Sine, cosine, tangent and cotangent - everything you need to know for the OGE and the Unified State Exam! Right triangle. The Complete Illustrated Guide (2019)

Instructions

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note

When calculating the sides of a right triangle, knowledge of its characteristics can play a role:
1) If the leg of a right angle lies opposite an angle of 30 degrees, then it is equal to half the hypotenuse;
2) The hypotenuse is always longer than any of the legs;
3) If a circle is circumscribed around a right triangle, then its center must lie in the middle of the hypotenuse.

The hypotenuse is the side in a right triangle that is opposite the 90 degree angle. In order to calculate its length, it is enough to know the length of one of the legs and the size of one of the acute angles of the triangle.

Instructions

Let us know one of the legs and the angle adjacent to it. To be specific, let these be the side |AB| and angle α. Then we can use the formula for the trigonometric cosine - cosine ratio of the adjacent leg to. Those. in our notation cos α = |AB| / |AC|. From this we obtain the length of the hypotenuse |AC| = |AB| / cos α.
If we know the side |BC| and angle α, then we will use the formula to calculate the sine of the angle - the sine of the angle is equal to the ratio of the opposite leg to the hypotenuse: sin α = |BC| / |AC|. We find that the length of the hypotenuse is |AC| = |BC| / cos α.

For clarity, let's look at an example. Let the length of the leg |AB| be given. = 15. And angle α = 60°. We get |AC| = 15 / cos 60° = 15 / 0.5 = 30.
Let's look at how you can check your result using the Pythagorean theorem. To do this, we need to calculate the length of the second leg |BC|. Using the formula for the tangent of the angle tan α = |BC| / |AC|, we get |BC| = |AB| * tan α = 15 * tan 60° = 15 * √3. Next, we apply the Pythagorean theorem, we get 15^2 + (15 * √3)^2 = 30^2 => 225 + 675 = 900. Check completed.

Helpful advice

After calculating the hypotenuse, check whether the resulting value satisfies the Pythagorean theorem.

Sources:

• Table of prime numbers from 1 to 10000

Legs are the two short sides of a right triangle that make up the vertex whose size is 90°. The third side in such a triangle is called the hypotenuse. All these sides and angles of the triangle are interconnected by certain relationships that make it possible to calculate the length of the leg if several other parameters are known.

Instructions

Use the Pythagorean theorem for leg (A) if you know the length of the other two sides (B and C) of the right triangle. This theorem states that the sum of the squared lengths of the legs is equal to the square of the hypotenuse. It follows from this that the length of each leg is equal to the square root of the lengths of the hypotenuse and the second leg: A=√(C²-B²).

Use the definition of the direct trigonometric function “sine” for an acute angle if you know the magnitude of the angle (α) lying opposite the leg being calculated and the length of the hypotenuse (C). This states that the sine of this known ratio of the length of the desired leg to the length of the hypotenuse. This means that the length of the desired leg is equal to the product of the length of the hypotenuse and the sine of the known angle: A=C∗sin(α). For the same known quantities, you can also use the cosecant and calculate the required length by dividing the length of the hypotenuse by the cosecant of the known angle A=C/cosec(α).

Use the definition of the direct trigonometric cosine function if, in addition to the length of the hypotenuse (C), the magnitude of the acute angle (β) adjacent to the desired one is also known. The cosine of this angle is the ratio of the lengths of the desired leg and the hypotenuse, and from this we can conclude that the length of the leg is equal to the product of the length of the hypotenuse and the cosine of the known angle: A=C∗cos(β). You can use the definition of the secant function and calculate the desired value by dividing the length of the hypotenuse by the secant of the known angle A=C/sec(β).

Derive the required formula from a similar definition for the derivative of the trigonometric function tangent, if in addition to the value of the acute angle (α) lying opposite the desired leg (A), the length of the second leg (B) is known. The tangent of the angle opposite to the desired leg is the ratio of the length of this leg to the length of the second leg. This means that the desired value will be equal to the product of the length of the known leg and the tangent of the known angle: A=B∗tg(α). From these same known quantities, another formula can be derived if we use the definition of the cotangent function. In this case, to calculate the length of the leg, it will be necessary to find the ratio of the length of the known leg to the cotangent of the known angle: A=B/ctg(α).

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The word “kathet” came into Russian from Greek. In exact translation, it means a plumb line, that is, perpendicular to the surface of the earth. In mathematics, legs are the sides that form a right angle of a right triangle. The side opposite this angle is called the hypotenuse. The term “cathet” is also used in architecture and welding technology.

Both legs are connected to each other and by a cotangent. In this case, the tangent will be the ratio of side a to side b, that is, the opposite side to the adjacent side. This relationship can be expressed by the formula tgCAB=a/b. Accordingly, the inverse ratio will be the cotangent: ctgCAB=b/a.

The relationship between the sizes of the hypotenuse and both legs was determined by the ancient Greek Pythagoras. People still use the theorem and his name. It says that the square of the hypotenuse is equal to the sum of the squares of the legs, that is, c2 = a2 + b2. Accordingly, each leg will be equal to the square root of the difference between the squares of the hypotenuse and the other leg. This formula can be written as b=√(c2-a2).

The length of the leg can also be expressed through the relationships known to you. According to the theorems of sines and cosines, a leg is equal to the product of the hypotenuse and one of these functions. It can be expressed as and or cotangent. Leg a can be found, for example, using the formula a = b*tan CAB. In exactly the same way, depending on the given tangent or , the second leg is determined.

The term "cathet" is also used in architecture. It is applied to the Ionic capital and plumb through the middle of its back. That is, in this case, this term is perpendicular to a given line.

In welding technology there is a “fillet weld leg”. As in other cases, this is the shortest distance. Here we are talking about the gap between one of the parts being welded to the border of the seam located on the surface of the other part.

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Sources:

• what are leg and hypotenuse in 2019

We will begin our study of trigonometry with the right triangle. Let's define what sine and cosine are, as well as tangent and cotangent of an acute angle. This is the basics of trigonometry.

Let us remind you that right angle is an angle equal to 90 degrees. In other words, half a turned angle.

Sharp corner- less than 90 degrees.

Obtuse angle- greater than 90 degrees. In relation to such an angle, “obtuse” is not an insult, but a mathematical term :-)

Let's draw a right triangle. A right angle is usually denoted by . Please note that the side opposite the corner is indicated by the same letter, only small. Thus, the side opposite angle A is designated .

The angle is denoted by the corresponding Greek letter.

Hypotenuse of a right triangle is the side opposite the right angle.

Legs- sides lying opposite acute angles.

The leg lying opposite the angle is called opposite(relative to angle). The other leg, which lies on one of the sides of the angle, is called adjacent.

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

Cosine acute angle in a right triangle - the ratio of the adjacent leg to the hypotenuse:

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

Another (equivalent) definition: the tangent of an acute angle is the ratio of the sine of the angle to its cosine:

Cotangent acute angle in a right triangle - the ratio of the adjacent side to the opposite (or, which is the same, the ratio of cosine to sine):

Note the basic relationships for sine, cosine, tangent, and cotangent below. They will be useful to us when solving problems.

Let's prove some of them.

Okay, we have given definitions and written down formulas. But why do we still need sine, cosine, tangent and cotangent?

We know that the sum of the angles of any triangle is equal to.

We know the relationship between parties right triangle. This is the Pythagorean theorem: .

It turns out that knowing two angles in a triangle, you can find the third. Knowing the two sides of a right triangle, you can find the third. This means that the angles have their own ratio, and the sides have their own. But what should you do if in a right triangle you know one angle (except the right angle) and one side, but you need to find the other sides?

This is what people in the past encountered when making maps of the area and the starry sky. After all, it is not always possible to directly measure all sides of a triangle.

Sine, cosine and tangent - they are also called trigonometric angle functions- give relationships between parties And corners triangle. Knowing the angle, you can find all its trigonometric functions using special tables. And knowing the sines, cosines and tangents of the angles of a triangle and one of its sides, you can find the rest.

We will also draw a table of the values ​​of sine, cosine, tangent and cotangent for “good” angles from to.

Please note the two red dashes in the table. At appropriate angle values, tangent and cotangent do not exist.

Let's look at several trigonometry problems from the FIPI Task Bank.

1. In a triangle, the angle is , . Find .

The problem is solved in four seconds.

Because the , .

2. In a triangle, the angle is , , . Find .

Let's find it using the Pythagorean theorem.

The problem is solved.

Often in problems there are triangles with angles and or with angles and. Remember the basic ratios for them by heart!

For a triangle with angles and the leg opposite the angle at is equal to half of the hypotenuse.

A triangle with angles and is isosceles. In it, the hypotenuse is times larger than the leg.

We looked at problems solving right triangles - that is, finding unknown sides or angles. But that's not all! There are many problems in the Unified State Examination in mathematics that involve sine, cosine, tangent or cotangent of an external angle of a triangle. More on this in the next article.

One of the areas of mathematics that students struggle with the most is trigonometry. It is not surprising: 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 use trigonometry when proving theorems, and this requires either a developed mathematical memory or the ability to derive complex logical chains.

Origins of trigonometry

Getting acquainted with this science should begin with the definition of sine, cosine and tangent of an angle, but first you need to understand what trigonometry does in general.

Historically, the main object of study in this branch of mathematical science was right triangles. 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 in question 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 in art.

First stage

Initially, people talked about the relationship between angles and sides exclusively using 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 branch of mathematics.

The study of trigonometry in school today begins with right triangles, after which students use the acquired knowledge in physics and solving abstract trigonometric equations, which begin in high school.

Spherical trigonometry

Later, when science reached the next level of development, formulas with sine, cosine, tangent, and 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 in 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 the thread. Attach the thread to any two points on the globe so that it is taut. Please note - it has taken on the shape of an arc. Spherical geometry deals with such forms, which is used in geodesy, astronomy and other theoretical and applied fields.

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. It 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 the 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, which 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 equal to 180 degrees.

Definition

Finally, with a firm understanding of the geometric basis, one can turn to the definition of 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 side 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 their ratio will always be less than one. Thus, if in your answer to a problem you get a sine or cosine with a value greater than 1, look for an error in the calculations or reasoning. This answer is clearly incorrect.

Finally, the tangent of an angle is the ratio of the opposite side to the adjacent side. Dividing the sine by the cosine will give the same result. Look: according to the formula, we divide the length of the side by the hypotenuse, then divide by the length of the second side and multiply by the hypotenuse. Thus, we get the same relationship as in the definition of tangent.

Cotangent, accordingly, is the ratio of the side adjacent to the corner to the opposite side. We get the same result by dividing one by the tangent.

So, we have looked at the definitions of what sine, cosine, tangent and cotangent are, and we can move on to formulas.

The simplest formulas

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

The first formula 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 need to know the size of the angle rather than 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: 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, transformation rules and several basic formulas, you can at any time derive the required more complex formulas on a sheet of paper.

Formulas for double angles and addition of arguments

Two more formulas that you need to learn are related to the values ​​of sine and cosine for the sum and difference of angles. They are presented in the figure below. Please note that in the first case, sine and cosine are multiplied both times, and in the second, the pairwise product of 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 by taking the alpha angle equal to the beta angle.

Finally, note that double angle formulas can be rearranged to reduce the power 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 ​​the figure, and the size of each side, etc.

The sine theorem states that dividing the length of each side of a triangle by the opposite angle results in the same number. Moreover, this number will be equal to two radii of the circumscribed circle, that is, the circle containing all the points of a 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 adjacent angle - 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.

Careless mistakes

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 take a look at the most popular ones.

First, you should not convert fractions to decimals until you get the final result - you can leave the answer as a fraction unless otherwise stated in the conditions. Such a transformation cannot be called a mistake, but it should be remembered that at each stage of the problem new roots may appear, which, according to the author’s idea, should be reduced. In this case, you will waste your time on unnecessary mathematical operations. This is especially true for values ​​such as the root of three or the root of two, because they are found in problems at every step. The same goes for 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 you will also demonstrate a complete lack of understanding 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 confuse them, 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 practical meaning. What is sine, cosine, tangent for an engineer or astronomer? These are concepts with which you can calculate the distance to distant stars, predict the fall of a meteorite, or send a research probe to another planet. Without them, it is impossible to build a building, design a car, calculate the load on a 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're sine, cosine, tangent. You can use them in calculations and successfully solve school problems.

The whole point of trigonometry comes down to the fact that using the known parameters of a triangle you need to calculate the unknowns. There are six parameters in total: the length of three sides and the size of three angles. The only difference in the tasks lies in the fact that different input data are given.

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

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

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

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Initially, the definitions of trigonometric functions whose argument is an angle were expressed in terms of the ratio of the sides of a right triangle.

Definitions of trigonometric functions

The sine of an angle (sin α) is the ratio of the leg opposite this angle to the hypotenuse.

Cosine of the angle (cos α) - the ratio of the adjacent leg to the hypotenuse.

Angle tangent (t g α) - the ratio of the opposite side to the adjacent side.

Angle cotangent (c t g α) - the ratio of the adjacent side to the opposite side.

These definitions are given for the acute angle of a right triangle!

Let's give an illustration.

In triangle ABC with right angle C, the sine of angle A is equal to the ratio of leg BC to hypotenuse AB.

The definitions of sine, cosine, tangent and cotangent allow you to calculate the values ​​of these functions from the known lengths of the sides of the triangle.

Important to remember!

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

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

In this context, we can define sine, cosine, tangent and cotangent of an angle of arbitrary magnitude. Let us imagine a unit circle with its center at the origin of the Cartesian coordinate system.

The initial point A with coordinates (1, 0) rotates around the center of the unit circle through a certain angle α and goes to point A 1. The definition is given in terms of the coordinates of point A 1 (x, y).

Sine (sin) of the rotation angle

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

Cosine (cos) of the rotation angle

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

Tangent (tg) of the rotation angle

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

Cotangent (ctg) of the rotation angle

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

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

Important to remember!

Sine and cosine are defined for any angles α.

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

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

When solving practical examples, do not say “sine of the angle of rotation α”. The words “angle of rotation” are simply omitted, implying that it is already clear from the context what is being discussed.

Numbers

What about the definition of sine, cosine, tangent and cotangent of a number, and not the angle of rotation?

Sine, cosine, tangent, cotangent of a number

Sine, cosine, tangent and cotangent of a number t is a number that is respectively equal to sine, cosine, tangent and cotangent in t radian.

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

There is another approach to determining the sine, cosine, tangent and cotangent of a number. Let's take a closer look at it.

Any real number t a point on the unit circle is associated with the center at the origin of the rectangular Cartesian coordinate system. Sine, cosine, tangent and cotangent are determined through the coordinates of this point.

The starting point on the circle is point A with coordinates (1, 0).

Positive number t

Negative number t corresponds to the point to which the starting point will go if it moves around the circle counterclockwise and passes the path t.

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

Sine (sin) of t

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

Cosine (cos) of t

Cosine of a number t- abscissa of the point of the unit circle corresponding to the number t. cos t = x

Tangent (tg) of t

Tangent of a number t- the ratio of the ordinate to the abscissa of a point on the unit circle corresponding to the number t. t g t = y x = sin t cos t

The latest definitions are in accordance with and do not contradict the definition given at the beginning of this paragraph. Point on the circle corresponding to the number t, coincides with the point to which the starting point goes after turning by an angle t radian.

Trigonometric functions of angular and numeric argument

Each value of the angle α corresponds to a certain value of the sine and cosine of this angle. Just like all angles α other than α = 90 ° + 180 ° k, k ∈ Z (α = π 2 + π k, k ∈ Z) correspond to a certain tangent value. Cotangent, as stated above, is defined for all α except α = 180° k, k ∈ Z (α = π k, k ∈ Z).

We can say that sin α, cos α, t g α, c t g α are functions of the angle alpha, or functions of the angular argument.

Similarly, we can talk about sine, cosine, tangent and cotangent as functions of a numerical argument. Every real number t corresponds to a certain value of the sine or cosine of a number t. All numbers other than π 2 + π · k, k ∈ Z, correspond to a tangent value. Cotangent, similarly, is defined for all numbers except π · k, k ∈ Z.

Basic functions of trigonometry

Sine, cosine, tangent and cotangent are the basic trigonometric functions.

It is usually clear from the context which argument of the trigonometric function (angular argument or numeric argument) we are dealing with.

Let's return to the definitions given at the very beginning and the alpha angle, which lies in the range from 0 to 90 degrees. The trigonometric definitions of sine, cosine, tangent, and cotangent are entirely consistent with the geometric definitions given by the aspect ratios of a right triangle. Let's show it.

Let's take a unit circle with a center in a rectangular Cartesian coordinate system. Let's rotate the starting point A (1, 0) by an angle of up to 90 degrees and draw a perpendicular to the abscissa axis from the resulting point A 1 (x, y). In the resulting right triangle, the angle A 1 O H is equal to the angle of rotation α, the length of the leg O H is equal to the abscissa of the point A 1 (x, y). The length of the leg opposite the angle is equal to the ordinate of the point A 1 (x, y), and the length of the hypotenuse is equal to one, since it is the radius of the unit circle.

In accordance with the definition from geometry, the sine of angle α is equal to the ratio of the opposite side to the hypotenuse.

sin α = A 1 H O A 1 = y 1 = y

This means that determining the sine of an acute angle in a right triangle through the aspect ratio is equivalent to determining the sine of the rotation angle α, with alpha lying in the range from 0 to 90 degrees.

Similarly, the correspondence of definitions can be shown for cosine, tangent and cotangent.

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Sinus acute angle α of a right triangle is the ratio opposite leg to hypotenuse.
It is denoted as follows: sin α.

Cosine The acute angle α of a right triangle is the ratio of the adjacent leg to the hypotenuse.
It is designated as follows: cos α.

Tangent acute angle α is the ratio of the opposite side to the adjacent side.
It is designated as follows: tg α.

Cotangent acute angle α is the ratio of the adjacent side to the opposite side.
It is designated as follows: ctg α.

The sine, cosine, tangent and cotangent of an angle depend only on the size of the angle.

Rules:

Basic trigonometric identities in a right triangle:

(α – acute angle opposite to the leg b and adjacent to the leg a . Side With – hypotenuse. β – second acute angle).

As the acute angle increasessin α andtan α increase, andcos α decreases.

For any acute angle α:

sin (90° – α) = cos α

cos (90° – α) = sin α

Example-explanation:

Let in a right triangle ABC
AB = 6,
BC = 3,
angle A = 30º.

Let's find out the sine of angle A and the cosine of angle B.

Solution .

1) First, we find the value of angle B. Everything is simple here: since in a right triangle the sum of the acute angles is 90º, then angle B = 60º:

B = 90º – 30º = 60º.

2) Let's calculate sin A. We know that the sine is equal to the ratio of the opposite side to the hypotenuse. For angle A, the opposite side is side BC. So:

BC 3 1
sin A = -- = - = -
AB 6 2

3) Now let's calculate cos B. We know that the cosine is equal to the ratio of the adjacent leg to the hypotenuse. For angle B, the adjacent leg is the same side BC. This means that we again need to divide BC by AB - that is, perform the same actions as when calculating the sine of angle A:

BC 3 1
cos B = -- = - = -
AB 6 2

The result is:
sin A = cos B = 1/2.

sin 30º = cos 60º = 1/2.

It follows from this that in a right triangle, the sine of one acute angle is equal to the cosine of the other acute angle - and vice versa. This is exactly what our two formulas mean:
sin (90° – α) = cos α
cos (90° – α) = sin α

Let's make sure of this again:

1) Let α = 60º. Substituting the value of α into the sine formula, we get:
sin (90º – 60º) = cos 60º.
sin 30º = cos 60º.

2) Let α = 30º. Substituting the value of α into the cosine formula, we get:
cos (90° – 30º) = sin 30º.
cos 60° = sin 30º.

(For more information about trigonometry, see the Algebra section)