How to prove the cosine theorem analytically?

Prove the cosine theorem.

Teacher: In the process of introduction, we not only found the relationship between the corners of oblique triangles, but also gave a proof, which was based on the method of classification discussion, which classified oblique triangle into the sum and difference of two right triangles, and then proved it by using the Pythagorean theorem and acute trigonometric functions. This is a good way to prove the cosine theorem, but it is more cumbersome.

Now that we have learned about trigonometric functions, whether a is an acute, right, or obtuse angle, we have a uniform definition, and we can avoid categorical discussions by using trigonometric functions and the distance between two fixed points to prove the cosine theorem.

We're still going to prove that C is the main thing.

We place the vertex c at the origin and ca falls on the positive semi-axis of the x-axis, since ac=b,cb=a,ab=c of abc, then a, b, c sit on a(b,0), b(acos c,asin c),c(0,0)

Please analyze how the coordinates of point b are obtained.

Birth: acb= c, cb is the terminal edge of acb, b is a point on cb, let the coordinates of b be (x,y), then sinc= =, cos c== so the coordinates of b point x=acosc,y=asinc

Teacher: Very accurate, how to find the distance between points A and B?

Raw: ab 2 = (acosc-b)2 + (asinc-0)2

a2cos2c-2abcosc+b2-a2sin2c

a2+b2-2abcos c., i.e., c2=a2+b2-2abcos c

Teacher: Everyone, please see, we have also derived the cosine theorem here, and this proof method is the analytic method. This method will be studied in detail in the future.

The cosine theorem can be described in language as follows: the square of one side of a triangle is equal to the sum of the squares of the other two sides, minus 2 times the product of the two sides and the cosine of the angle, i.e.

a2=b2+c2-2bccos a.

c2=a2+b2-2abcos c.

b2=a2+c2-2accos b.

The upstairs is already nice.

The formula of the cosine theorem is proved to be: vector method, trigonometric function method, auxiliary circle method for graphing.

The cosine theorem is an important theorem to reveal the relationship between the corners of triangles, and it can be used directly to solve a class of problems of finding the third side or finding the angle of the two sides and the angles of a known triangle.

1. Vector method; Vector cosine formula: cosa=b c, can also be written as cosa=ac ab. Cosine is a type of trigonometric function.

In RT abc (right triangle), c = 90°, the cosine of a is the hypotenuse of the triangle that accommodates its adjacent edge. The cosine theorem, the fundamental theorem of Euclidean plane geometry. The cosine theorem is a mathematical theorem describing the relationship between the length of three sides in a triangle and the cosine value of an angle, and is a generalization of the Pythagorean theorem in the case of a general triangle, and the Pythagorean theorem is a special case of the cosine theorem.

2. Trigonometric function method; The formula of the cosine theorem of trigonometric function is cosa=(b +c -a) 2bc; cosa = adjacent edge than hypotenuse. The cosine theorem of trigonometric functions formulas the elimination of years: f(x)=cosx(xer).

Cosine (cosine function), a type of trigonometric function. In RT ABC (Right Triangle), Zc = 90°, the cosine of ZA is the hypotenuse of its adjacent edge than the triangle, i.e. COSA=BLC, which can also be written as COSA=ACIAB.

3. Assist in the drawing of circular bridge sheds; Auxiliary circle method is a commonly used method of drawing, through the introduction of auxiliary circles to solve a method of drawing problems, for some drawing problems, in the analysis or drawing, it is necessary to introduce auxiliary circles to determine the relative position of certain points, line segments or angles, and use this method to solve the drawing problems, which is called auxiliary circle method drawing. A special case of auxiliary circle plotting is the drifting tangent method (see "Drifting tangent plotting").

The four methods of proving the cosine theorem are introduced as follows:

The cosine theorem formula proves that there are only three methods: vector method, trigonometric function method, and auxiliary circle method for graphing.

The cosine theorem is an important theorem to reveal the relationship between the corners of triangles, and it can be used directly to solve a class of problems of finding the third side or finding the angle of the two sides and the angles of a known triangle.

1. Vector method; Vector cosine formula: cosa=b c, can also be written as cosa=ac ab. Cosine is a type of trigonometric function.

In RT ABC (right triangle), c = 90°, the cosine of a is the hypotenuse of its adjacent edge than the triangle. The cosine theorem, the fundamental theorem of Euclidean plane geometry. The cosine cavity burial theorem is a mathematical theorem describing the relationship between the length of three sides and the cosine value of an angle in a triangle, which is a generalization of the Pythagorean theorem in the case of a general triangle, and the Pythagorean theorem is a special case of the cosine theorem.

2. Trigonometric function method; The formula of the cosine theorem of trigonometric function is cosa=(b +c -a) 2bc; cosa = adjacent edge than hypotenuse. Trigonometric cosine theorem formula: f(x)=cosx(xer).

Cosine (cosine function), a type of trigonometric function. In RT ABC (right triangle), Zc = 90°, the cosine of ZA is the hypotenuse of its adjacent side than the triangle, i.e. cosa=blc, or cosa=aciab.

3. Auxiliary circle method drawing; Auxiliary circle method is a commonly used method of drawing, through the introduction of auxiliary circles to solve a method of drawing problems, for some drawing problems, in the analysis or drawing, it is necessary to introduce auxiliary circles to determine the relative position of certain points, line segments or angles, and use this method to solve the drawing problems, which is called auxiliary circle method drawing. A special case of auxiliary circle plotting is the wandering tangent method.