Instruction

1

To find

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

Solution:

According to the problem conditions we have:

a=3,

b=4,

C=5.

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.

**the cosine**of an angle in**a triangle**, the lengths of the sides are known, we can use the theorem**of the cosine of**s. 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.Solution:

According to the problem conditions we have:

a=3,

b=4,

C=5.

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.

2

If the triangle is rectangular, then to find

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:

c?=b?+huh?,

C=v(b?+huh?)

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.

a=3,

b=4.

Doing elementary calculations, we get:

cos?=0,8.

**the cosine of**a 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:

c?=b?+huh?,

C=v(b?+huh?)

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.

a=3,

b=4.

Doing elementary calculations, we get:

cos?=0,8.

3

Similarly, is

Example 3.

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

b?=with?-huh?,

b=v(c?-huh?)

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:

cos?=0,8.

**the cosine of**the rectangular**triangle**in other cases:Example 3.

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

b?=with?-huh?,

b=v(c?-huh?)

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:

cos?=0,8.

4

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

cos?=b/C.

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:

cos?=0,8

So all our formulas are correct.

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:cos?=b/C.

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:

cos?=0,8

So all our formulas are correct.