Summary of knowledge points of trigonometric function in high school mathematics 1. Acute angle formula of trigonometric function
Opposite side/hypotenuse of sine =
Adjacent edge/hypotenuse of cos=
Opposite side of tan =/adjacent side of tan =
Adjacent side of cot =/opposite side of cot =
Second, the double angle formula
Sin2A=2SinA？ Kosa
Cos2A = cos a2-Sina 2 = 1-2 Sina 2 = 2 cos a2- 1
Tan2A=(2tanA)/( 1-tanA2) (Note: SinA2 is the square of Sina sin2(A)).
Triple or triple angle formula
sin 3 = 4 sinin(/3+)sin(/3-)
cos3=4coscos(/3+)cos(/3-)
tan3a=tanatan(/3+a)tan(/3-a)
Derivation of triple angle formula
sin3a
=sin(2a+a)
=sin2acosa+cos2asina
Auxiliary angle formula
Asin+bcos = (A2+B2) (1/2) sin (+t), where
Sinter =B/(A2+B2)( 1/2)
Cost =A/(A2+B2)( 1/2)
tant=B/A
asin+Bcos =(A2+B2)( 1/2)cos(-t)，tant=A/B
Fourthly, the power decreasing formula.
sin 2()=( 1-cos(2))/2 = versin(2)/2
cos 2()=( 1+cos(2))/2 = covers(2)/2
tan 2()=( 1-cos(2))/( 1+cos(2))
Derived formula
tan+cot=2/sin2
tan-cot=-2cot2
1+cos2=2cos2
1-cos2=2sin2
1+ sine = (sine /2+ cosine /2)2
= 2 Sina( 1-Sina)+( 1-2 Sina)Sina
= 3 sinar -4 sinar
cos3a
=cos(2a+a)
=cos2acosa-sin2asina
=(2 cosa- 1)cosa-2( 1-Sina)cosa
=4cosa-3cosa
sin3a=3sina-4sina
=4sina(3/4-sina)
= 4 Sina[(3/2)- Sina]
=4sina(sin60-sina)
=4sina(sin60+sina)(sin60-sina)
= 4 Sina * 2 sin[(60+a)/2]cos[(60-a)/2]* 2 sin[(60-a)/2]cos[(60-a)/2]
=4sinasin(60+a)sin(60-a)
cos3a=4cosa-3cosa
= 4 Xhosa (Xhosa -3/4)
=4cosa[cosa-(3/2)]
=4cosa(cosa-cos30)
=4cosa(cosa+cos30)(cosa-cos30)
= 4 cosa * 2cos[(a+30)/2]cos[(a-30)/2]* {-2 sin[(a+30)/2]sin[(a-
30)/2]}
=-4 eicosapentaenoic acid (a+30) octyl (a-30)
=-4 Coxsacin [90-(60-a)] Xin [-90+(60+a)]
=-4 cos(60-a)[-cos(60+a)]
= 4 cos(60-a)cos(60+a)
Comparing the above two formulas, we can get
tan3a=tanatan(60-a)tan(60+a)
Verb (abbreviation of verb) half-angle formula
tan(A/2)=( 1-cosA)/sinA = sinA/( 1+cosA)；
cot(A/2)= sinA/( 1-cosA)=( 1+cosA)/sinA。
sin2(a/2)=( 1-cos(a))/2
cos2(a/2)=( 1+cos(a))/2
tan(a/2)=( 1-cos(a))/sin(a)= sin(a)/( 1+cos(a))
Six, triangle and
sin(++)= sincoscos+cossincos+coscoscossin
Xin Xin Xin
cos(++)= coscoscos-cossinsin-sincossin-sinsincos
tan(++)=(tan+tan+tan-tantan tan)/( 1-tantan-tantan-tantan)
Seven, the sum and difference of two angles
cos(+)=coscos-sinsin
cos(-)=coscos+sinsin
sin()=sincoscossin
tan(+)=(tan+tan)/( 1-tantan)
tan(-)=(tan-tan)/( 1+tantan)
Eight, sum and difference product
sin+sin=2sin[(+)/2]cos[(-)/2]
sin-sin=2cos[(+)/2]sin[(-)/2]
cos+cos=2cos[(+)/2]cos[(-)/2]
cos-cos=-2sin[(+)/2]sin[(-)/2]
tanA+tanB = sin(A+B)/cosa cosb = tan(A+B)( 1-tanA tanB)
tanA-tanB = sin(A-B)/cosa cosb = tan(A-B)( 1+tanA tanB)
Nine, product and difference
sinsin =[cos(-)-cos(+)/2
coscos=[cos(+)+cos(-)]/2
sincos=[sin(+)+sin(-)]/2
cossin=[sin(+)-sin(-)]/2
X. inductive formula
Sin (-) =-Sin
cos(-)=cos
tan(—a)=-tan
sin(/2-)=cos
cos(/2-)=sin
sin(/2+)=cos
cos(/2+)=-sin
Sin (-) = sin
cos(-)=-cos
Sin (+) =-sin
cos(+)=-cos
tanA=sinA/cosA
tan(/2+)=-cot
tan(/2-)=cot
Brown (-) =-Brown
Brown (+) = Brown
Inductive formula memory skills: odd variables are unchanged, and symbols look at quadrants.
XI。 General formula
sin=2tan(/2)/[ 1+tan(/2)]
cos =[ 1-tan(/2)]/ 1+tan(/2)]
tan=2tan(/2)/[ 1-tan(/2)]
Twelve. Other formulas
( 1)(sin)2+(cos)2= 1
1+(tan)2= (seconds) 2
(3) 1+(cot)^2=(csc)^2
(4) For any non-right triangle, there is always
tanA+tanB+tanC=tanAtanBtanC
Certificate:
A+B=-C
tan(A+B)=tan(-C)
(tanA+tanB)/( 1-tanA tanB)=(tan-tanC)/( 1+tantanC)
Surface treatment can be carried out.
tanA+tanB+tanC=tanAtanBtanC
Obtain a certificate
It can also be proved that this relationship holds when x+y+z=n(nZ).
The following conclusions can be drawn from tana+tanbtana+tanb+tanc = tanatanbtanc.
(5)cotAcotB+cotAcotC+cotbctc = 1
(6) Cost (A/2)+ Cost (B/2)+ Cost (C/2)= Cost (A/2) Cost (B/2)
(7)(cosA)2+(cosB)2+(cosC)2 = 1-2 cosacasbcosc
(8)(sinA)2+(sinB)2+(sinC)2 = 2+2cosAcosBcosC
(9)sin+sin(+2/n)+sin(+2 * 2/n)+sin(+2 * 3/n)++ sin[+2 *(n- 1)/n]= 0
Cos+cos (+2/n)+cos (+2 * 2/n)+cos (+2 * 3/n)+cos [+2 * (n-1)/n] = 0 and
sin 2()+sin 2(-2/3)+sin 2(+2/3)= 3/2
tanAtanBtan(A+B)+tanA+tan B- tan(A+B)= 0
Extended reading: a way to learn functions well 1. Learning math is like playing games. If you want to play a good game, you must first be familiar with the rules of the game.
In mathematics, the rules of the game are the so-called basic definitions. If you want to learn functions well, you must first master the basic definitions and corresponding image features, such as definition domain, value domain, parity, monotonicity, periodicity, symmetry axis and so on.
Many students have entered the misunderstanding of learning function, thinking that as long as they master the problem-solving methods, they can learn mathematics well. In fact, they should master the most basic definitions before they can learn how to solve problems well. In the final analysis, all problem-solving methods should start from the basic definitions, and it is best to master the algebraic expressions and image characteristics of these definitions and properties.
Second, keep in mind several basic elementary functions and their related properties, images and transformations.
There are several basic elementary functions in middle school: linear function (linear equation), quadratic function, inverse proportional function, exponential function, logarithmic function, sine and cosine function and tangent cotangent function. All function problems are based on these functions, but in different forms, and finally they can be solved through basic knowledge.
There are also three kinds of functions, which are not found in textbooks. They often appear in college entrance examination and self-enrollment examination: y=ax+b/x, a function with absolute value and a cubic function. The properties of these functions, such as domain, range, monotonicity, parity, etc., and the characteristics of images should be studied well.
Third, image is the soul of function! If we want to learn and do function problems well, we must pay full attention to the problem of function image.
Looking through the function questions of college entrance examination over the years, we can find that there is a calculation, and almost 80% of the function questions are related to images. This requires students to pay more attention to the image of the function when learning the function, and to be able to draw, see and use pictures! Translation, scaling, flipping, rotation, composition and superposition of multi-attention function images.