[Boutique] Narrative and prove the cosine theorem

Part 1: Describe and prove the cosine theorem

Plane vector proof

As shown in the figure, there is a+b=c ∴c·c=·

∴c^2=a·a+2a·b+b·b∴c^2=a^2+b^2+2|a||b|Cos

Also ∵Cos=-Cosθ

∴c2=a2+b2-2|a||b|Cosθ

Then disassemble, get c2=a2+b2-2*a*b*CosC

That is, CosC=/2*a*b

The same can be said to be the other, and the following CosC=/2ab is to move CosC to the left to indicate it.

Plane geometry proof

In any △ABC

Do AD⊥BC.

The side to which ∠C is is c, the side to which ∠B is opposite is b, and the side to which ∠A is is a

Then there is BD=cosB*c, AD=sinB*c, DC=BC-BD=a-cosB*c

According to the Pythagorean theorem:

AC2=AD2+DC2

B2=2+2

B2=2+a2-2ac*cosB+2*c2

B2=*c2-2ac*cosB+a2

B2=c2+a2-2ac*cosB

cosB=/2ac

PART 2: Describe and prove the cosine theorem

For any triangle, the square of either side is equal to the sum of the squares of the other two sides minus the two-fold product of the cosines of the two sides with their angles. If the three sides are a, b, and the c triangle is A, B, C, then the property is satisfied -

a^2 = b^2+ c^2 - 2·b·c·cosA

b^2 = a^2 + c^2 - 2·a·c·cosB

c^2 = a^2 + b^2 - 2·a·b·cosC

cosC = /

cosB = /

cosA = /

First cosine theorem

Let the three sides of △ABC be a, b, and c, and the angles they are facing are A, B, and C, respectively.

a=b·cos C+c·cos B, b=c·cos A+a·cos C, c=a·cos B+b·cos A.

PART 3: Describe and prove the cosine theorem

Knowing the three sides of a triangle, you can find three interior angles.

The third side is obtained by knowing the two sides and the angle of the triangle.

It is known that the two sides of the triangle and one side of the triangle are opposite to each other, and the other corners and the third side can be obtained.

Decision theorem one:

If m is c, the two values ​​are the number of positive roots, c1 is the expression of c in the expression of the root before the plus sign, and c2 is the expression of c before the root number

Minus value

1 If m=2, there are two solutions

2 If m=1, there is a solution

3 If m=0, there is a zero solution.

Note: If c1 is equal to c2 and c1 or c2 is greater than 0, this case counts to the second case, which is a solution.

Decision Theorem 2:

When a>bsinA

1 When b>a and cosA>0, there are two solutions

2 When b>a and cosA<=0, there is a zero solution

3 When b=a and cosA>0, there is a solution

4 When b=a and cosA<=0, there is a zero solution

5 when b two when a=bsinA

1 When cosA>0, there is a solution

2 When cosA<=0, there is a zero solution

For example, it is known that the ratio of the three sides of ΔABC is 5:4:3, and the maximum internal angle is obtained.

Solve the three sides of the triangle as a, b, c and a: b: c = 5: 4: 3.

From the big side of the triangle to the big angle, ∠A is the largest angle. Cosine theorem

Cos A=0

So ∠A=90°.

Another example is △ABC, AB=2, AC=3, ∠A=60 degrees, and find the length of BC.

The solution is known by the cosine theorem

BC2=AB2+AC2-2AB×AC·cos A

=4+9-2×2×3×cos60

=13-12x0.5

=13-6

=7

So BC=√7.

The above two small examples simply illustrate the role of the cosine theorem.