Advice 1: How to find the cosine of the angle

Cosine is one of the main trigonometric functions. The cosineom acute angle in a right triangle is the ratio of the adjacent leg to the hypotenuse. The definition of cosine is tied to a rectangular triangle, but often the angle whose cosine is necessary to determine, in a right triangle is not located. How to find the cosine of any angle?
How to find the cosine of the angle
If you want to find the cosine of an angle in a right triangle, you must use the definition of cosine to find the ratio of adjacent leg to the hypotenuse:
cos? = a/c where a is the length of the leg, the length of the hypotenuse.
If you want to find the cosine of angle in an arbitrary triangle, use the theorem of cosines:
if the angle is acute: cos? = (a2 + b2 – c2)/(2ab);
if the angle is blunt: cos? = (C2 – a2 – b2)/(2ab), where a and b are the lengths of the sides adjacent to the angle, side length opposite corner.
If you want to find the cosine of angle in an arbitrary geometrical figure, determine the magnitude of the angle in degrees or radians, and the cosine of the angle to find its value using scientific calculator, tables Bradis or any other mathematical application.
Useful advice
Mathematical notation of cosine cos.
The value of a cosine cannot be greater than 1 and less than -1.

Advice 2: How to find the cosine in the triangle

Often the geometric (trigonometric) tasks required to find the cosine of the angle in the triangle, because the cosine of the angle allows to determine the value of the angle.
Triangle ABC
To find the cosine of an angle in a triangle, the lengths of the sides are known, we can use the theorem of the cosine ofs. According to this theorem, the squared length of an arbitrary side of a triangle equals the sum of the squares of its two other sides without twice the product of the lengths of these sides into the cosine of the angle between them:

and?=b?+c?-2*b*c*cos?, where:

a, b, C be the sidelengths of a triangle (or rather their lengths),

? – the angle opposite the side a (its value).

From these equalities easily is cos?:

cos?=( b?+c?-huh? )/(2*b*c)

Example 1.

There is a triangle with sides a, b, C equal 3, 4, 5 mm respectively.

Find the cosine of the angle between the long sides.


According to the problem conditions we have:




We denote the opposite side and the angle across the? then, according to the formula derived above, we have:

cos?=(b?+c?-huh? )/(2*b*c)=(4?+5?-3?)/(2*4*5)=(16+25-9)/40=32/40=0,8

The answer of 0.8.
If the triangle is rectangular, then to find the cosine ofa angle is enough to know only the lengths of any two sides of (the cosine of a right angle is equal to 0).

Suppose you have a rectangular triangle with sides a, b, C, where C is the hypotenuse.

Consider all the options:

Example 2.

Find cos?, if you know the lengths of the sides a and b (sides of triangle)

We use advanced Pythagorean theorem:



cos?=(b?+c?-huh? )/(2*b*c)=(b?+b?+huh?-huh?)/(2*b*v(b?+a?))=(2*b?)/(2*b*v(b?+a?))=b/v(b?+huh?)

To check the correctness of the formula, substitute in the values from example 1, i.e.



Doing elementary calculations, we get:

Similarly, is the cosine of the rectangular triangle in other cases:

Example 3.

Known a and C (hypotenuse and opposite side), find cos?



cos?=(b?+c?-huh? )/(2*b*c)=(C?-a?+with?-huh?)/(2*s*v (? -a?))=(2*s?-2*a?)/(2*s*v (? -a?))=v (? -huh?)/C.

Substituting the values a=3 and C=5 from the first example, we get:

Example 4.

Famous b and C (the hypotenuse and adjacent side).

Find cos?

Producing the same (shown in examples 2 and 3 of the transition), we obtain that in this case the cosine in the triangle is calculated by a simple formula:


The simplicity of obtained formula is explained simple: in fact, adjacent to the corner ? side is the projection of the hypotenuse, so its length is equal to the length of the hypotenuse multiplied by cos?.

Substituting the values b=4 and C=5 from the first example, will get:


So all our formulas are correct.

Advice 3: How to find acute angle in a right triangle

Directlythe coal , the triangle is probably one of the most famous, from a historical point of view, geometric shapes. Pythagorean pants" competition may be only a "Eureka!" Archimedes.
How to find acute angle in a right triangle
You will need
  • drawing a triangle;
  • - the range;
  • - protractor.
As a rule, the vertices of the angles of a triangle are denoted by capital Latin letters (A, B, C), and the opposite side of the small Latin letters (a, b, c) or by the names of the vertices of the triangle constituting that side (AC, BC, AB).
The sum of the angles of a triangle is 180 degrees. In a rectangular triangle, one angle (straight) will always be 90 degrees and the other acute, i.e. less than 90 degrees each. To determine the angle in a rectangular triangle is straight, measure with a ruler the sides of the triangle and determine the greatest. It is called the hypotenuse (AB) and is located opposite the right angle (C). The other two sides form a right angle are called legs (AC, BC).
When it is determined what the angle is sharp, you can either measure the angle with a protractor, or calculated using mathematical formulas.
To determine the measure of the angle with a protractor, align the top (let's denote it by the letter A) with a special mark on the ruler in the center of the protractor, the side AC must coincide with its upper edge. Note on the circular part of the protractor to the point, through which the hypotenuse AB. The value at this point corresponds to the angle in degrees. If the protractor provided (2) the values for acute angle need to choose smaller, for the stupid - big.
The angle can be calculated by making simple mathematical calculations. You will need a basic knowledge of trigonometry. If you know the length of the hypotenuse AB and leg sun, calculate the value of the sine of the angle A: sin (A) = BC / AB.
The resulting value find the reference tables Bradis and determine what angle corresponds to the obtained number. This method was used by our grandmothers.
In our time, it is sufficient to take the calculator with the function of calculating trigonometric formulas. For example, the built-in Windows calculator. Run the application "Calculator" in the menu "View" select "Engineering". Calculate the sine of the desired angle, for example, sin (A) = BC/AB = 2/4 = 0.5
Switch calculator mode to inverse functions, click INV on the scoreboard calculator, then click calculate arcsine functions (on the scoreboard indicated, as sin to the minus one). In the window calculation will appear the following inscription: asind (0.5) = 30. I.e., the value of the desired angle of 30 degrees.

Advice 4: How to determine angles in a right triangle

The rectangular triangle is characterized by certain relationships between the angles and sides. Knowing the meaning of some of them, you can calculate the other. For this purpose, formula, based, in turn on the axioms and theorems of geometry.
How to determine angles in a right triangle
From the name of a right triangle it is clear that one of its angles is a direct. Regardless of whether it is an isosceles right triangle or not, it always has one angle equal to 90 degrees. If given a right triangle that is both isosceles and then, based on the fact that the figure is a straight angle, find two angles at its base. These angles are equal, so each of them has a value equal to:

α=180°- 90°/2=45°
In addition to the above, it is also possible the other case, when the triangle is rectangular, but is not isosceles. In many problems the angle of the triangle is 30° and the other 60°, because the sum of all angles in a triangle must equal 180°. If given the hypotenuse of a right triangle and side angle can be found from matching the two sides:

sin α=a/c where a is the side opposite to the hypotenuse of a triangle, with the hypotenuse of the triangle

Accordingly, α=arcsin(a/c)

Also the angle you can find the formula for cosine:

cos α=b/c where b is the adjacent side to the hypotenuse of the triangle
If you know only two sides, the angle α can be found according to the formula of the tangent. The tangent of this angle is equal to the ratio of opposite over adjacent:

tg α=a/b

From this it follows that α=arctg(a/b)

When the straight angle and one of the corners found by the above method, the second is as follows:

Is the advice useful?