The derivation process of cosine theorem

The derivation process of cosine theorem includes the application of basic properties, the derivation of cosine theorem and the triangle area formula.

1. Application of basic properties: We know that in any triangle ABC, the opposite sides of angles A, B and C are A, B and C respectively. According to the triangle interior angle sum theorem, A+B+C=π. Using the inductive formula, we can get that cos(π minus A)= negative cosA.

2. Derivation of Cosine Theorem: According to the above basic properties, we can write the formula of Cosine Theorem: a=b times cosA+c times cosB, b=a times cosB+c times cosA, and c=a times cosC+b times cosA. These formulas can be obtained by expressing the side length of a triangle as a function of angle, and then substituting these functions into the triangle area formula.

3. Triangle area formula: We use the triangle area formula S= 1/2bc multiplied by sinA= 1/2ac multiplied by sinB= 1/2ab multiplied by sinC to prove the cosine theorem. By comparing the cosine theorem with the triangle area formula, we can see that the two forms are the same, but the function of angle and the function of side length are interchanged. This proves the correctness of cosine theorem.

Application of cosine theorem:

1. Judging the shape of triangle: We can judge the shape of triangle by cosine theorem. If the lengths of three sides of a triangle are A, B and C respectively, and we know the lengths of two sides and the included angle A between them, then we can calculate the length of the third side C by cosine theorem ... If the length of C is equal to the sum of the lengths of A and B, then this triangle is an equilateral triangle. If the length of C is greater than or less than the sum of the lengths of A and B, then this triangle is an obtuse triangle or an acute triangle.

2. Solving the unknown quantity in triangle: Cosine theorem can be used to solve the unknown quantity in triangle, such as angle and side length. For example, if we know the lengths of three sides of a triangle are A, B and C, and the size of angle A, we can use cosine theorem to calculate the size of angle B. ..

3. Application in practical problems: Cosine theorem is also widely used in practical problems, such as physics, engineering, navigation and astronomy. For example, in physics, cosine theorem can be used to calculate the composition and decomposition of forces; In engineering, cosine theorem can be used to calculate stress and displacement in structures; In navigation, cosine theorem can be used to calculate the sailing direction and distance.