Formula for finding the tangent of an angle in a right triangle. Sine, cosine, tangent and cotangent: definitions in trigonometry, examples, formulas
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 recall 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 indicated by the corresponding Greek letter.
Hypotenuse of a right triangle is the side opposite 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 acute angle in right triangle- this is an attitude opposite leg 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 of it trigonometric functions according to 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! IN Unified State Exam options in mathematics there are many problems where the sine, cosine, tangent or cotangent of the external angle of a triangle appears. More on this in the next article.
Trigonometry is a topic that many people avoid. Despite this, if you find her the right approach it will become very interesting for you. Trigonometric formulas, including formulas for finding the tangent, are used in many areas real life. This article will talk about ways to find the tangent of an angle and give examples of using this quantity in life. This will give you motivation to study this topic.
Despite the opinion that exists among most schoolchildren, trigonometry is used quite often in life. A good example practical application will give you an incentive not to be lazy. Here are several areas of activity where trigonometric calculations are used, including finding the tangent of an angle:
- Economy.
- Astronomy.
- Aviation.
- Engineering.
So, below are ways to find tg.
How to find the tg of an angle
Finding the tangent of an angle is quite simple. You can study this topic and just memorize the rules, but all this can fly out of your head during the exam. Therefore it is worth approaching this issue meaningful. Basic formulas to remember:
- tg0° = 0
- tg30° = 1/√3
- tg45° = 1
- tg60° = √3
- tg90° = ∞ (infinity/indefinite)
Please note that the values are ascending: the larger the angle, the greater the tangent value. Accordingly, with a degree value of an angle of 0°, we get 0. With a value of thirty degrees, we get one divided by the root of three, etc., until we reach 90°. With it, the tangent value is equal to infinity or uncertainty (based on the specific situation).
These expressions follow from the rule for finding the tangent through a right triangle. Thus, the tangent of angle A (tgA) is equal to the ratio of the opposite side to the adjacent side. Imagine that you are given a right triangle in which all the sides are known, but the corner is not known. To solve the problem, you need to find the tangent of angle A. The size of the side that lies opposite the angle is 1, and the size of the adjacent leg is √3. Their ratio gives 1/√3. We already know that the angle at this indicator is 30 degrees. Accordingly, angle A = 30°.
In a right triangle right angle both tangents are adjacent. The opposite side of this angle is the hypotenuse. Precisely because we cannot divide two legs into each other (the condition of finding will be violated), the tangent is 90° in in this case does not exist.
In addition to all this, you often have to find the tangent of an obtuse angle. Typically, problems involve obtuse angles of 120 or 150 degrees. The formula for finding the tangent of an obtuse angle is as follows: tg(180-a) = tga.
For example, we need to find the tangent of 120°. The question you need to ask yourself is: how much do you need to subtract from 180 to get 120? Definitely 60°. It follows that tangent 120° and tangent 60° are equal to each other and tan120° = √3. Using the same logic, you can find a tangent of 150 and 180 degrees. Their values will be equal to 1/√3 and 0, respectively. The values of the tangents of other angles are given in trigonometric table, but they are used extremely rarely.
How to find tg of an angle online
There are many online resources for finding the tangent of an angle. One of these is the FXYZ website. Follow this link. You will see a page where the basic formulas related to tangent will be given, as well as a calculator. Using the calculator is quite simple. You must enter the appropriate ones and the calculator will calculate the answer. This simple algorithm will help you if you forgot something. There are two calculators on this site. One is for finding the tangent value based on the lengths of the legs of the triangle, and the second is based on the angle value. Use the computer that the task requires.
As you may have noticed, finding the tangent and other trigonometric indicators is very often used in real life, and finding these values is not at all difficult. If you understand the essence of the finding, then you won’t have to memorize anything - you yourself will be able to reach the correct answer. If something still doesn’t work out, use a calculator, but don’t overuse it. On an exam, test or school test work No one will give you such an opportunity. Moreover, if you enroll in a department where trigonometry is studied higher mathematics, without basic knowledge you will have to work hard to avoid getting cut off.
The straight line y=f(x) will be tangent to the graph shown in the figure at point x0 if it passes through the point with coordinates (x0; f(x0)) and has an angular coefficient f"(x0). Find such a coefficient, Knowing the features of a tangent, it’s not difficult.
You will need
- - mathematical reference book;
- - a simple pencil;
- - notebook;
- - protractor;
- - compass;
- - pen.
Instructions
If the value f‘(x0) does not exist, then either there is no tangent, or it runs vertically. In view of this, the presence of a derivative of the function at the point x0 is due to the existence of a non-vertical tangent tangent to the graph of the function at the point (x0, f(x0)). In this case, the angular coefficient of the tangent will be equal to f "(x0). Thus, the geometric meaning of the derivative becomes clear - the calculation of the angular coefficient of the tangent.
Draw additional tangents that would be in contact with the graph of the function at points x1, x2 and x3, and also mark the angles formed by these tangents with the x-axis (this angle is counted in the positive direction from the axis to the tangent line). For example, the angle, that is, α1, will be acute, the second (α2) will be obtuse, and the third (α3) equal to zero, since the tangent line is parallel to the OX axis. In this case, the tangent of an obtuse angle is negative, the tangent of an acute angle is positive, and at tg0 the result is zero.
note
Correctly determine the angle formed by the tangent. To do this, use a protractor.
Two inclined lines will be parallel if their angular coefficients are equal to each other; perpendicular if the product angular coefficients these tangents are equal to -1.
Sources:
- Tangent to the graph of a function
Cosine, like sine, is classified as a “direct” trigonometric function. Tangent (together with cotangent) is classified as another pair called “derivatives”. There are several definitions of these functions that make it possible to find the tangent given by known value cosine of the same value.
Instructions
Subtract the quotient of unity by the value raised to the cosine of the given angle, and extract the square root from the result - this will be the tangent value of the angle, expressed by its cosine: tan(α)=√(1-1/(cos(α))²) . Please note that in the formula the cosine is in the denominator of the fraction. The impossibility of dividing by zero precludes the use of this expression for angles equal to 90°, as well as those differing from this value by numbers that are multiples of 180° (270°, 450°, -90°, etc.).
There is also alternative way calculating the tangent from a known cosine value. It can be used if there is no restriction on the use of others. To implement this method, first determine the angle value from a known cosine value - this can be done using the arc cosine function. Then simply calculate the tangent for the angle of the resulting value. IN general view this algorithm can be written as follows: tg(α)=tg(arccos(cos(α))).
There is also an exotic option using the definition of cosine and tangent through the acute angles of a right triangle. In this definition, cosine corresponds to the ratio of the length of the leg adjacent to the angle under consideration to the length of the hypotenuse. Knowing the value of the cosine, you can select the corresponding lengths of these two sides. For example, if cos(α) = 0.5, then the adjacent can be taken equal to 10 cm, and the hypotenuse - 20 cm. The specific numbers do not matter here - you will get the same and correct numbers with any values that have the same . Then, using the Pythagorean theorem, determine the length of the missing side - the opposite leg. It will be equal square root from the difference between the lengths of the squared hypotenuse and the known leg: √(20²-10²)=√300. By definition, tangent corresponds to the ratio of the lengths of the opposite and adjacent legs (√300/10) - calculate it and get the tangent value found using the classical definition of cosine.
Sources:
- cosine through tangent formula
One of the trigonometric functions, most often denoted by the letters tg, although tan is also used. The easiest way to represent the tangent is as a sine ratio angle to its cosine. This is an odd periodic and non-continuous function, each cycle of which is equal to the number Pi, and the break point corresponds to half of this number.
Tangent- this is one of the trigonometric functions . Initially, trigonometric functions express the dependencies of the elements of right triangles - sides and angles. In a right triangle legs - these are the sides forming a right angle, hypotenuse - Third side. Then tangent of the angle- this is the ratio of the opposite side to the adjacent side. Thus, it is a dimensionless quantity, i.e. it is not measured in degrees or meters, it is just a number. Denoted as tg . To solve many geometric and mathematical problems, you need to calculate the tangent of an angle. You can find it in different ways.
Necessary:
- calculator;
— MS Excel;
— basic knowledge in mathematics, geometry and trigonometry.
Instructions:
- This quantity can be defined as the ratio sine angle to cosine the same angle. If they are known, then the required characteristic can be calculated using the formula tg(a)=sin(a)/cos(a).
- The value can be calculated using engineering calculator. To do this, dial the number and press the key tg. The tangent value can be as large or small as desired, but for angles that are multiples of 90 degrees, this characteristic does not exist.
- The value of tg can be determined from the graph of the function Y=tg(x). To do this you need on the axis X find the value of the angle for which you are looking this characteristic, draw a perpendicular from this point to the x-axis ( OX axis) until it intersects with the graph, then from the intersection point draw a perpendicular to the ordinate axis ( OY axis). Meaning Y at this point will be the desired tangent value.
- How to find the tangent of an angle if you don’t have a calculator at hand? You can calculate it in the program Excel . Type in any cell =tan(radians(a)), Where A— the number from which the characteristic value is sought, press Enter. The value of this value will appear in the cell.
- Also, trigonometric functions are sometimes defined in terms of ranks . This allows you to calculate their value with any accuracy. For example, if we expand the tangent into Taylor series , then the first terms of this series will be x+1/3*x^2+2/15*x^5+… The sum of this infinite series can be calculated using properties of limits .
The tangent of an angle is a number that is determined by the ratio of the opposite and adjacent legs in a triangle to this angle. Knowing only this relationship, it is possible to find out the magnitude of the angle, say, using the trigonometric function inverse to the tangent - arctangent.
Instructions
1. If you have Bradis tables on hand in paper or electronic form, then determining the angle will come down to searching for the value in the tangent table. The angle value will be compared to it - that is, what needs to be detected.
2. If there are no tables, then you will have to calculate the arctangent value. You can use, say, a standard calculator from the Windows OS for this. Open the main menu by clicking the “Start” button or pressing the WIN key, go to the “All programs” section, then to the “Typical” subsection and select “Calculator”. The same can be done through the program launch dialog - press the WIN + R key combination or select the “Run” line in the main menu, type the calc command and press the Enter key or click the “OK” button.
3. Switch the calculator to the mode that allows you to calculate trigonometric functions. To do this, open the “View” section in its menu and select “Engineer” or “Scientist” (depending on the version of the operating system used).
4. Enter famous meaning tangent This can be done either from the keyboard or by clicking the necessary buttons on the calculator interface.
5. Make sure that the “Degrees” field is checked so that you get the calculation result in degrees, and not in radians or grads.
6. Check the checkbox labeled Inv - this will invert the values of the calculated functions indicated on the calculator buttons.
7. Click the button labeled tg (tangent) and the calculator will calculate the value of the inverse tangent function - arctangent. This will be the desired angle.
8. All this can be done using online trigonometric function calculators. Finding such services on the Internet is quite easy with help search engines. And some of the search engines (say, Google) themselves have built-in calculators.
Websites have such a complex system that it is sometimes difficult to detect them The main thing menu. More often than not, such an item is located in the “header” of the site for a quick transition to it. In some cases, the transition is carried out by opening the main page, it all depends on the type of site.
You will need
- – browser;
- - Internet connection.
Instructions
1. Go to the main page of the site and find a link to menu. It can also be located directly on it. Occasionally The main thing menu may be hidden in a drop-down list, to view it you will need to click on the link to expand it. Sometimes it looks like an ordinary Windows Explorer, and to move through its items or to view the table of contents, you will need to click on the plus sign next to the directory name.
2. If you are on a certain page of the site and cannot find a link to go to the main page, carefully look at its table of contents and find a link in the form of a logo or an ordinary text name of the source. You can also go to the main page by entering the main site address in the appropriate line of your browser.
3. Please note that many sites may contain multiple menu, let's say menu user profile settings, where it is indicated personal information and login details, and menu site to navigate through its content. In the first case, this may be a link to manage your profile or edit personal data, parameters account and so on. In the second - ordinary menu, which organizes content so that you can navigate through sections according to their purpose.
4. If you need to locate a sitemap, check the main page for a link to it. Many of them easily do not contain a site map, due to the fact that they are rarely used. To go to main menu site, also pay attention to its main functions, links to which are saved as you navigate through the pages. When you are in a certain thread of a forum, you can follow the links at the top or bottom of the block with topics; usually the folder tree of the subforum in which you are located is written there.
Helpful advice
Use the menu on the main page.
The tangent of an angle, like other trigonometric functions, expresses the relationship between the sides and angles of a right triangle. The use of trigonometric functions allows you to replace quantities in degree measurement with linear parameters in calculations.
Instructions
1. If you have a protractor, you can measure this angle of the triangle and use the Bradis table to determine the tangent value. If it is not possible to determine the degree value of the angle, determine its tangent using measurements of the linear values of the figure. To do this, make auxiliary constructions: from an arbitrary point on one side of the angle, lower a perpendicular to the other side. Measure the distance between the ends of the perpendicular on the sides of the angle, write the result of the measurement in the numerator of the fraction. Now measure the distance from the vertex of the given angle to the vertex of the right angle, that is, to the point on the side of the angle at which the perpendicular was dropped. Write the resulting number into the denominator of the fraction. The fraction compiled based on the results of measurements is equal to the tangent of the angle.
2. The tangent of an angle can be determined by calculation as the ratio of the opposite leg to the adjacent one. It is also possible to calculate the tangent through direct trigonometric functions of the angle in question - sine and cosine. Tangent of the angle equal to the ratio the sine of this angle to its cosine. Unlike the constant functions of sine and cosine, the tangent has a break and is not defined at an angle of 90 degrees. When the angle is zero, its tangent is zero. From the relations of a right triangle, it is clear that an angle of 45 degrees has a tangent, equal to one, from the fact that the legs of such a right triangle are equal.
3. For angle values from 0 to 90 degrees, its tangent has positive meaning, from the fact that the sine and cosine in this interval are positive. The limits of tangent metamorphosis in this area are from zero to infinitely large values at angles close to right. For negative angle values, its tangent also changes sign. Graph of the function Y=tg(x) on the interval -90°
