The first round review teaching plan of new mathematics in senior three (Lecture 24) —— Triangle identity deformation and its application
1. Curriculum standard requirements:
1. Experience the process of deriving the cosine formula of the difference between two angles by the product of vectors, and further understand the function of vector method;
2. We can deduce the sine, cosine and tangent formulas of the sum and difference of two angles and the sine, cosine and tangent formulas of two angles from the cosine formula of the difference of two angles, and understand their internal relations;
3. You can use the above formula to carry out simple identity transformation (including guiding and deducing product sum and difference, product sum and difference, and half-angle formula, but you don't need to remember).
Two. Propositional trend
Judging from the direction of college entrance examination in recent years, there are more opportunities to choose and solve these questions, sometimes in the form of fill-in-the-blank questions. They are often combined with the properties of trigonometric functions, trigonometric solutions and vectors. The main problem is the evaluation of trigonometric function, and the properties of trigonometric function are studied through trigonometric transformation.
The content of this lecture is one of the key points of college entrance examination review, and the simplification, evaluation and proof of trigonometric identities are the basic problems of trigonometric transformation. In the college entrance examination over the years, while studying the mastery and application of trigonometric formula, we also pay attention to the flexibility and divergence of thinking, as well as the ability of observation, operation and observation, operational reasoning and comprehensive analysis.
Step 3 get to the point
1. trigonometric function of sum and difference of two angles
2. Double angle formula
3. Simplification of trigonometric functions
Common methods: ① directly apply the formula to reduce the order and eliminate the term; (2) String cutting, homonym, homonym and abnormity; ③ Reverse use of trigonometric formula, etc. (2) Simplify requirements: ① Find the value if you can find it; ② Make the number of trigonometric functions as small as possible; (3) the number of items should be as small as possible; ④ Try to make the denominator not contain trigonometric function; ⑤ Try to make the root sign not contain trigonometric function.
(1) power reduction formula
; ; .
(2) Auxiliary angle formula
4. There are three types of trigonometric function evaluation.
Evaluation of (1) angle: generally, given angles are non-special angles, so it is necessary to observe the relationship between the given angle and the special angle, and eliminate the non-special angle through trigonometric transformation, which is transformed into the problem of finding the trigonometric function value of the special angle;
(2) Value evaluation: give trigonometric function values of some angles and find trigonometric function values of other angles. The key to solving problems lies in "changing angles", such as expressing angles with formulas containing known angles, and paying attention to the discussion of angle range when solving problems;
(3) Finding the angle with a given value: in essence, it is transformed into a problem of finding a given value, and the obtained function value of the angle is combined with the range of the angle and the monotonicity of the function to get the angle.
5. Proof of trigonometric equality
The idea of proving (1) trigonometric identity is to change "different" at both ends of the equation into "same" by transforming trigonometric identity and using the method of simplifying the complex and making the left and right equal.
(2) The idea of proving the trigonometric conditional equation is to find the relationship between the known condition and the equation to be proved through observation, and prove it by substitution method, parameter elimination method or analysis method.
Four. Typical case analysis
Question 1: trigonometric function of sum and difference of two angles
Example 1. Known, looking for cos.
Analysis: Because it can be regarded as a double angle, the following two solutions can be obtained.
The solution 1: given by the known sin+sin =1........................ ①,
cos+cos=0…………②，
① 2+② 2 to obtain 2+2co;
Company.
12-22 get cos 2+cos 2+2 cos()=- 1，
That is 2cos () [] =- 1.
∴。
Solution 2: From ①, we can get ③.
From ② to ④.
④