cos(A+B)= cosa cosb-Sina sinb cos(A-B)= cosa cosb+Sina sinb
tan(A+B)=(tanA+tanB)/( 1-tanA tanB)tan(A-B)=(tanA-tanB)/( 1+tanA tanB)ctg(A+B)=(ctgActgB- 1)/(ctg B+ctgA)ctg(A-B)=(ctgActgB+ 1)/(ctg B-ctgA)
tan2A = 2 tana/( 1-tan2A)ctg2A =(ctg2A- 1)/2 ctga cos2a = cos2a-sin2a = 2 cos2a- 1 = 1-2 sin2a
sin(A/2)=√(( 1-cosA)/2)sin(A/2)=-√(( 1-cosA)/2)cos(A/2)=√(( 1+cosA)/2)cos(A/2)=√(( 1+cosA)/2)tan(A/2)=√(( 1-cosA)/(( 1+cosA
2sinAcosB=sin(A+B)+sin(A-B)
2cosAsinB=sin(A+B)-sin(A-B)
2cosAcosB=cos(A+B)-sin(A-B)
-2sinAsinB=cos(A+B)-cos(A-B)
sinA+sinB = 2 sin((A+B)/2)cos((A-B)/2
cosA+cosB = 2cos((A+B)/2)sin((A-B)/2)
tanA+tanB=sin(A+B)/cosAcosB
tanA-tanB=sin(A-B)/cosAcosB
ctgA+ctgB=sin(A+B)/sinAsinB
-ctgA+ctgB=sin(A+B)/sinAsin
The concepts of set and function
I. Collection of related concepts
1, meaning of set: some specified objects are set together into a set, where each object is called an element.
2. Collective
Three characteristics of elements:
1. element determinism; 2. Basic
; 3. Basic
Description: (1) For a given set, the elements in the set are certain, and any object is either an element of the given set or not.
(2) In any given set, any two elements are different objects. When the same object is contained in a set, it has only one element.
(3) The elements in the set are equal and have no order. So to judge whether two sets are the same, we only need to compare whether their elements are the same, and we don't need to examine whether the arrangement order is the same.
(4) The three characteristics of set elements make the set itself deterministic and holistic.
1) Formula for sum and difference of two angles (remember everything written)
sin(A+B)=sinAcosB+cosAsinB
sin(A-B)=sinAcosB-sinBcosA？
cos(A+B)=cosAcosB-sinAsinB
cos(A-B)=cosAcosB+sinAsinB
tan(A+B)=(tanA+tanB)/( 1-tanA tanB)
tan(A-B)=(tanA-tanB)/( 1+tanA tanB)
2) Using the above formula, the following contents can be deduced.
tan2A=2tanA/[ 1-(tanA)^2]
cos2a=(cosa)^2-(sina)^2=2(cosa)^2 - 1= 1-2(sina)^2
(The cosine above is very important)
sin2A=2sinA*cosA
(3)
Just remember this:
Tan(A/2)=( 1-cosA)/ Sina = Sina /( 1+cosA)
4) Cosine in double angles can be used to derive.
(sinA)^2=( 1-cos2A)/2
(cosA)^2=( 1+cos2A)/2
5) Use the words above
The following commonly used simplified formulas can be derived
1-cosA=sin^(A/2)*2
1-sinA=cos^(A/2)*2
+
1) Formula for sum and difference of two angles (remember everything written)
sin(A+B)=sinAcosB+cosAsinB
sin(A-B)=sinAcosB-sinBcosA？
cos(A+B)=cosAcosB-sinAsinB
cos(A-B)=cosAcosB+sinAsinB
tan(A+B)=(tanA+tanB)/( 1-tanA tanB)
tan(A-B)=(tanA-tanB)/( 1+tanA tanB)
2) Using the above formula, the following contents can be deduced.
tan2A=2tanA/[ 1-(tanA)^2]
cos2a=(cosa)^2-(sina)^2=2(cosa)^2 - 1= 1-2(sina)^2
(The cosine above is very important)
sin2A=2sinA*cosA
(3)
Just remember this:
Tan(A/2)=( 1-cosA)/ Sina = Sina /( 1+cosA)
4) Cosine in double angles can be used to derive.
(sinA)^2=( 1-cos2A)/2
(cosA)^2=( 1+cos2A)/2
5) Use the words above
The formula can be derived from the following commonly used simplified formulas.
1-cosA=sin^(A/2)*2
1-sinA=cos^(A/2)*2
3. Expression of assembly: {…} such as {basketball player of our school}, {Pacific,
, Indian Ocean,
}
1. Use
Exhibition set: a={ basketball players in our school}, b={ 1, 2, 3, 4, 5}
2. Representation method of set:
and
.
Note: Commonly used number sets and their symbols:
(i.e.
) remember: n
N* or n+
z
q
r
On the concept of "belonging"
Elements in a collection are usually lowercase.
For example, if A is an element of set A, it is said that A belongs to set A and is marked as A ∈ A. On the contrary, A does not belong to set A and is marked as a(a
List the elements in the collection one by one, and then use the
Attached.
: describes the common properties of the elements in the collection and writes them to.
A method of representing an internal set. A method to indicate whether some objects belong to this set under certain conditions.
① language
Example: {A triangle that is not a right triangle}
② Description of mathematical formula: If the solution set of inequality x-3]2 is {x(r| x-3]2} or {x| x-3]2}.
4, the classification of the set:
1.
A collection with a finite number of elements.
2.
A collection containing an infinite number of elements.
3. An example of an empty set without any elements: {x|x2=-5}
Second, the basic relationship between sets
1. "Inclusion" relation-
Note: There are two possibilities that A is a part of B (1); (2)a and B are the same set.
On the other hand, set A is not included in set B, or set B does not include set A, which is marked as ab or ba.
2. "Equality" relationship (5≥5, and 5≤5, then 5=5)
Example: let a = {x | x2-1= 0} b = {-1,1} "The elements are the same".
Conclusion: For two sets A and B, if any element of set A is an element of set B and any element of set B is an element of set A, we say that set A is equal to set B, that is, A = B.
(1) any setting is its own.
. one
②
If a(b) and a( b), then set A is set B.
, recorded as ab (or ba)
③ If a(b, b(c), then a(c
(4) if a(b) and b(a), then a = b.
3. A set without any elements is called
Set, recorded as φ.
Rule: An empty set is any set.
An empty set is arbitrary
about
.
Third, the operation set
Definition of 1. intersection: Generally speaking, the set consisting of all elements belonging to A and B is called the intersection of A and B. 。
Write it as a∩b (pronounced as "A crosses B"), that is, a∩b={x|x∈a, x∈b}.
2,
Definition: Generally speaking, a set consisting of all elements belonging to set A or set B is called A and B..
Note: a∪b (pronounced as "A and B") means a∪b={x|x∈a, or x∈b}.
3. Intersection sum
A ∩ A = A，A ∩ φ = φ，A ∩ B = B ∩ A，A ∪ φ = A，A ∪ B = B ∪ A .
4. Complete works and
( 1)
Let s be a set, and A is a subset of S (that is, a subset of S), which consists of all that do not belong to A..
A set named s
Setting a
(or remainder)
Note: csa is csa ={x (x(s) and x (a)}
(2) Complete set: If the set S contains all the elements of each set we want to study, it can be regarded as a complete set, usually expressed by U. 。
(3) Properties: (1) cu (cua) = a2 (cua) ∩ a = φ 3 (cua) ∪ a = u.