hypotenuse of the triangle. Accordingly, the cosine is the ratio of the adjacent leg and the hypotenuse. Both of these values always have a value less than one, since the hypotenuse is always longer than the leg.

The tangent of an angle is a value equal to the ratio of the opposite leg to the adjacent leg of the desired angle, or sine to cosine. The cotangent, in turn, is the ratio of the adjacent leg of the desired angle to the opposite cactet. The cotangent of an angle can also be obtained by dividing the unit by the value of the tangent.

unit circle

A unit circle in geometry is a circle whose radius is equal to one. Such a circle is constructed in the Cartesian coordinate system, with the center of the circle coinciding with the point of origin, and the initial position of the radius vector is determined by the positive direction of the X axis (abscissa axis). Each point of the circle has two coordinates: XX and YY, that is, the coordinates of the abscissa and ordinate. Selecting any point on the circle in the XX plane, and dropping the perpendicular from it to the abscissa axis, we get a right triangle formed by a radius to the selected point (let us denote it by the letter C), a perpendicular drawn to the X axis (the intersection point is denoted by the letter G), and a segment the abscissa axis between the origin (the point is denoted by the letter A) and the intersection point G. The resulting triangle ACG is a right triangle inscribed in a circle, where AG is the hypotenuse, and AC and GC are the legs. The angle between the radius of the circle AC and the segment of the abscissa axis with the designation AG, we define as α (alpha). So, cos α = AG/AC. Given that AC is the radius of the unit circle, and it is equal to one, it turns out that cos α=AG. Similarly, sin α=CG.

In addition, knowing these data, it is possible to determine the coordinate of point C on the circle, since cos α=AG, and sin α=CG, which means that point C has the given coordinates (cos α; sin α). Knowing that the tangent is equal to the ratio of the sine to the cosine, we can determine that tg α \u003d y / x, and ctg α \u003d x / y. Considering angles in a negative coordinate system, one can calculate that the sine and cosine values of some angles can be negative.

Calculations and basic formulas

Values of trigonometric functions

Having considered the essence of trigonometric functions through the unit circle, we can derive the values of these functions for some angles. The values are listed in the table below.

The simplest trigonometric identities

Equations in which there is an unknown value under the sign of the trigonometric function are called trigonometric. Identities with the value sin x = α, k is any integer:

sin x = 0, x = πk.
2. sin x \u003d 1, x \u003d π / 2 + 2πk.
sin x \u003d -1, x \u003d -π / 2 + 2πk.
sin x = a, |a| > 1, no solutions.
sin x = a, |a| ≦ 1, x = (-1)^k * arcsin α + πk.

Identities with the value cos x = a, where k is any integer:

cos x = 0, x = π/2 + πk.
cos x = 1, x = 2πk.
cos x \u003d -1, x \u003d π + 2πk.
cos x = a, |a| > 1, no solutions.
cos x = a, |a| ≦ 1, х = ±arccos α + 2πk.

Identities with the value tg x = a, where k is any integer:

tg x = 0, x = π/2 + πk.
tg x \u003d a, x \u003d arctg α + πk.

Identities with value ctg x = a, where k is any integer:

ctg x = 0, x = π/2 + πk.
ctg x \u003d a, x \u003d arcctg α + πk.

Cast formulas

This category of constant formulas denotes methods by which you can go from trigonometric functions of the form to functions of the argument, that is, convert the sine, cosine, tangent and cotangent of an angle of any value to the corresponding indicators of the angle of the interval from 0 to 90 degrees for greater convenience of calculations.

The formulas for reducing functions for the sine of an angle look like this:

sin(900 - α) = α;
sin(900 + α) = cos α;
sin(1800 - α) = sin α;
sin(1800 + α) = -sin α;
sin(2700 - α) = -cos α;
sin(2700 + α) = -cos α;
sin(3600 - α) = -sin α;
sin(3600 + α) = sin α.

For the cosine of an angle:

cos(900 - α) = sin α;
cos(900 + α) = -sin α;
cos(1800 - α) = -cos α;
cos(1800 + α) = -cos α;
cos(2700 - α) = -sin α;
cos(2700 + α) = sin α;
cos(3600 - α) = cos α;
cos(3600 + α) = cos α.

The use of the above formulas is possible subject to two rules. First, if the angle can be represented as a value (π/2 ± a) or (3π/2 ± a), the value of the function changes:

from sin to cos;
from cos to sin;
from tg to ctg;
from ctg to tg.

The value of the function remains unchanged if the angle can be represented as (π ± a) or (2π ± a).

Secondly, the sign of the reduced function does not change: if it was initially positive, it remains so. The same is true for negative functions.

Addition Formulas

These formulas express the values of the sine, cosine, tangent, and cotangent of the sum and difference of two rotation angles in terms of their trigonometric functions. Angles are usually denoted as α and β.

The formulas look like this:

sin(α ± β) = sin α * cos β ± cos α * sin.
cos(α ± β) = cos α * cos β ∓ sin α * sin.
tan(α ± β) = (tan α ± tan β) / (1 ∓ tan α * tan β).
ctg(α ± β) = (-1 ± ctg α * ctg β) / (ctg α ± ctg β).

These formulas are valid for any angles α and β.

Double and triple angle formulas

The trigonometric formulas of a double and triple angle are formulas that relate the functions of the angles 2α and 3α, respectively, to the trigonometric functions of the angle α. Derived from addition formulas:

sin2α = 2sinα*cosα.
cos2α = 1 - 2sin^2α.
tg2α = 2tgα / (1 - tg^2 α).
sin3α = 3sinα - 4sin^3α.
cos3α = 4cos^3α - 3cosα.
tg3α = (3tgα - tg^3 α) / (1-tg^2 α).

Transition from sum to product

Considering that 2sinx*cosy = sin(x+y) + sin(x-y), simplifying this formula, we obtain the identity sinα + sinβ = 2sin(α + β)/2 * cos(α − β)/2. Similarly, sinα - sinβ = 2sin(α - β)/2 * cos(α + β)/2; cosα + cosβ = 2cos(α + β)/2 * cos(α − β)/2; cosα - cosβ = 2sin(α + β)/2 * sin(α − β)/2; tgα + tgβ = sin(α + β) / cosα * cosβ; tgα - tgβ = sin(α - β) / cosα * cosβ; cosα + sinα = √2sin(π/4 ∓ α) = √2cos(π/4 ± α).

Transition from product to sum

These formulas follow from the identities for the transition of the sum to the product:

sinα * sinβ = 1/2*;
cosα * cosβ = 1/2*;
sinα * cosβ = 1/2*.

Reduction Formulas

In these identities, the square and cubic powers of the sine and cosine can be expressed in terms of the sine and cosine of the first power of a multiple angle:

sin^2 α = (1 - cos2α)/2;
cos^2α = (1 + cos2α)/2;
sin^3 α = (3 * sinα - sin3α)/4;
cos^3 α = (3 * cosα + cos3α)/4;
sin^4 α = (3 - 4cos2α + cos4α)/8;
cos^4 α = (3 + 4cos2α + cos4α)/8.

Universal substitution

The universal trigonometric substitution formulas express trigonometric functions in terms of the tangent of a half angle.

sin x \u003d (2tgx / 2) * (1 + tg ^ 2 x / 2), while x \u003d π + 2πn;
cos x = (1 - tg^2 x/2) / (1 + tg^2 x/2), where x = π + 2πn;
tg x \u003d (2tgx / 2) / (1 - tg ^ 2 x / 2), where x \u003d π + 2πn;
ctg x \u003d (1 - tg ^ 2 x / 2) / (2tgx / 2), while x \u003d π + 2πn.

Special cases

Special cases of the simplest trigonometric equations are given below (k is any integer).

Private for sine:

sin x value	x value
0	pk
1	π/2 + 2πk
-1	-π/2 + 2πk
1/2	π/6 + 2πk or 5π/6 + 2πk
-1/2	-π/6 + 2πk or -5π/6 + 2πk
√2/2	π/4 + 2πk or 3π/4 + 2πk
-√2/2	-π/4 + 2πk or -3π/4 + 2πk
√3/2	π/3 + 2πk or 2π/3 + 2πk
-√3/2	-π/3 + 2πk or -2π/3 + 2πk

Cosine quotients:

cos x value	x value
0	π/2 + 2πk
1	2πk
-1	2 + 2πk
1/2	±π/3 + 2πk
-1/2	±2π/3 + 2πk
√2/2	±π/4 + 2πk
-√2/2	±3π/4 + 2πk
√3/2	±π/6 + 2πk
-√3/2	±5π/6 + 2πk

Private for tangent:

tg x value	x value
0	pk
1	π/4 + πk
-1	-π/4 + πk
√3/3	π/6 + πk
-√3/3	-π/6 + πk
√3	π/3 + πk
-√3	-π/3 + πk

Cotangent quotients:

ctg x value	x value
0	π/2 + πk
1	π/4 + πk
-1	-π/4 + πk
√3	π/6 + πk
-√3	-π/3 + πk
√3/3	π/3 + πk
-√3/3	-π/3 + πk

Theorems

Sine theorem

There are two versions of the theorem - simple and extended. Simple theorem sinuses: a/sin α = b/sin β = c/sin γ. In this case, a, b, c are the sides of the triangle, and α, β, γ are the opposite angles, respectively.

Extended sine theorem for an arbitrary triangle: a/sin α = b/sin β = c/sin γ = 2R. In this identity, R denotes the radius of the circle in which the given triangle is inscribed.

Cosine theorem

The identity is displayed in this way: a^2 = b^2 + c^2 - 2*b*c*cos α. In the formula, a, b, c are the sides of the triangle, and α is the angle opposite side a.

Tangent theorem

The formula expresses the relationship between the tangents of two angles, and the length of the sides opposite them. The sides are labeled a, b, c, and the corresponding opposite angles are α, β, γ. The formula of the tangent theorem: (a - b) / (a+b) = tg((α - β)/2) / tg((α + β)/2).

Cotangent theorem

Associates the radius of a circle inscribed in a triangle with the length of its sides. If a, b, c are the sides of a triangle, and A, B, C, respectively, are their opposite angles, r is the radius of the inscribed circle, and p is the half-perimeter of the triangle, the following identities hold:

ctg A/2 = (p-a)/r;
ctg B/2 = (p-b)/r;
ctg C/2 = (p-c)/r.

Applications

Trigonometry is not only a theoretical science associated with mathematical formulas. Its properties, theorems and rules are used in practice by various branches of human activity - astronomy, air and sea navigation, music theory, geodesy, chemistry, acoustics, optics, electronics, architecture, economics, mechanical engineering, measuring work, computer graphics, cartography, oceanography, and many others.

Sine, cosine, tangent and cotangent are the basic concepts of trigonometry, with which you can mathematically express the relationship between angles and lengths of sides in a triangle, and find the desired quantities through identities, theorems and rules.

Trigonometry is a branch of mathematics that studies trigonometric functions and their use in geometry. The development of trigonometry began in the days of ancient Greece. During the Middle Ages, scientists from the Middle East and India made an important contribution to the development of this science.

This article is about basic concepts and definitions of trigonometry. It discusses the definitions of the main trigonometric functions: sine, cosine, tangent and cotangent. Their meaning in the context of geometry is explained and illustrated.

Initially, the definitions of trigonometric functions, whose argument is an angle, were expressed through 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.

The cosine of the angle (cos α) is the ratio of the adjacent leg to the hypotenuse.

The tangent of the angle (t g α) is the ratio of the opposite leg to the adjacent one.

The cotangent of the angle (c t g α) is the ratio of the adjacent leg to the opposite one.

These definitions are given for an 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 make it possible to calculate the values of these functions from the known lengths of the sides of a triangle.

Important to remember!

The range of sine and cosine values: from -1 to 1. In other words, sine and cosine take values from -1 to 1. The range of tangent and cotangent values is the entire number line, that is, these functions can take any value.

The definitions given above refer to acute angles. In trigonometry, the concept of the angle of rotation is introduced, the value of which, unlike an acute angle, is not limited by frames from 0 to 90 degrees. The angle of rotation in degrees or radians is expressed by any real number from - ∞ to + ∞.

In this context, one can define the sine, cosine, tangent and cotangent of an angle of arbitrary magnitude. Imagine a unit circle centered at the origin of the Cartesian coordinate system.

The starting point A with coordinates (1 , 0) rotates around the center of the unit circle by some angle α and goes to point A 1 . The definition is given through the coordinates of the point A 1 (x, y).

Sine (sin) of the rotation angle

The sine of the rotation angle α is the ordinate of the point A 1 (x, y). sinα = y

Cosine (cos) of the angle of rotation

The cosine of the angle of rotation α is the abscissa of the point A 1 (x, y). cos α = x

Tangent (tg) of rotation angle

The tangent of the angle of rotation α is the ratio of the ordinate of the point A 1 (x, y) to its abscissa. t g α = y x

Cotangent (ctg) of rotation angle

The cotangent of the angle of rotation α is the ratio of the abscissa of the point A 1 (x, y) to its ordinate. c t g α = x y

Sine and cosine are defined for any angle of rotation. This is logical, because the abscissa and ordinate of the point after the rotation can be determined at any angle. The situation is different with tangent and cotangent. The tangent is not defined when the point after the rotation goes to the point with zero abscissa (0 , 1) and (0 , - 1). In such cases, the expression for the tangent t g α = y x simply does not make sense, since it contains division by zero. The situation is similar with the cotangent. The difference is that the cotangent is not defined in cases where the ordinate of the point vanishes.

Important to remember!

Sine and cosine are defined for any angles α.

The tangent is defined for all angles except α = 90° + 180° k , k ∈ Z (α = π 2 + π k , k ∈ Z)

The cotangent is defined for all angles except α = 180° k, k ∈ Z (α = π k, k ∈ Z)

When deciding practical examples don't say "sine of the angle of rotation α". The words "angle of rotation" are simply omitted, implying that from the context it is already clear what is at stake.

Numbers

What about the definition of the 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 a number is called, which is respectively equal to the sine, cosine, tangent and cotangent in t radian.

For example, the sine of 10 π is equal to the sine of the rotation angle of 10 π rad.

There is another approach to the definition of the sine, cosine, tangent and cotangent of a number. Let's consider it in more detail.

Anyone real number t a point on the unit circle is put in correspondence with the center at the origin of the rectangular Cartesian coordinate system. Sine, cosine, tangent and cotangent are defined in terms of 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 move if it moves counterclockwise around the circle and passes the path t .

Now that the connection between the number and the point on the circle has been established, we proceed to the definition of sine, cosine, tangent and cotangent.

Sine (sin) of the number t

Sine of a number t- ordinate of the point of 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 the point of the unit circle corresponding to the number t. t g t = y x = sin t cos t

The latter definitions are consistent with and do not contradict the definition given at the beginning of this section. Point on a circle corresponding to a number t, coincides with the point to which the starting point passes after turning through the angle t radian.

Trigonometric functions of angular and numerical 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) corresponds to a certain value of the tangent. The cotangent, as mentioned above, is defined for all α, except for 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, one can speak of sine, cosine, tangent and cotangent as functions of a numerical argument. Every real number t corresponds to a specific value of the sine or cosine of a number t. All numbers other than π 2 + π · k , k ∈ Z, correspond to the value of the tangent. The cotangent is similarly 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 data at the very beginning of the definitions and the angle alpha, which lies in the range from 0 to 90 degrees. Trigonometric definitions sine, cosine, tangent and cotangent are fully consistent with the geometric definitions given using the ratios of the sides of a right triangle. Let's show it.

Take a unit circle centered on a rectangular Cartesian coordinate system. Let's rotate the starting point A (1, 0) by an angle of up to 90 degrees and draw from the resulting point A 1 (x, y) perpendicular to the x-axis. 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 corner 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 the angle α is equal to the ratio of the opposite leg to the hypotenuse.

sin α \u003d A 1 H O A 1 \u003d y 1 \u003d y

This means that the definition of the sine of an acute angle in a right triangle through the aspect ratio is equivalent to the definition of the sine of the angle of rotation 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.

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I think you deserve more than that. Here is my key to trigonometry:

Draw the dome, wall and ceiling
Trigonometric functions is nothing but the percentage of these three forms.

Metaphor for sine and cosine: dome

Instead of just looking at the triangles themselves, imagine them in action by finding some particular real-life example.

Imagine that you are in the middle of a dome and want to hang up a movie projector screen. You point your finger at the dome at some "x" angle, and a screen should be hung from that point.

The angle you point to determines:

sine(x) = sin(x) = screen height (floor to dome mounting point)
cosine(x) = cos(x) = distance from you to the screen (by floor)
hypotenuse, the distance from you to the top of the screen, always the same, equal to the radius of the dome

Do you want the screen to be as big as possible? Hang it right above you.

Do you want the screen to hang to the maximum long distance from you? Hang it straight perpendicular. The screen will have a zero height in this position and will hang as far back as you requested.

The height and distance from the screen are inversely proportional: the closer the screen hangs, the higher its height will be.

Sine and cosine are percentages

No one in my years of study, alas, explained to me that the trigonometric functions sine and cosine are nothing but percentages. Their values range from +100% to 0 to -100%, or from a positive maximum to zero to a negative maximum.

Let's say I paid a tax of 14 rubles. You don't know how much it is. But if you say that I paid 95% in tax, you will understand that I was simply skinned like a sticky.

Absolute height means nothing. But if the sine value is 0.95, then I understand that the TV is hanging almost on top of your dome. Very soon it will reach its maximum height in the center of the dome, and then begin to decline again.

How can we calculate this percentage? Very simple: divide the current screen height by the maximum possible (the radius of the dome, also called the hypotenuse).

That's why we are told that “cosine = opposite leg / hypotenuse”. This is all in order to get a percentage! The best way to define the sine is “the percentage of the current height from the maximum possible”. (The sine becomes negative if your angle points "underground". The cosine becomes negative if the angle points to the dome point behind you.)

Let's simplify the calculations by assuming we are at the center of the unit circle (radius = 1). We can skip the division and just take the sine equal to the height.

Each circle, in fact, is a single, enlarged or reduced in scale to the desired size. So determine the relationships on the unit circle and apply the results to your particular circle size.

Experiment: take any corner and see what percentage of height to width it displays:

The graph of the growth of the value of the sine is not just a straight line. The first 45 degrees cover 70% of the height, and the last 10 degrees (from 80° to 90°) cover only 2%.

This will make it clearer to you: if you go in a circle, at 0 ° you rise almost vertically, but as you approach the top of the dome, the height changes less and less.

Tangent and secant. Wall

One day a neighbor built a wall right back to back to your dome. Cried your window view and good resale price!

But is it possible to somehow win in this situation?

Of course yes. What if we hang a movie screen right on the neighbor's wall? You aim at the corner (x) and get:

tan(x) = tan(x) = screen height on the wall
distance from you to the wall: 1 (this is the radius of your dome, the wall doesn't move anywhere from you, right?)
secant(x) = sec(x) = “length of ladder” from you standing in the center of the dome to the top of the suspended screen

Let's clarify a couple of things about the tangent, or screen height.

it starts at 0, and can go infinitely high. You can stretch the screen higher and higher on the wall to get just an endless canvas for watching your favorite movie! (For such a huge one, of course, you will have to spend a lot of money).
tangent is just an enlarged version of sine! And while the growth of the sine slows down as you move towards the top of the dome, the tangent continues to grow!

Sekansu also has something to brag about:

the secant starts at 1 (the ladder is on the floor, away from you towards the wall) and starts going up from there
The secant is always longer than the tangent. The sloped ladder you hang your screen with needs to be longer than the screen itself, right? (In unrealistic sizes, when the screen is sooooo long and the ladder needs to be placed almost vertically, their sizes are almost the same. But even then the secant will be a little longer).

Remember the values are percent. If you decide to hang the screen at a 50 degree angle, tan(50)=1.19. Your screen is 19% larger than the distance to the wall (dome radius).

(Enter x=0 and test your intuition - tan(0) = 0 and sec(0) = 1.)

Cotangent and cosecant. Ceiling

Incredibly, your neighbor has now decided to build a ceiling over your dome. (What's the matter with him? He apparently doesn't want you to peep on him while he walks around the yard naked...)

Well, it's time to build an exit to the roof and talk to the neighbor. You choose the angle of inclination, and start building:

the vertical distance between the roof outlet and the floor is always 1 (radius of the dome)
cotangent(x) = cot(x) = distance between dome top and exit point
cosecant(x) = csc(x) = length of your path to the roof

The tangent and secant describe the wall, while the cotangent and cosecant describe the floor.

Our intuitive conclusions this time are similar to the previous ones:

If you take an angle of 0°, your exit to the roof will take forever as it will never reach the ceiling. Problem.
The shortest “stairway” to the roof will be obtained if you build it at an angle of 90 degrees to the floor. The cotangent will be equal to 0 (we don’t move along the roof at all, we exit strictly perpendicularly), and the cosecant will be equal to 1 (“the length of the ladder” will be minimal).

Visualize Connections

If all three cases are drawn in a dome-wall-floor combination, the following will be obtained:

Well, wow, it's all the same triangle, enlarged in size to reach the wall and the ceiling. We have vertical sides (sine, tangent), horizontal sides (cosine, cotangent), and “hypotenuses” (secant, cosecant). (You can see from the arrows how far each element reaches. The cosecant is the total distance from you to the roof).

A little magic. All triangles share the same equalities:

From the Pythagorean theorem (a 2 + b 2 = c 2) we see how the sides of each triangle are connected. In addition, height-to-width ratios must also be the same for all triangles. (Just step back from the largest triangle to the smaller one. Yes, the size has changed, but the proportions of the sides will remain the same).

Knowing which side in each triangle is 1 (the radius of the dome), we can easily calculate that "sin/cos = tan/1".

I have always tried to remember these facts through simple visualization. In the picture you can clearly see these dependencies and understand where they come from. This technique is much better than memorizing dry formulas.

Don't Forget Other Angles

Shh… No need to get hung up on one graph, thinking that the tangent is always less than 1. If you increase the angle, you can reach the ceiling without reaching the wall:

Pythagorean connections always work, but the relative sizes can be different.

(You've probably noticed that the ratio of sine and cosine is always the smallest because they are enclosed within a dome.)

To summarize: what do we need to remember?

For most of us, I would say that this will be enough:

trigonometry explains the anatomy of mathematical objects such as circles and repeating intervals
the dome/wall/roof analogy shows the relationship between different trigonometric functions
the result of the trigonometric functions are the percentages that we apply to our scenario.

You don't need to memorize formulas like 1 2 + cot 2 = csc 2 . They are suitable only for stupid tests in which knowledge of a fact is presented as understanding it. Take a minute to draw a semicircle in the form of a dome, a wall and a roof, sign the elements, and all the formulas will be asked for you on paper.

Application: Inverse Functions

Any trigonometric function takes an angle as input and returns the result as a percentage. sin(30) = 0.5. This means that a 30 degree angle takes up 50% of the maximum height.

The inverse trigonometric function is written as sin -1 or arcsin (“arxine”). It is also common to write asin in various languages programming.

If our height is 25% of the dome's height, what is our angle?

In our table of proportions, you can find the ratio where the secant is divided by 1. For example, the secant by 1 (the hypotenuse to the horizontal) will be equal to 1 divided by the cosine:

Let's say our secant is 3.5, i.e. 350% of the unit circle radius. What angle of inclination to the wall does this value correspond to?

Appendix: Some examples

Example: Find the sine of angle x.

Boring task. Let's complicate the banal “find the sine” to “What is the height as a percentage of the maximum (hypotenuse)?”.

First, notice that the triangle is rotated. There is nothing wrong with this. The triangle also has a height, it is shown in green in the figure.

What is the hypotenuse equal to? By the Pythagorean theorem, we know that:

3 2 + 4 2 = hypotenuse 2 25 = hypotenuse 2 5 = hypotenuse

Fine! The sine is the percentage of the height from the longest side of the triangle, or the hypotenuse. In our example, the sine is 3/5 or 0.60.

Of course, we can go in several ways. Now we know that the sine is 0.60 and we can simply find the arcsine:

Asin(0.6)=36.9

And here is another approach. Note that the triangle is "face to face with the wall", so we can use tangent instead of sine. The height is 3, the distance to the wall is 4, so the tangent is ¾ or 75%. We can use the arc tangent to go from percentage back to angle:

Tan = 3/4 = 0.75 atan(0.75) = 36.9 Example: Will you swim to shore?

You are in a boat and you have enough fuel to sail 2 km. You are now 0.25 km from the coast. At what maximum angle to the shore can you swim to it so that you have enough fuel? Addition to the condition of the problem: we only have a table of arc cosine values.

What we have? The coastline can be represented as a “wall” in our famous triangle, and the “length of the stairs” attached to the wall can be represented as the maximum possible distance by boat to the shore (2 km). A secant emerges.

First, you need to switch to percentages. We have 2 / 0.25 = 8, which means we can swim 8 times the straight distance to the shore (or to the wall).

The question arises “What is the secant 8?”. But we cannot give an answer to it, since we only have arc cosines.

We use our previously derived dependencies to map the secant to the cosine: “sec/1 = 1/cos”

The secant of 8 is equal to the cosine of ⅛. An angle whose cosine is ⅛ is acos(1/8) = 82.8. And this is the largest angle that we can afford on a boat with the specified amount of fuel.

Not bad, right? Without the dome-wall-ceiling analogy, I would be confused in a bunch of formulas and calculations. Visualization of the problem greatly simplifies the search for a solution, besides, it is interesting to see which trigonometric function will eventually help.

For each task, think like this: am I interested in a dome (sin/cos), a wall (tan/sec), or a ceiling (cot/csc)?

And trigonometry will become much more pleasant. Easy calculations for you!

Trigonometry is a branch of mathematics that studies trigonometric functions, as well as their use in practice. These features include sinus, cosine, tangent and cotangent.

Sine is a trigonometric function, the ratio of the magnitude of the opposite leg to the magnitude of the hypotenuse.

Sine in trigonometry.

As mentioned above, the sine is directly related to trigonometry and trigonometric functions. Its function is determined by

help to calculate the angle, provided that the dimensions of the sides of the triangle are known;
help to calculate the size of the side of the triangle, provided that the angle is known.

It must be remembered that the value of the sine will always be the same for any size of the triangle, since the sine is not a measurement, but a ratio.

Consequently, in order not to calculate this constant value for each solution of a particular problem, special trigonometric tables were created. In them, the values of sines, cosines, tangents and cotangents have already been calculated and fixed. Usually these tables are given on the flyleaf of textbooks on algebra and geometry. They can also be found on the Internet.

Sine in geometry.

Geometry requires visualization, therefore, in order to understand in practice, what is the sine of an angle, you need to draw a triangle with a right angle.

Let us assume that the sides forming a right angle are named a, c, the opposite angle X.

Usually the length of the sides is indicated in the tasks. Let's say a=3, b=4. In this case, the aspect ratio will look like ¾. Moreover, if we lengthen the sides of the triangle adjacent to the acute angle X, then the sides will increase A And V, and the hypotenuse is the third side of a right triangle that is not at right angles to the base. Now the sides of the triangle can be called differently, for example: m, n, k.

With this modification, the law of trigonometry worked: the lengths of the sides of the triangle changed, but their ratio did not.

The fact that if you change the length of the sides of a triangle as many times as you like and while maintaining the value of the angle x, the ratio between its sides will still remain unchanged, ancient scientists noticed. In our case, the length of the sides could change like this: a / b \u003d ¾, when the side is lengthened A up to 6 cm, and V- up to 8 cm we get: m/n = 6/8 = 3/4.

The ratios of the sides in a right-angled triangle in this regard are called:

the sine of the angle x is the ratio of the opposite leg to the hypotenuse: sinx = a/c;
the cosine of the angle x is the ratio of the adjacent leg to the hypotenuse: cosx = w/s;
the tangent of the angle x is the ratio of the opposite leg to the adjacent one: tgx \u003d a / b;
the cotangent of the angle x is the ratio of the adjacent leg to the opposite one: ctgx \u003d in / a.

What is the sine, cosine, tangent, cotangent of an angle will help you understand a right triangle.

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 \ (AC \) ); the legs are the two remaining sides \ (AB \) and \ (BC \) (those that are adjacent to right angle), moreover, if we consider the legs with respect to the angle \ (BC \) , then the leg \ (AB \) is the adjacent leg, and the leg \ (BC \) 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- this is the ratio of the opposite (far) leg to the hypotenuse.

In our triangle:

\[ \sin \beta =\dfrac(BC)(AC) \]

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

In our triangle:

\[ \cos \beta =\dfrac(AB)(AC) \]

Angle tangent- this is the ratio of the opposite (far) leg to the adjacent (close).

In our triangle:

\[ tg\beta =\dfrac(BC)(AB) \]

Cotangent of an angle- this is the ratio of the adjacent (close) leg to the opposite (far).

In our triangle:

\[ ctg\beta =\dfrac(AB)(BC) \]

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 believe? Then make sure by looking at the picture:

Consider, for example, the cosine of the angle \(\beta \) . By definition, from a triangle \(ABC \) : \(\cos \beta =\dfrac(AB)(AC)=\dfrac(4)(6)=\dfrac(2)(3) \), but we can calculate the cosine of the angle \(\beta \) from the triangle \(AHI \) : \(\cos \beta =\dfrac(AH)(AI)=\dfrac(6)(9)=\dfrac(2)(3) \). 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 \(ABC \) , shown in the figure below, we find \(\sin \ \alpha ,\ \cos \ \alpha ,\ tg\ \alpha ,\ ctg\ \alpha \).

\(\begin(array)(l)\sin \ \alpha =\dfrac(4)(5)=0.8\\\cos \ \alpha =\dfrac(3)(5)=0.6\\ tg\ \alpha =\dfrac(4)(3)\\ctg\ \alpha =\dfrac(3)(4)=0.75\end(array) \)

Well, did you get it? Then try it yourself: calculate the same for the angle \(\beta \) .

Answers: \(\sin \ \beta =0.6;\ \cos \ \beta =0.8;\ tg\ \beta =0.75;\ ctg\ \beta =\dfrac(4)(3) \).

Unit (trigonometric) circle

Understanding the concepts of degree and radian, we considered a circle with a radius equal to \ (1 \) . 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 the Cartesian coordinate system. 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 \(x \) axis (in our example, this is the radius \(AB \) ).

Each point on the circle corresponds to two numbers: the coordinate along the axis \(x \) and the coordinate along the axis \(y \) . 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 the triangle \(ACG \) . It's rectangular because \(CG \) is perpendicular to the \(x \) axis.

What is \(\cos \ \alpha \) from the triangle \(ACG \) ? That's right \(\cos \ \alpha =\dfrac(AG)(AC) \). Besides, we know that \(AC \) is the radius of the unit circle, so \(AC=1 \) . Substitute this value into our cosine formula. Here's what happens:

\(\cos \ \alpha =\dfrac(AG)(AC)=\dfrac(AG)(1)=AG \).

And what is \(\sin \ \alpha \) from the triangle \(ACG \) ? Well, of course, \(\sin \alpha =\dfrac(CG)(AC) \)! Substitute the value of the radius \ (AC \) in this formula and get:

\(\sin \alpha =\dfrac(CG)(AC)=\dfrac(CG)(1)=CG \)

So, can you tell me what are the coordinates of the point \(C \) , which belongs to the circle? Well, no way? But what if you realize that \(\cos \ \alpha \) and \(\sin \alpha \) are just numbers? What coordinate does \(\cos \alpha \) correspond to? Well, of course, the coordinate \(x \) ! And what coordinate does \(\sin \alpha \) correspond to? That's right, the \(y \) coordinate! So the point \(C(x;y)=C(\cos \alpha ;\sin \alpha) \).

What then are \(tg \alpha \) and \(ctg \alpha \) ? That's right, let's use the appropriate definitions of tangent and cotangent and get that \(tg \alpha =\dfrac(\sin \alpha )(\cos \alpha )=\dfrac(y)(x) \), A \(ctg \alpha =\dfrac(\cos \alpha )(\sin \alpha )=\dfrac(x)(y) \).

What if the angle is larger? Here, for example, as in this picture:

What has changed in this example? Let's figure it out. To do this, we again turn to a right-angled triangle. Consider a right triangle \(((A)_(1))((C)_(1))G \) : an angle (as adjacent to the angle \(\beta \) ). What is the value of sine, cosine, tangent and cotangent for an angle \(((C)_(1))((A)_(1))G=180()^\circ -\beta \ \)? That's right, we adhere to the corresponding definitions of trigonometric functions:

\(\begin(array)(l)\sin \angle ((C)_(1))((A)_(1))G=\dfrac(((C)_(1))G)(( (A)_(1))((C)_(1)))=\dfrac(((C)_(1))G)(1)=((C)_(1))G=y; \\\cos \angle ((C)_(1))((A)_(1))G=\dfrac(((A)_(1))G)(((A)_(1)) ((C)_(1)))=\dfrac(((A)_(1))G)(1)=((A)_(1))G=x;\\tg\angle ((C )_(1))((A)_(1))G=\dfrac(((C)_(1))G)(((A)_(1))G)=\dfrac(y)( x);\\ctg\angle ((C)_(1))((A)_(1))G=\dfrac(((A)_(1))G)(((C)_(1 ))G)=\dfrac(x)(y)\end(array) \)

Well, as you can see, the value of the sine of the angle still corresponds to the coordinate \ (y \) ; the value of the cosine of the angle - the coordinate \ (x \) ; 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 \(x \) 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 the whole revolution of the radius vector around the circle is \(360()^\circ \) or \(2\pi \) . Is it possible to rotate the radius vector by \(390()^\circ \) or by \(-1140()^\circ \) ? Well, of course you can! In the first case, \(390()^\circ =360()^\circ +30()^\circ \), so the radius vector will make one full rotation and stop at \(30()^\circ \) or \(\dfrac(\pi )(6) \) .

In the second case, \(-1140()^\circ =-360()^\circ \cdot 3-60()^\circ \), that is, the radius vector will make three complete revolutions and stop at the position \(-60()^\circ \) or \(-\dfrac(\pi )(3) \) .

Thus, from the above examples, we can conclude that angles that differ by \(360()^\circ \cdot m \) or \(2\pi \cdot m \) (where \(m \) is any integer ) correspond to the same position of the radius vector.

The figure below shows the angle \(\beta =-60()^\circ \) . The same image corresponds to the corner \(-420()^\circ ,-780()^\circ ,\ 300()^\circ ,660()^\circ \) etc. This list can be continued indefinitely. All these angles can be written with the general formula \(\beta +360()^\circ \cdot m\) or \(\beta +2\pi \cdot m \) (where \(m \) is any integer)

\(\begin(array)(l)-420()^\circ =-60+360\cdot (-1);\\-780()^\circ =-60+360\cdot (-2); \\300()^\circ =-60+360\cdot 1;\\660()^\circ =-60+360\cdot 2.\end(array) \)

Now, knowing the definitions of the basic trigonometric functions and using the unit circle, try to answer what the values \u200b\u200bare equal to:

\(\begin(array)(l)\sin \ 90()^\circ =?\\\cos \ 90()^\circ =?\\\text(tg)\ 90()^\circ =? \\\text(ctg)\ 90()^\circ =?\\\sin \ 180()^\circ =\sin \ \pi =?\\\cos \ 180()^\circ =\cos \ \pi =?\\\text(tg)\ 180()^\circ =\text(tg)\ \pi =?\\\text(ctg)\ 180()^\circ =\text(ctg)\ \pi =?\\\sin \ 270()^\circ =?\\\cos \ 270()^\circ =?\\\text(tg)\ 270()^\circ =?\\\text (ctg)\ 270()^\circ =?\\\sin \ 360()^\circ =?\\\cos \ 360()^\circ =?\\\text(tg)\ 360()^ \circ =?\\\text(ctg)\ 360()^\circ =?\\\sin \ 450()^\circ =?\\\cos \ 450()^\circ =?\\\text (tg)\ 450()^\circ =?\\\text(ctg)\ 450()^\circ =?\end(array) \)

Here's a unit circle to help you:

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

\(\begin(array)(l)\sin \alpha =y;\\cos\alpha =x;\\tg\alpha =\dfrac(y)(x);\\ctg\alpha =\dfrac(x )(y).\end(array) \)

From here, we determine the coordinates of the points corresponding to certain measures of the angle. Well, let's start in order: the corner in \(90()^\circ =\dfrac(\pi )(2) \) corresponds to a point with coordinates \(\left(0;1 \right) \) , therefore:

\(\sin 90()^\circ =y=1 \) ;

\(\cos 90()^\circ =x=0 \) ;

\(\text(tg)\ 90()^\circ =\dfrac(y)(x)=\dfrac(1)(0)\Rightarrow \text(tg)\ 90()^\circ \)- does not exist;

\(\text(ctg)\ 90()^\circ =\dfrac(x)(y)=\dfrac(0)(1)=0 \).

Further, adhering to the same logic, we find out that the corners in \(180()^\circ ,\ 270()^\circ ,\ 360()^\circ ,\ 450()^\circ (=360()^\circ +90()^\circ)\ \ ) correspond to points with coordinates \(\left(-1;0 \right),\text( )\left(0;-1 \right),\text( )\left(1;0 \right),\text( )\left(0 ;1 \right) \), 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:

\(\displaystyle \sin \ 180()^\circ =\sin \ \pi =0 \)

\(\displaystyle \cos \ 180()^\circ =\cos \ \pi =-1 \)

\(\text(tg)\ 180()^\circ =\text(tg)\ \pi =\dfrac(0)(-1)=0 \)

\(\text(ctg)\ 180()^\circ =\text(ctg)\ \pi =\dfrac(-1)(0)\Rightarrow \text(ctg)\ \pi \)- does not exist

\(\sin \ 270()^\circ =-1 \)

\(\cos \ 270()^\circ =0 \)

\(\text(tg)\ 270()^\circ =\dfrac(-1)(0)\Rightarrow \text(tg)\ 270()^\circ \)- does not exist

\(\text(ctg)\ 270()^\circ =\dfrac(0)(-1)=0 \)

\(\sin \ 360()^\circ =0 \)

\(\cos \ 360()^\circ =1 \)

\(\text(tg)\ 360()^\circ =\dfrac(0)(1)=0 \)

\(\text(ctg)\ 360()^\circ =\dfrac(1)(0)\Rightarrow \text(ctg)\ 2\pi \)- does not exist

\(\sin \ 450()^\circ =\sin \ \left(360()^\circ +90()^\circ \right)=\sin \ 90()^\circ =1 \)

\(\cos \ 450()^\circ =\cos \ \left(360()^\circ +90()^\circ \right)=\cos \ 90()^\circ =0 \)

\(\text(tg)\ 450()^\circ =\text(tg)\ \left(360()^\circ +90()^\circ \right)=\text(tg)\ 90() ^\circ =\dfrac(1)(0)\Rightarrow \text(tg)\ 450()^\circ \)- does not exist

\(\text(ctg)\ 450()^\circ =\text(ctg)\left(360()^\circ +90()^\circ \right)=\text(ctg)\ 90()^ \circ =\dfrac(0)(1)=0 \).

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:

\(\left. \begin(array)(l)\sin \alpha =y;\\cos \alpha =x;\\tg \alpha =\dfrac(y)(x);\\ctg \alpha =\ dfrac(x)(y).\end(array) \right\)\ \text(Need to remember or be able to output!! \) !}

And here are the values of the trigonometric functions of the angles in and \(30()^\circ =\dfrac(\pi )(6),\ 45()^\circ =\dfrac(\pi )(4) \) given in the table below, you must remember:

No need to be scared, now we will show one of the examples of a fairly simple memorization of the corresponding values:

To use this method, it is vital to remember the sine values \u200b\u200bfor all three angle measures ( \(30()^\circ =\dfrac(\pi )(6),\ 45()^\circ =\dfrac(\pi )(4),\ 60()^\circ =\dfrac(\pi )(3) \)), as well as the value of the tangent of the angle in \(30()^\circ \) . Knowing these \(4\) values, it is quite easy to restore the entire table - the cosine values are transferred in accordance with the arrows, that is:

\(\begin(array)(l)\sin 30()^\circ =\cos \ 60()^\circ =\dfrac(1)(2)\ \ \\\sin 45()^\circ = \cos \ 45()^\circ =\dfrac(\sqrt(2))(2)\\\sin 60()^\circ =\cos \ 30()^\circ =\dfrac(\sqrt(3 ))(2)\ \end(array) \)

\(\text(tg)\ 30()^\circ \ =\dfrac(1)(\sqrt(3)) \), knowing this, it is possible to restore the values for \(\text(tg)\ 45()^\circ , \text(tg)\ 60()^\circ \). The numerator “\(1 \) ” will match \(\text(tg)\ 45()^\circ \ \) , and the denominator “\(\sqrt(\text(3)) \) ” will match \(\text (tg)\ 60()^\circ \ \) . Cotangent values are transferred in accordance with the arrows shown in the figure. If you understand this and remember the scheme with arrows, then it will be enough to remember only \(4 \) 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 bring out general formula to find the coordinates of a point. Here, for example, we have such a circle:

We are given that point \(K(((x)_(0));((y)_(0)))=K(3;2) \) is the center of the circle. The radius of the circle is \(1,5 \) . It is necessary to find the coordinates of the point \(P \) obtained by rotating the point \(O \) by \(\delta \) degrees.

As can be seen from the figure, the coordinate \ (x \) of the point \ (P \) corresponds to the length of the segment \ (TP=UQ=UK+KQ \) . The length of the segment \ (UK \) corresponds to the coordinate \ (x \) of the center of the circle, that is, it is equal to \ (3 \) . The length of the segment \(KQ \) can be expressed using the definition of cosine:

\(\cos \ \delta =\dfrac(KQ)(KP)=\dfrac(KQ)(r)\Rightarrow KQ=r\cdot \cos \ \delta \).

Then we have that for the point \(P \) the coordinate \(x=((x)_(0))+r\cdot \cos \ \delta =3+1,5\cdot \cos \ \delta \).

By the same logic, we find the value of the y coordinate for the point \(P\) . Thus,

\(y=((y)_(0))+r\cdot \sin \ \delta =2+1,5\cdot \sin \delta \).

So in general view point coordinates are determined by the formulas:

\(\begin(array)(l)x=((x)_(0))+r\cdot \cos \ \delta \\y=((y)_(0))+r\cdot \sin \ \delta \end(array) \), Where

\(((x)_(0)),((y)_(0)) \) - coordinates of the center of the circle,

\(r\) - circle radius,

\(\delta \) - 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 zero, and the radius is equal to one:

\(\begin(array)(l)x=((x)_(0))+r\cdot \cos \ \delta =0+1\cdot \cos \ \delta =\cos \ \delta \\y =((y)_(0))+r\cdot \sin \ \delta =0+1\cdot \sin \ \delta =\sin \ \delta \end(array) \)

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