The trigonometric function relationship in triangle is one of the key contents of college entrance examination over the years. This section mainly helps candidates to deeply understand sine and cosine theorems and master the methods and skills of solving oblique triangles.
● Difficult magnetic field
(★★★★★★★) It is known that the three internal angles A, B and C of △ABC satisfy A+C=2B. Find the value of cos.
● Case studies
[example1] There is a mountain at an altitude of 1 km on Island A, and there is an observation station P at the top. 1 1 am, a ship was detected at 30° B east of the island, with a dip angle of 60, and arrived at 1 1 am.
(1) Find the ship speed, in km/h;
(2) after a period of time, the ship arrived in the west direction of D Island. How far is the ship from island a at this time?
Proposition intention: This topic mainly examines the basic knowledge of triangles and students' ability to comprehensively use triangle knowledge to read maps and solve practical problems.
Knowledge support: the triangle relation of triangle is mainly used, and the key is to find the right orientation and make rational use of the angle relation.
Analysis of wrong questions: the examinee's orientation identification is inaccurate and the calculation is easy to make mistakes.
Skills and methods: mainly according to the angle relationship in the triangle, use sine theorem to solve the problem.
Solution: (1) in Rt△PAB, ∠ APB = 60 Pa = 1, ∴AB= (km).
In Rt△PAC, ∠ APC = 30, ∴AC= (km).
In △ACB, ∠ cab = 30+60 = 90.
(2)DAC = 90-60 = 30
sin DCA = sin( 180-∠ACB)= Sina CB =
sinCDA=sin(∠ACB-30 )=sinACB？ cos30 -cosACB？ sin30。
In delta delta δ△ACD, according to sine theorem,
∴
Answer: At this time, the ship is 0/000 km away from Island A/KLOC.
[Example 2] It is known that the three internal angles A, B and C of △ABC satisfy A+C=2B, let x=cos and f(x)=cosB ().
(1) Try to find the analytical formula of function f(x) and its definition domain;
(2) judging its monotonicity and proving it;
(3) Find the range of this function.
Proposition intention: This question mainly examines the examinee's ability to solve comprehensive problems by using triangular knowledge, and examines the examinee's flexible application of basic knowledge and the examinee's computing ability, which belongs to the category of ★★★★★★.
Knowledge dependence: mainly based on the related formulas and properties of trigonometric functions and the related properties of functions to solve problems.
Analysis of wrong questions: it is difficult for candidates to flexibly use the relevant formulas in trigonometric functions, and it is not easy to think of using the monotonicity of functions to find the range of functions.
Skills and methods: The key of this question is to find the analytical formula of f(x) by using the related formula of trigonometric function. The formula is mainly sum-difference product and sum-difference formula. Pay attention to the scope of || when solving the domain.
Solution: (1)∵A+C=2B, ∴ B = 60, A+C = 120.
∵0 ≤| | 0，x 1-x2 < 0，∴ f (x2
That is, f (x2) < f (x 1), if x 1, x2∈ (,1), then 4x 12-3 > 0.
4x22-3>0,4x 1x2+3>0,x 1-x2