Where the problems to solve were considered right triangle, I promised to outline a technique for memorizing the definitions of sine and cosine. Using it, you will always quickly remember which side belongs to the hypotenuse (adjacent or opposite). I decided not to put it off for too long, required material below, please read 😉

The fact is that I have repeatedly observed how students in grades 10-11 have difficulty remembering these definitions. They remember very well that the leg refers to the hypotenuse, but which one- they forget and confused. The price of a mistake, as you know in an exam, is a lost point.

The information I will present directly has nothing to do with mathematics. She is connected with imaginative thinking, and with methods of verbal-logical communication. That's exactly how I remember it, once and for alldefinition data. If you do forget them, you can always easily remember them using the techniques presented.

Let me remind you of the definitions of sine and cosine in a right triangle:

Cosine acute angle in a right triangle, this is the ratio of the adjacent leg to the hypotenuse:

Sinus The acute angle in a right triangle is the ratio of the opposite side to the hypotenuse:

So, what associations do you have with the word cosine?

Probably everyone has their own 😉Remember the link:

Thus, the expression will immediately appear in your memory -

«… ratio of the ADJACENT leg to the hypotenuse».

The problem with determining cosine has been solved.

If you need to remember the definition of sine in a right triangle, then remembering the definition of cosine, you can easily establish that the sine of an acute angle in a right triangle is the ratio of the opposite side to the hypotenuse. After all, there are only two legs; if the adjacent leg is “occupied” by the cosine, then only the opposite leg remains with the sine.

What about tangent and cotangent? The confusion is the same. Students know that this is a relationship of legs, but the problem is to remember which one refers to which - either the opposite to the adjacent, or vice versa.

Definitions:

Tangent The acute angle in a right triangle is the ratio of the opposite side to the adjacent side:

Cotangent The acute angle in a right triangle is the ratio of the adjacent side to the opposite:

How to remember? There are two ways. One also uses a verbal-logical connection, the other uses a mathematical one.

MATHEMATICAL METHOD

There is such a definition - the tangent of an acute angle is the ratio of the sine of the angle to its cosine:

*Having memorized the formula, you can always determine that the tangent of an acute angle in a right triangle is the ratio of the opposite side to the adjacent side.

Likewise.The cotangent of an acute angle is the ratio of the cosine of the angle to its sine:

So! By remembering these formulas, you can always determine that:

- the tangent of an acute angle in a right triangle is the ratio of the opposite side to the adjacent side

— the cotangent of an acute angle in a right triangle is the ratio of the adjacent side to the opposite side.

WORD-LOGICAL METHOD

About tangent. Remember the link:

That is, if you need to remember the definition of tangent, using this logical connection, you can easily remember what it is

“... the ratio of the opposite side to the adjacent side”

If we talk about cotangent, then remembering the definition of tangent you can easily voice the definition of cotangent -

“... the ratio of the adjacent side to the opposite side”

There is an interesting trick for remembering tangent and cotangent on the website " Mathematical tandem " , look.

UNIVERSAL METHOD

You can just memorize it.But as practice shows, thanks to verbal-logical connections, a person remembers information for a long time, and not only mathematical ones.

I hope the material was useful to you.

Sincerely, Alexander Krutitskikh

P.S: I would be grateful if you tell me about the site on social networks.

Table of values ​​of trigonometric functions

Note. This table of trigonometric function values ​​uses the √ sign to indicate square root. To indicate a fraction, use the symbol "/".

see also useful materials:

For determining the value of a trigonometric function, find it at the intersection of the line indicating the trigonometric function. For example, sine 30 degrees - we look for the column with the heading sin (sine) and find the intersection of this table column with the row “30 degrees”, at their intersection we read the result - one half. Similarly we find cosine 60 degrees, sine 60 degrees (once again, at the intersection of the column sin (sine) and the row 60 degrees we find sin value 60 = √3/2), etc. The values ​​of sines, cosines and tangents of other “popular” angles are found in the same way.

Sine pi, cosine pi, tangent pi and other angles in radians

The table below of cosines, sines and tangents is also suitable for finding the value of trigonometric functions whose argument is given in radians. To do this, use the second column of angle values. Thanks to this, you can convert the value of popular angles from degrees to radians. For example, let's find the angle of 60 degrees in the first line and read its value in radians under it. 60 degrees is equal to π/3 radians.

The number pi unambiguously expresses the dependence of the circumference on the degree measure of the angle. Thus, pi radians are equal to 180 degrees.

Any number expressed in terms of pi (radians) can be easily converted to degrees by replacing pi (π) with 180.

Examples:
1. Sine pi.
sin π = sin 180 = 0
thus, the sine of pi is the same as the sine of 180 degrees and it is equal to zero.

2. Cosine pi.
cos π = cos 180 = -1
thus, the cosine of pi is the same as the cosine of 180 degrees and it is equal to minus one.

3. Tangent pi
tg π = tg 180 = 0
thus, tangent pi is the same as tangent 180 degrees and it is equal to zero.

Table of sine, cosine, tangent values ​​for angles 0 - 360 degrees (common values)

angle α value (degrees)	angle α value in radians (via pi)	sin (sinus)	cos (cosine)	tg (tangent)	ctg (cotangent)	sec (secant)	cosec (cosecant)
0	0	0	1	0	-	1	-
15	π/12			2 - √3	2 + √3
30	π/6	1/2	√3/2	1/√3	√3	2/√3	2
45	π/4	√2/2	√2/2	1	1	√2	√2
60	π/3	√3/2	1/2	√3	1/√3	2	2/√3
75	5π/12			2 + √3	2 - √3
90	π/2	1	0	-	0	-	1
105	7π/12		-	- 2 - √3	√3 - 2
120	2π/3	√3/2	-1/2	-√3	-√3/3
135	3π/4	√2/2	-√2/2	-1	-1	-√2	√2
150	5π/6	1/2	-√3/2	-√3/3	-√3
180	π	0	-1	0	-	-1	-
210	7π/6	-1/2	-√3/2	√3/3	√3
240	4π/3	-√3/2	-1/2	√3	√3/3
270	3π/2	-1	0	-	0	-	-1
360	2π	0	1	0	-	1	-

If in the table of values ​​of trigonometric functions a dash is indicated instead of the function value (tangent (tg) 90 degrees, cotangent (ctg) 180 degrees), then for a given value of the degree measure of the angle the function does not have a specific value. If there is no dash, the cell is empty, which means we have not yet entered desired value. We are interested in what queries users come to us for and supplement the table with new values, despite the fact that current data on the values ​​of cosines, sines and tangents of the most common angle values ​​is quite sufficient to solve most problems.

Table of values ​​of trigonometric functions sin, cos, tg for the most popular angles
0, 15, 30, 45, 60, 90 ... 360 degrees
(numeric values ​​“as per Bradis tables”)

angle α value (degrees)	angle α value in radians	sin (sine)	cos (cosine)	tg (tangent)	ctg (cotangent)
0	0
15		0,2588	0,9659	0,2679
30		0,5000		0,5774
45		0,7071
		0,7660
60		0,8660	0,5000	1,7321
	7π/18

Examples:

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

Argument and meaning

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 we need to determine the cosine of this angle.

2) Let us complete any right triangle on this angle.

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

Cosine of a number

The number circle allows you to determine the cosine of any number, but usually you 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 \(-0 ,71\)).

For cosine for other numbers often encountered in practice, see.

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

Cosine of any angle

Thanks to the number circle, you can determine the cosine of not only an acute angle, but also an obtuse, negative, and even greater than \(360°\) (full revolution). How to do this is easier to see once than to hear \(100\) times, so look at the picture.

Now an explanation: suppose we need to determine the cosine of the angle KOA with degree measure in \(150°\). Combining the point ABOUT with the center of the circle, and the side OK– with the \(x\) axis. After this, set aside \(150°\) counterclockwise. Then the ordinate of the point A 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), do the same, but set \(60°\) clockwise.

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

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

As you could replace, both the cosine of a number and the cosine of an arbitrary angle are defined almost identically. Only the way the point is found on the circle changes.

Cosine signs by 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 the 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 - 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).

Relation to other trigonometric functions:

- the same angle (or number): main trigonometric identity\(\sin^2⁡x+\cos^2⁡x=1\)
- the same angle (or number): by the formula \(1+tg^2⁡x=\)\(\frac(1)(\cos^2⁡x)\)
- and the sine of the same angle (or number): the formula \(ctgx=\)\(\frac(\cos(x))(\sin⁡x)\)
For other most commonly used formulas, see.

Solution of the equation \(\cos⁡x=a\)

The solution to the equation \(\cos⁡x=a\), where \(a\) is a number no greater than \(1\) and no less than \(-1\), i.e. \(a∈[-1;1]\):

\(\cos ⁡x=a\) \(⇔\) \(x=±\arccos⁡a+2πk, k∈Z\)

If \(a>1\) or \(a<-1\), то решений у уравнения нет.

Example . Solve the trigonometric equation \(\cos⁡x=\)\(\frac(1)(2)\).
Solution:

Let's solve the equation using the number circle. For this:
1) Let's build the axes.
2) Let's construct a circle.
3) On the cosine axis (axis \(y\)) mark the point \(\frac(1)(2)\) .
4) Draw a perpendicular to the cosine axis through this point.
5) Mark the intersection points of the perpendicular and the circle.
6) Let's sign the values ​​of these points: \(\frac(π)(3)\) ,\(-\)\(\frac(π)(3)\) .
7) Let’s write down all the values ​​corresponding to these points using the formula \(x=t+2πk\), \(k∈Z\):
\(x=±\)\(\frac(π)(3)\) \(+2πk\), \(k∈Z\);

Answer: \(x=±\frac(π)(3)+2πk\) \(k∈Z\)

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

If we plot the angles in radians along the \(x\) axis, and the cosine values ​​corresponding 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 coordinate axes:
abscissa axis: \((\)\(\frac(π)(2)\) \(+πn\),\(;0)\), where \(n ϵ Z\)
Y axis: \((0;1)\)
- intervals of constancy of sign:
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 \(n ϵ Z\)
- maximums and minimums of the function:
the function has a maximum value \(y=1\) at points \(x=2πn\), where \(n ϵ Z\)
the function has a minimum value \(y=-1\) at points \(x=π+2πn\), where \(n ϵ Z\).

Let's understand simple concepts: sine and cosine and calculation cosine squared and sine squared.

Sine and cosine are studied in trigonometry (the study of right-angle triangles).

Therefore, first, let’s remember the basic concepts of a right triangle:

Hypotenuse- the side that always lies opposite the right angle (90 degree angle). The hypotenuse is the longest side of a right angle triangle.

The remaining two sides in a right triangle are called legs.

You should also remember that three angles in a triangle always add up to 180°.

Now let's move on to cosine and sine of the angle alpha (∠α)(this can be called any indirect angle in a triangle or used as a designation x - "x", which does not change the essence).

Sine of angle alpha (sin ∠α)- this is an attitude opposite leg (the side opposite the corresponding angle) to the hypotenuse. If you look at the figure, then sin ∠ABC = AC / BC

Cosine of angle alpha (cos ∠α)- attitude adjacent to the angle of the leg to the hypotenuse. Looking again at the figure above, cos ∠ABC = AB / BC

And just as a reminder: cosine and sine will never be greater than one, since any roll is shorter than the hypotenuse (and the hypotenuse is the longest side of any triangle, because the longest side is located opposite the largest angle in the triangle).

Cosine squared, sine squared

Now let's move on to the basic trigonometric formulas: calculating cosine squared and sine squared.

To calculate them, you should remember the basic trigonometric identity:

sin 2 α + cos 2 α = 1(sine square plus cosine square of one angle always equals one).

From the trigonometric identity we draw conclusions about the sine:

sin 2 α = 1 - cos 2 α

sine square alpha is equal to one minus the cosine of the double angle alpha and divide all this by two.

sin 2 α = (1 – cos(2α)) / 2

​​​​​​​From the trigonometric identity we draw conclusions about the cosine:

cos 2 α = 1 - sin 2 α

or a more complex version of the formula: cosine square alpha is equal to one plus the cosine of the double angle alpha and also divide everything by two.

cos 2 α = (1 + cos(2α)) / 2

These two more complex formulas for sine squared and cosine squared are also called “reducing the power for squared trigonometric functions.” Those. there was a second degree, they lowered it to the first and the calculations became more convenient.

I won't convince you not to write cheat sheets. Write! Including cheat sheets on trigonometry. Later I plan to explain why cheat sheets are needed and why cheat sheets are useful. And here is information on how not to learn, but to remember some trigonometric formulas. So - trigonometry without a cheat sheet! We use associations for memorization.

1. Addition formulas:

Cosines always “come in pairs”: cosine-cosine, sine-sine. And one more thing: cosines are “inadequate”. “Everything is not right” for them, so they change the signs: “-” to “+”, and vice versa.

Sinuses - “mix”: sine-cosine, cosine-sine.

2. Sum and difference formulas:

cosines always “come in pairs”. By adding two cosines - “koloboks”, we get a pair of cosines - “koloboks”. And by subtracting, we definitely won’t get any koloboks. We get a couple of sines. Also with a minus ahead.

Sinuses - “mix” :

3. Formulas for converting a product into a sum and difference.

When do we get a cosine pair? When we add cosines. That's why

When do we get a couple of sines? When subtracting cosines. From here:

“Mixing” is obtained both when adding and subtracting sines. What's more fun: adding or subtracting? That's right, fold. And for the formula they take addition:

In the first and third formulas, the sum is in parentheses. Rearranging the places of the terms does not change the sum. The order is important only for the second formula. But, in order not to get confused, for ease of remembering, in all three formulas in the first brackets we take the difference

and secondly - the amount

Cheat sheets in your pocket give you peace of mind: if you forget the formula, you can copy it. And they give you confidence: if you fail to use the cheat sheet, you can easily remember the formulas.