Definition of sine, cosine, tangent and cotangent. Right triangle

21.10.2019

The ratio of the opposite leg to the hypotenuse is called sine of an acute angle right triangle.

\sin \alpha = \frac(a)(c)

Cosine of an acute angle of a right triangle

The ratio of the nearest leg to the hypotenuse is called cosine of an acute angle right triangle.

\cos \alpha = \frac(b)(c)

Tangent of an acute angle of a right triangle

The ratio of the opposite leg to the adjacent leg is called acute angle tangent right triangle.

tg \alpha = \frac(a)(b)

Cotangent of an acute angle of a right triangle

The ratio of the adjacent leg to the opposite leg is called cotangent of an acute angle right triangle.

ctg \alpha = \frac(b)(a)

Sine of an arbitrary angle

The ordinate of the point on the unit circle to which the angle \alpha corresponds is called sine of an arbitrary angle rotation \alpha .

\sin \alpha=y

Cosine of an arbitrary angle

The abscissa of a point on the unit circle to which the angle \alpha corresponds is called cosine of an arbitrary angle rotation \alpha .

\cos \alpha=x

Tangent of an arbitrary angle

The ratio of the sine of an arbitrary rotation angle \alpha to its cosine is called tangent of an arbitrary angle rotation \alpha .

tg \alpha = y_(A)

tg \alpha = \frac(\sin \alpha)(\cos \alpha)

Cotangent of an arbitrary angle

The ratio of the cosine of an arbitrary rotation angle \alpha to its sine is called cotangent of an arbitrary angle rotation \alpha .

ctg \alpha =x_(A)

ctg \alpha = \frac(\cos \alpha)(\sin \alpha)

An example of finding an arbitrary angle

If \alpha is some angle AOM , where M is a point on the unit circle, then

\sin \alpha=y_(M) , \cos \alpha=x_(M) , tg \alpha=\frac(y_(M))(x_(M)), ctg \alpha=\frac(x_(M))(y_(M)).

For example, if \angle AOM = -\frac(\pi)(4), then: the ordinate of the point M is -\frac(\sqrt(2))(2), the abscissa is \frac(\sqrt(2))(2) and that's why

\sin \left (-\frac(\pi)(4) \right)=-\frac(\sqrt(2))(2);

\cos \left (\frac(\pi)(4) \right)=\frac(\sqrt(2))(2);

tg;

ctg \left (-\frac(\pi)(4) \right)=-1.

Table of values of sines of cosines of tangents of cotangents

The values of the main frequently encountered angles are given in the table:

	0^(\circ) (0)	30^(\circ)\left(\frac(\pi)(6)\right)	45^(\circ)\left(\frac(\pi)(4)\right)	60^(\circ)\left(\frac(\pi)(3)\right)	90^(\circ)\left(\frac(\pi)(2)\right)	180^(\circ)\left(\pi\right)	270^(\circ)\left(\frac(3\pi)(2)\right)	360^(\circ)\left(2\pi\right)
\sin\alpha	0	\frac12	\frac(\sqrt 2)(2)	\frac(\sqrt 3)(2)	1	0	−1	0
\cos\alpha	1	\frac(\sqrt 3)(2)	\frac(\sqrt 2)(2)	\frac12	0	−1	0	1
tg\alpha	0	\frac(\sqrt 3)(3)	1	\sqrt3	—	0	—	0
ctg\alpha	—	\sqrt3	1	\frac(\sqrt 3)(3)	0	—	0	—

In this article, we will show how definitions of sine, cosine, tangent and cotangent of angle and number in trigonometry. Here we will talk about notation, give examples of records, give graphic illustrations. In conclusion, we draw a parallel between the definitions of sine, cosine, tangent and cotangent in trigonometry and geometry.

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Definition of sine, cosine, tangent and cotangent

Let's follow how the concept of sine, cosine, tangent and cotangent is formed in the school mathematics course. In geometry lessons, the definition of sine, cosine, tangent and cotangent of an acute angle in a right triangle is given. And later trigonometry is studied, which refers to the sine, cosine, tangent and cotangent of the angle of rotation and the number. We give all these definitions, give examples and give the necessary comments.

Acute angle in a right triangle

From the course of geometry, the definitions of the sine, cosine, tangent and cotangent of an acute angle in a right triangle are known. They are given as the ratio of the sides of a right triangle. We present their formulations.

Definition.

Sine of an acute angle in a right triangle is the ratio of the opposite leg to the hypotenuse.

Definition.

Cosine of an acute angle in a right triangle is the ratio of the adjacent leg to the hypotenuse.

Definition.

Tangent of an acute angle in a right triangle is the ratio of the opposite leg to the adjacent leg.

Definition.

Cotangent of an acute angle in a right triangle is the ratio of the adjacent leg to the opposite leg.

The notation of sine, cosine, tangent and cotangent is also introduced there - sin, cos, tg and ctg, respectively.

For example, if ABC is a right triangle with a right angle C, then the sine of the acute angle A is equal to the ratio of the opposite leg BC to the hypotenuse AB, that is, sin∠A=BC/AB.

These definitions allow you to calculate the values of the sine, cosine, tangent and cotangent of an acute angle from the known lengths of the sides of a right triangle, as well as from known values sine, cosine, tangent, cotangent and the length of one of the sides to find the lengths of the other sides. For example, if we knew that in a right triangle the leg AC is 3 and the hypotenuse AB is 7 , then we could calculate the cosine of the acute angle A by definition: cos∠A=AC/AB=3/7 .

Angle of rotation

In trigonometry, they begin to look at the angle more widely - they introduce the concept of angle of rotation. The angle of rotation, unlike an acute angle, is not limited to frames from 0 to 90 degrees, the angle of rotation in degrees (and in radians) can be expressed by any real number from −∞ to +∞.

In this light, the definitions of the sine, cosine, tangent and cotangent are no longer an acute angle, but an angle of arbitrary magnitude - the angle of rotation. They are given through the x and y coordinates of the point A 1 , into which the so-called initial point A(1, 0) passes after it rotates through an angle α around the point O - the beginning of a rectangular Cartesian coordinate system and the center of the unit circle.

Definition.

Sine of rotation angleα is the ordinate of the point A 1 , that is, sinα=y .

Definition.

cosine of the angle of rotationα is called the abscissa of the point A 1 , that is, cosα=x .

Definition.

Tangent of rotation angleα is the ratio of the ordinate of point A 1 to its abscissa, that is, tgα=y/x .

Definition.

The cotangent of the angle of rotationα is the ratio of the abscissa of the point A 1 to its ordinate, that is, ctgα=x/y .

The sine and cosine are defined for any angle α, since we can always determine the abscissa and ordinate of the point, which is obtained by rotating the starting point by the angle α. And tangent and cotangent are not defined for any angle. The tangent is not defined for such angles α at which the initial point goes to a point with zero abscissa (0, 1) or (0, −1) , and this takes place at angles 90°+180° k , k∈Z (π /2+π k rad). Indeed, at such angles of rotation, the expression tgα=y/x does not make sense, since it contains division by zero. As for the cotangent, it is not defined for such angles α at which the starting point goes to a point with zero ordinate (1, 0) or (−1, 0) , and this is the case for angles 180° k , k ∈Z (π k rad).

So, the sine and cosine are defined for any rotation angles, the tangent is defined for all angles except 90°+180° k , k∈Z (π/2+π k rad), and the cotangent is for all angles except 180° ·k , k∈Z (π·k rad).

The notations already known to us appear in the definitions sin, cos, tg and ctg, they are also used to denote the sine, cosine, tangent and cotangent of the angle of rotation (sometimes you can find the notation tan and cotcorresponding to tangent and cotangent). So the sine of the rotation angle of 30 degrees can be written as sin30°, the records tg(−24°17′) and ctgα correspond to the tangent of the rotation angle −24 degrees 17 minutes and the cotangent of the rotation angle α . Recall that when writing the radian measure of an angle, the notation "rad" is often omitted. For example, the cosine of a rotation angle of three pi rads is usually denoted cos3 π .

In conclusion of this paragraph, it is worth noting that in talking about the sine, cosine, tangent and cotangent of the angle of rotation, the phrase “angle of rotation” or the word “rotation” is often omitted. That is, instead of the phrase "sine of the angle of rotation alpha", the phrase "sine of the angle of alpha" is usually used, or even shorter - "sine of alpha". The same applies to cosine, and tangent, and cotangent.

Let's also say that the definitions of the sine, cosine, tangent, and cotangent of an acute angle in a right triangle are consistent with the definitions just given for the sine, cosine, tangent, and cotangent of a rotation angle ranging from 0 to 90 degrees. We will substantiate this.

Numbers

Definition.

Sine, cosine, tangent and cotangent of a number t is a number equal to the sine, cosine, tangent and cotangent of the angle of rotation in t radians, respectively.

For example, the cosine of 8 π is, by definition, a number equal to the cosine of an angle of 8 π rad. And the cosine of the angle in 8 π rad is equal to one, therefore, the cosine of the number 8 π is equal to 1.

There is another approach to the definition of the sine, cosine, tangent and cotangent of a number. It consists in the fact that each real number t is assigned a point of the unit circle centered at the origin of the rectangular coordinate system, and the sine, cosine, tangent and cotangent are determined through the coordinates of this point. Let's dwell on this in more detail.

Let us show how the correspondence between real numbers and points of the circle is established:

the number 0 is assigned the starting point A(1, 0) ;
a positive number t is associated with a point on the unit circle, which we will get to if we move around the circle from the starting point in a counterclockwise direction and go through a path of length t;
a negative number t is associated with a point on the unit circle, which we will get to if we move around the circle from the starting point in a clockwise direction and go through a path of length |t| .

Now let's move on to the definitions of sine, cosine, tangent and cotangent of the number t. Let us assume that the number t corresponds to a point of the circle A 1 (x, y) (for example, the number &pi/2; corresponds to the point A 1 (0, 1) ).

Definition.

The sine of a number t is the ordinate of the unit circle point corresponding to the number t , that is, sint=y .

Definition.

The cosine of a number t is called the abscissa of the point of the unit circle corresponding to the number t , that is, cost=x .

Definition.

Tangent of a number t is the ratio of the ordinate to the abscissa of the point of the unit circle corresponding to the number t, that is, tgt=y/x. In another equivalent formulation, the tangent of the number t is the ratio of the sine of this number to the cosine, that is, tgt=sint/cost .

Definition.

Cotangent of a number t is the ratio of the abscissa to the ordinate of the point of the unit circle corresponding to the number t, that is, ctgt=x/y. Another formulation is as follows: the tangent of the number t is the ratio of the cosine of the number t to the sine of the number t : ctgt=cost/sint .

Here we note that the definitions just given agree with the definition given at the beginning of this subsection. Indeed, the point of the unit circle corresponding to the number t coincides with the point obtained by rotating the starting point through an angle of t radians.

It is also worth clarifying this point. Let's say we have a sin3 entry. How to understand whether the sine of the number 3 or the sine of the rotation angle of 3 radians is in question? This is usually clear from the context, otherwise it probably doesn't matter.

Trigonometric functions of angular and numerical argument

According to the definitions given in the previous paragraph, each rotation angle α corresponds to a well-defined value sin α , as well as the value cos α . In addition, all rotation angles other than 90°+180° k , k∈Z (π/2+π k rad) correspond to the values tgα , and other than 180° k , k∈Z (π k rad ) are the values of ctgα . Therefore sinα, cosα, tgα and ctgα are functions of the angle α. In other words, these are functions of the angular argument.

Similarly, we can talk about the functions sine, cosine, tangent and cotangent of a numerical argument. Indeed, each real number t corresponds to a well-defined value of sint , as well as cost . In addition, all numbers other than π/2+π·k , k∈Z correspond to the values tgt , and the numbers π·k , k∈Z correspond to the values ctgt .

The functions sine, cosine, tangent and cotangent are called basic trigonometric functions.

It is usually clear from the context that we are dealing with trigonometric functions of an angular argument or a numerical argument. Otherwise, we can consider the independent variable as both a measure of the angle (the angle argument) and a numeric argument.

However, the school mainly studies numeric functions, that is, functions whose arguments, as well as their corresponding function values, are numbers. Therefore, if we are talking about functions, then it is advisable to consider trigonometric functions as functions of numerical arguments.

Connection of definitions from geometry and trigonometry

If we consider the angle of rotation α from 0 to 90 degrees, then the data in the context of trigonometry of the definition of the sine, cosine, tangent and cotangent of the angle of rotation are fully consistent with the definitions of the sine, cosine, tangent and cotangent of an acute angle in a right triangle, which are given in the geometry course. Let's substantiate this.

Draw a unit circle in the rectangular Cartesian coordinate system Oxy. Note the starting point A(1, 0) . Let's rotate it by an angle α ranging from 0 to 90 degrees, we get the point A 1 (x, y) . Let's drop the perpendicular A 1 H from the point A 1 to the Ox axis.

It is easy to see that in a right triangle the angle A 1 OH is equal to the angle of rotation α, the length of the leg OH adjacent to this angle is equal to the abscissa of the point A 1, that is, |OH|=x, the length of the leg A 1 H opposite to the angle is equal to the ordinate of the point A 1 , that is, |A 1 H|=y , and the length of the hypotenuse OA 1 is equal to one, since it is the radius of the unit circle. Then, by definition from geometry, the sine of an acute angle α in a right triangle A 1 OH is equal to the ratio of the opposite leg to the hypotenuse, that is, sinα=|A 1 H|/|OA 1 |=y/1=y . And by definition from trigonometry, the sine of the angle of rotation α is equal to the ordinate of the point A 1, that is, sinα=y. This shows that the definition of the sine of an acute angle in a right triangle is equivalent to the definition of the sine of the angle of rotation α for α from 0 to 90 degrees.

Similarly, it can be shown that the definitions of the cosine, tangent, and cotangent of an acute angle α are consistent with the definitions of the cosine, tangent, and cotangent of the angle of rotation α.

Bibliography.

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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 this branch 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 different 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.

Firstly, you should not convert ordinary fractions to decimals until the final result is obtained - you can leave the answer in the form common 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.

How to find the sine?

The study of geometry helps to develop thinking. This subject is included in the curriculum. In life, knowledge of this subject can be useful - for example, when planning an apartment.

From the history

As part of the geometry course, trigonometry is also studied, which explores trigonometric functions. In trigonometry, we study the sines, cosines, tangents, and cotangents of an angle.

But for now, let's start with the simplest - sine. Let's take a closer look at the very first concept - the sine of an angle in geometry. What is a sine and how to find it?

The concept of "sine of the angle" and sinusoids

The sine of an angle is the ratio of the values of the opposite leg and the hypotenuse of a right triangle. This is a direct trigonometric function, which is written in writing as "sin (x)", where (x) is the angle of the triangle.

On the graph, the sine of an angle is indicated by a sinusoid with its own characteristics. A sinusoid looks like a continuous wavy line that lies within certain limits on the coordinate plane. The function is odd, therefore it is symmetric with respect to 0 on the coordinate plane (it leaves the origin of the coordinates).

The domain of this function lies in the range from -1 to +1 on the Cartesian coordinate system. The period of the sine angle function is 2 Pi. This means that every 2 Pi the pattern is repeated and the sine wave goes through a full cycle.

Sinusoid equation

sin x = a / c
where a is the leg opposite to the angle of the triangle
c - hypotenuse of a right triangle

Properties of the sine of an angle

sin(x) = - sin(x). This feature demonstrates that the function is symmetrical, and if the values x and (-x) are set aside on the coordinate system in both directions, then the ordinates of these points will be opposite. They will be at an equal distance from each other.
Another feature of this function is that the graph of the function increases on the segment [- P / 2 + 2 Pn]; [P/2 + 2Pn], where n is any integer. A decrease in the graph of the sine of the angle will be observed on the segment: [P / 2 + 2 Pn]; [ 3P/2 + 2Pn].
sin (x) > 0 when x is in the range (2Pn, P + 2Pn)
(x)< 0, когда х находится в диапазоне (-П+2Пn, 2Пn)

The values of the sines of the angle are determined by special tables. Such tables have been created to facilitate the counting process. complex formulas and equations. It is easy to use and contains the values of not only the sin(x) function, but also the values of other functions.

Moreover, the table of standard values of these functions is included in the mandatory memory study, like the multiplication table. This is especially true for classes with a physical and mathematical bias. In the table you can see the values of the main angles used in trigonometry: 0, 15, 30, 45, 60, 75, 90, 120, 135, 150, 180, 270 and 360 degrees.

There is also a table that defines the values of the trigonometric functions of non-standard angles. Using different tables, you can easily calculate the sine, cosine, tangent and cotangent of some angles.

Equations are made with trigonometric functions. Solving these equations is easy if you know simple trigonometric identities and reductions of functions, for example, such as sin (P / 2 + x) \u003d cos (x) and others. A separate table has also been compiled for such casts.

How to find the sine of an angle

When the task is to find the sine of an angle, and by condition we have only the cosine, tangent, or cotangent of the angle, we can easily calculate what we need using trigonometric identities.

sin 2 x + cos 2 x = 1

From this equation, we can find both sine and cosine, depending on which value is unknown. We get a trigonometric equation with one unknown:

sin 2 x = 1 - cos 2 x
sin x = ± √ 1 - cos 2 x
ctg 2 x + 1 = 1 / sin 2 x

From this equation, you can find the value of the sine, knowing the value of the cotangent of the angle. To simplify, replace sin 2 x = y, and then you have a simple equation. For example, the value of the cotangent is 1, then:

1 + 1 = 1/y
2 = 1 / y
2y = 1
y = 1/2

Now we perform the reverse replacement of the player:

sin 2 x = ½
sin x = 1 / √2

Since we took the value of the cotangent for the standard angle (45 0), the obtained values \u200b\u200bcan be checked against the table.

If you have a tangent value, but you need to find the sine, another trigonometric identity will help:

tg x * ctg x = 1

It follows that:

ctg x = 1 / tg x

In order to find the sine of a non-standard angle, for example, 240 0, you need to use the angle reduction formulas. We know that π corresponds to 180 0 for us. Thus, we will express our equality using standard angles by expansion.