Advice 1: How to solve problems with the cosines of the

Most of the problems with the cosines need to be solved in geometry. If this concept is used in other Sciences, e.g. in physics, then apply geometric methods. Usually applies the theorem of the cosines or ratios in a right triangle.
How to solve problems with the cosines of the
You will need
  • - knowledge of the Pythagorean theorem, theorem of cosines;
  • - trigonometric identities;
  • calculator or tables Bradis.
With cosine, you can find any of the sides of a right triangle. To do this, use the mathematical relationship that States that the cosine of an acute angle triangle is the ratio of adjacent sides to the hypotenuse. Therefore, knowing the acute angle of a right triangle, find the side of it.
For example, the hypotenuse of a right triangle is 5 cm and the acute angle 60º with her. Locate adjacent to the acute corner side. To do this, use the definition of cosine cos(α)= b/a, where a is the hypotenuse of a right triangle, b is the side adjacent to the angle α. Then its length will be equal to b=a∙cos(α). Substitute values b=5∙cos(60º)= 5∙0,5=2,5 cm
The third side C, which is the second leg find using the Pythagorean theorem c=√(52-2,52)≈4.33 cm
Using the spherical law of cosines, you can find sides of triangles if you know two sides and the angle between them. To find the third side, find the sum of the squares of the two known sides, subtract from it twice the product multiplied by the cosine of the angle between them. From the result, extract the square root.
Example In triangle two sides are equal a=12 cm, b=9 cm Angle between them is 45º. Find the third side c. To find the third side use the cosine theorem c=√(a2+b2-a∙b∙cos(α)). Performing a lookup will get c=√(122+92-12∙9∙cos(45º))≈12,2 cm
When solving problems with cosines, use of identity, allowing to pass away from this trigonometric function to another, and Vice versa. The basic trigonometric identity: cos2(α)+sin2(α)=1; the ratio of tangent and cotangent: tg(α)=sin(α)/cos(α), ctg(α)=cos(α)/sin(α), etc. To find the value of the cosines of the angles use a special calculator or table Bradis.

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.
Is the advice useful?