The meaning of the tangent of an angle. Sine, cosine, tangent: what is it? How to find sine, cosine and tangent? Sine in trigonometry

Examples:

\(\cos(⁡30^°)=\)\(\frac(\sqrt(3))(2)\)
\(\cos⁡\)\(\frac(π)(3)\) \(=\)\(\frac(1)(2)\)
\(\cos⁡2=-0.416…\)

Argument and value

Cosine of an acute angle

Cosine of an acute angle can be determined using a right triangle - it is equal to the ratio of the adjacent leg to the hypotenuse.

Example :

1) Let an angle be given and you need to determine the cosine of this angle.

2) Let's complete any right-angled triangle on this corner.

3) Having measured the necessary sides, we can calculate the cosine.

The cosine of an acute angle is greater than \(0\) and less than \(1\)

If, when solving the problem, the cosine of an acute angle turned out to be greater than 1 or negative, then somewhere in the solution there is an error.

Cosine of a number

The number circle allows you to determine the cosine of any number, but usually find the cosine of numbers somehow related to : \(\frac(π)(2)\) , \(\frac(3π)(4)\) , \(-2π\ ).

For example, for the number \(\frac(π)(6)\) - the cosine will be equal to \(\frac(\sqrt(3))(2)\) . And for the number \(-\)\(\frac(3π)(4)\) it will be equal to \(-\)\(\frac(\sqrt(2))(2)\) (approximately \(-0 ,71\)).

Cosine for other numbers often encountered in practice, see.

The cosine value always lies between \(-1\) and \(1\). In this case, the cosine can be calculated for absolutely any angle and number.

Cosine of any angle

Thanks to the numerical circle, it is possible to determine the cosine of not only an acute angle, but also an obtuse, negative, and even greater than \ (360 ° \) (full turn). How to do it - it's easier to see once than to hear \(100\) times, so look at the picture.

Now an explanation: let it be necessary to determine the cosine of the angle KOA with degree measure in \(150°\). We combine the point O with the center of the circle, and the side OK- with the \(x\) axis. After that, set aside \ (150 ° \) counterclockwise. Then the ordinate of the point BUT will show us the cosine of this angle.

If we are interested in an angle with a degree measure, for example, in \ (-60 ° \) (angle KOV), we do the same, but \(60°\) set aside clockwise.

And finally, the angle is greater than \(360°\) (the angle KOS) - everything is similar to blunt, only after passing a full turn clockwise, we go to the second round and “get the lack of degrees”. Specifically, in our case, the angle \(405°\) is plotted as \(360° + 45°\).

It is easy to guess that to set aside an angle, for example, in \ (960 ° \), you need to make two turns (\ (360 ° + 360 ° + 240 ° \)), and for an angle in \ (2640 ° \) - whole seven.

It is worth remembering that:

The cosine of a right angle is zero. The cosine of an obtuse angle is negative.

Cosine signs in quarters

Using the cosine axis (that is, the abscissa axis, highlighted in red in the figure), it is easy to determine the signs of the cosines along a numerical (trigonometric) circle:

Where the values ​​on the axis are from \(0\) to \(1\), the cosine will have a plus sign (I and IV quarters are the green area),
- where the values ​​on the axis are from \(0\) to \(-1\), the cosine will have a minus sign (II and III quarters - purple area).

Example. Define the sign \(\cos 1\).
Solution: Let's find \(1\) on the trigonometric circle. We will start from the fact that \ (π \u003d 3,14 \). This means that one is approximately three times closer to zero (the "start" point).

If we draw a perpendicular to the cosine axis, it becomes obvious that \(\cos⁡1\) is positive.
Answer: a plus.

Relation to other trigonometric functions:

Function \(y=\cos(x)\)

If we plot the angles in radians along the \(x\) axis, and the cosine values ​​\u200b\u200bcorresponding to these angles along the \(y\) axis, we get the following graph:

This graph is called and has the following properties:

The domain of definition is any value of x: \(D(\cos(⁡x))=R\)
- range of values ​​- from \(-1\) to \(1\) inclusive: \(E(\cos(x))=[-1;1]\)
- even: \(\cos⁡(-x)=\cos(x)\)
- periodic with period \(2π\): \(\cos⁡(x+2π)=\cos(x)\)
- points of intersection with the coordinate axes:
abscissa: \((\)\(\frac(π)(2)\) \(+πn\),\(;0)\), where \(n ϵ Z\)
y-axis: \((0;1)\)
- character intervals:
the function is positive on the intervals: \((-\)\(\frac(π)(2)\) \(+2πn;\) \(\frac(π)(2)\) \(+2πn)\), where \(n ϵ Z\)
the function is negative on the intervals: \((\)\(\frac(π)(2)\) \(+2πn;\)\(\frac(3π)(2)\) \(+2πn)\), where \(n ϵ Z\)
- intervals of increase and decrease:
the function increases on the intervals: \((π+2πn;2π+2πn)\), where \(n ϵ Z\)
the function decreases on the intervals: \((2πn;π+2πn)\), where \(n ϵ Z\)
- maxima and minima of the function:
the function has a maximum value \(y=1\) at the points \(x=2πn\), where \(n ϵ Z\)
the function has a minimum value \(y=-1\) at the points \(x=π+2πn\), where \(n ϵ Z\).

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 the right angle), moreover, if we consider the legs with respect to the angle \ (BC \) , then the leg \ (AB \) is adjacent leg, and the leg \ (BC \) is opposite. 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 trust? 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 derive a general formula for finding 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 \(P\) . In this way,

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

So, in general terms, the coordinates of points 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:

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

Cosine is a well-known trigonometric function, which is also one of the main functions of trigonometry. The cosine of an angle in a right angled triangle is the ratio of the triangle's adjacent leg to the triangle's hypotenuse. Most often, the definition of cosine is associated with a triangle of exactly a rectangular type. But it also happens that the angle for which it is necessary to calculate the cosine in a triangle of a rectangular type is not located in this very triangle of a rectangular type. What then to do? How to find the cosine of the angle of a triangle?

If you want to calculate the cosine of an angle in a right-angled triangle, then everything is very simple. You just need to remember the definition of cosine, in which lies the solution to this problem. You just need to find the same ratio between the adjacent leg, as well as the hypotenuse of the triangle. Indeed, here it is not difficult to express the cosine of an angle. The formula looks like this: - cosα = a/c, here "a" is the length of the leg, and side "c", respectively, is the length of the hypotenuse. For example, the cosine of an acute angle of a right triangle can be found using this formula.

If you are interested in what the cosine of an angle in an arbitrary triangle is equal to, then the cosine theorem comes to the rescue, which should be used in such cases. The cosine theorem states that the square of a side of a triangle is a priori equal to the sum of the squares of the other sides of the same triangle, but without twice the product of these sides by the cosine of the angle between them.

1. If you need to find the cosine of an acute angle in a triangle, then you need to use the following formula: cosα \u003d (a 2 + b 2 - c 2) / (2ab).
2. If in a triangle it is necessary to find the cosine of an obtuse angle, then you need to use the following formula: cosα \u003d (c 2 - a 2 - b 2) / (2ab). The designations in the formula - a and b - are the lengths of the sides that are adjacent to the desired angle, c is the length of the side that is opposite the desired angle.

Also, the cosine of an angle can be calculated using the sine theorem. It says that all sides of a triangle are proportional to the sines of the angles that are opposite. Using the sine theorem, you can calculate the remaining elements of a triangle, knowing only two sides and an angle that is opposite one side, or two angles and one side. Consider an example. Problem conditions: a=1; b=2; c=3. The angle that is opposite to side "A", we denote - α, then, according to the formulas, we have: cosα \u003d (b² + c²-a²) / (2 * b * c) \u003d (2² + 3² -1²) / (2 * 2 *3)=(4+9-1)/12=12/12=1. Answer: 1.

If the cosine of the angle needs to be calculated not in a triangle, but in some other arbitrary geometric figure, then everything becomes a little more complicated. The value of the angle must first be determined in radians or degrees, and only then calculate the cosine from this value. Cosine by numerical value is determined using Bradis tables, engineering calculators, or special mathematical applications.

Special mathematical applications may have functions such as automatic calculation of the cosines of angles in a given figure. The beauty of such applications is that they give the right answer, and the user does not spend his time on solving sometimes quite complex problems. On the other hand, with the constant use of exclusively applications for solving problems, all skills for working with solving mathematical problems for finding the cosines of angles in triangles, as well as other arbitrary figures, are lost.

One of the branches of mathematics with which schoolchildren cope with the greatest difficulties is trigonometry. No wonder: in order to freely master this area of ​​knowledge, you need spatial thinking, the ability to find sines, cosines, tangents, cotangents using formulas, simplify expressions, and be able to use the number pi in calculations. In addition, you need to be able to apply trigonometry when proving theorems, and this requires either a developed mathematical memory or the ability to deduce complex logical chains.

Origins of trigonometry

Acquaintance with this science should begin with the definition of the sine, cosine and tangent of the angle, but first you need to figure out what trigonometry does in general.

Historically, right triangles have been the main object of study in this section of mathematical science. The presence of an angle of 90 degrees makes it possible to carry out various operations that allow one to determine the values ​​of all parameters of the figure under consideration using two sides and one angle or two angles and one side. In the past, people noticed this pattern and began to actively use it in the construction of buildings, navigation, astronomy, and even art.

First stage

Initially, people talked about the relationship of angles and sides exclusively on the example of right triangles. Then special formulas were discovered that made it possible to expand the boundaries of use in everyday life of this section of mathematics.

The study of trigonometry at school today begins with right-angled triangles, after which the acquired knowledge is used by students in physics and solving abstract trigonometric equations, work with which begins in high school.

Spherical trigonometry

Later, when science reached the next level of development, formulas with sine, cosine, tangent, cotangent began to be used in spherical geometry, where other rules apply, and the sum of the angles in a triangle is always more than 180 degrees. This section is not studied at school, but it is necessary to know about its existence, at least because the earth's surface, and the surface of any other planet, is convex, which means that any surface marking will be "arc-shaped" in three-dimensional space.

Take the globe and thread. Attach the thread to any two points on the globe so that it is taut. Pay attention - it has acquired the shape of an arc. It is with such forms that spherical geometry, which is used in geodesy, astronomy, and other theoretical and applied fields, deals.

Right triangle

Having learned a little about the ways of using trigonometry, let's return to basic trigonometry in order to further understand what sine, cosine, tangent are, what calculations can be performed with their help and what formulas to use.

The first step is to understand the concepts related to a right triangle. First, the hypotenuse is the side opposite the 90 degree angle. She is the longest. We remember that, according to the Pythagorean theorem, its numerical value is equal to the root of the sum of the squares of the other two sides.

For example, if two sides are 3 and 4 centimeters respectively, the length of the hypotenuse will be 5 centimeters. By the way, the ancient Egyptians knew about this about four and a half thousand years ago.

The two remaining sides that form a right angle are called legs. In addition, we must remember that the sum of the angles in a triangle in a rectangular coordinate system is 180 degrees.

Definition

Finally, with a solid understanding of the geometric base, we can turn to the definition of the sine, cosine and tangent of an angle.

The sine of an angle is the ratio of the opposite leg (i.e., the side opposite the desired angle) to the hypotenuse. The cosine of an angle is the ratio of the adjacent leg to the hypotenuse.

Remember that neither sine nor cosine can be greater than one! Why? Because the hypotenuse is by default the longest. No matter how long the leg is, it will be shorter than the hypotenuse, which means that their ratio will always be less than one. Thus, if you get a sine or cosine with a value greater than 1 in the answer to the problem, look for an error in calculations or reasoning. This answer is clearly wrong.

Finally, the tangent of an angle is the ratio of the opposite side to the adjacent side. The same result will give the division of the sine by the cosine. Look: in accordance with the formula, we divide the length of the side by the hypotenuse, after which we divide by the length of the second side and multiply by the hypotenuse. Thus, we get the same ratio as in the definition of tangent.

The cotangent, respectively, is the ratio of the side adjacent to the corner to the opposite side. We get the same result by dividing the unit by the tangent.

So, we have considered the definitions of what sine, cosine, tangent and cotangent are, and we can deal with formulas.

The simplest formulas

In trigonometry, one cannot do without formulas - how to find sine, cosine, tangent, cotangent without them? And this is exactly what is required when solving problems.

The first formula that you need to know when starting to study trigonometry says that the sum of the squares of the sine and cosine of an angle is equal to one. This formula is a direct consequence of the Pythagorean theorem, but it saves time if you want to know the value of the angle, not the side.

Many students cannot remember the second formula, which is also very popular when solving school problems: the sum of one and the square of the tangent of an angle is equal to one divided by the square of the cosine of the angle. Take a closer look: after all, this is the same statement as in the first formula, only both sides of the identity were divided by the square of the cosine. It turns out that a simple mathematical operation makes the trigonometric formula completely unrecognizable. Remember: knowing what sine, cosine, tangent and cotangent are, the conversion rules and a few basic formulas, you can at any time independently derive the required more complex formulas on a sheet of paper.

Double angle formulas and addition of arguments

Two more formulas that you need to learn are related to the values ​​\u200b\u200bof the sine and cosine for the sum and difference of the angles. They are shown in the figure below. Please note that in the first case, the sine and cosine are multiplied both times, and in the second, the pairwise product of the sine and cosine is added.

There are also formulas associated with double angle arguments. They are completely derived from the previous ones - as a practice, try to get them yourself, taking the angle of alpha equal to the angle of beta.

Finally, note that the double angle formulas can be converted to lower the degree of sine, cosine, tangent alpha.

Theorems

The two main theorems in basic trigonometry are the sine theorem and the cosine theorem. With the help of these theorems, you can easily understand how to find the sine, cosine and tangent, and therefore the area of ​​\u200b\u200bthe figure, and the size of each side, etc.

The sine theorem states that as a result of dividing the length of each of the sides of the triangle by the value of the opposite angle, we get the same number. Moreover, this number will be equal to two radii of the circumscribed circle, that is, the circle containing all points of the given triangle.

The cosine theorem generalizes the Pythagorean theorem, projecting it onto any triangles. It turns out that from the sum of the squares of the two sides, subtract their product, multiplied by the double cosine of the angle adjacent to them - the resulting value will be equal to the square of the third side. Thus, the Pythagorean theorem turns out to be a special case of the cosine theorem.

Mistakes due to inattention

Even knowing what sine, cosine and tangent are, it is easy to make a mistake due to absent-mindedness or an error in the simplest calculations. To avoid such mistakes, let's get acquainted with the most popular of them.

First, you should not convert ordinary fractions to decimals until the final result is obtained - you can leave the answer as an ordinary fraction, unless the condition states otherwise. Such a transformation cannot be called a mistake, but it should be remembered that at each stage of the task, new roots may appear, which, according to the author's idea, should be reduced. In this case, you will waste time on unnecessary mathematical operations. This is especially true for values ​​such as the root of three or two, because they occur in tasks at every step. The same applies to rounding "ugly" numbers.

Further, note that the cosine theorem applies to any triangle, but not the Pythagorean theorem! If you mistakenly forget to subtract twice the product of the sides multiplied by the cosine of the angle between them, you will not only get a completely wrong result, but also demonstrate a complete misunderstanding of the subject. This is worse than a careless mistake.

Thirdly, do not confuse the values ​​​​for angles of 30 and 60 degrees for sines, cosines, tangents, cotangents. Remember these values, because the sine of 30 degrees is equal to the cosine of 60, and vice versa. It is easy to mix them up, as a result of which you will inevitably get an erroneous result.

Application

Many students are in no hurry to start studying trigonometry, because they do not understand its applied meaning. What is sine, cosine, tangent for an engineer or astronomer? These are concepts thanks to which you can calculate the distance to distant stars, predict the fall of a meteorite, send a research probe to another planet. Without them, it is impossible to build a building, design a car, calculate the load on the surface or the trajectory of an object. And these are just the most obvious examples! After all, trigonometry in one form or another is used everywhere, from music to medicine.

Finally

So you are sine, cosine, tangent. You can use them in calculations and successfully solve school problems.

The whole essence of trigonometry boils down to the fact that unknown parameters must be calculated from the known parameters of the triangle. There are six parameters in total: the lengths of three sides and the magnitudes of three angles. The whole difference in the tasks lies in the fact that different input data are given.

How to find the sine, cosine, tangent based on the known lengths of the legs or the hypotenuse, you now know. Since these terms mean nothing more than a ratio, and a ratio is a fraction, the main goal of the trigonometric problem is to find the roots of an ordinary equation or a system of equations. And here you will be helped by ordinary school mathematics.

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 a right triangle and an angle in a circle are, and why all 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 origin point, 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:

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

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

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

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

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

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

1. ctg x = 0, x = π/2 + πk.
2. 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:

1. sin(α ± β) = sin α * cos β ± cos α * sin.
2. cos(α ± β) = cos α * cos β ∓ sin α * sin.
3. tan(α ± β) = (tan α ± tan β) / (1 ∓ tan α * tan β).
4. 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:

1. sin2α = 2sinα*cosα.
2. cos2α = 1 - 2sin^2α.
3. tg2α = 2tgα / (1 - tg^2 α).
4. sin3α = 3sinα - 4sin^3α.
5. cos3α = 4cos^3α - 3cosα.
6. 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

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

Private for sine:

Cosine quotients:

Private for tangent:

Cotangent quotients:

Theorems

Sine theorem

There are two versions of the theorem - simple and extended. Simple sine theorem: 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.