Power function diagram

Exponential function diagram

Logarithmic function diagram

Trigonometric function diagram

Sign of trigonometric function value in each quadrant

sinα cscα cosα secα tanα cotα

Properties of trigonometric functions

The function y = sinxy = cosxy = tanxy = cotx.

The region r r {x | x ∈ r and x ≠ kπ+, k ∈ z} {x | x ∈ r and x ≠ kπ, k ∈ z}

When the range [- 1, 1] x = 2kπ+, Ymax= 1

When x=2kπ-, Ymin=- 1

[- 1, 1]

Ymax= 1 when x=2kπ.

When x=2kπ+π, Ymin=- 1.

rare

There is no maximum value

There is no minimum R.

There is no maximum value

There is no minimum

Periodicity is 2π, periodicity is 2π, periodicity is π, periodicity is π.

Parity odd function even function odd function odd function

Monotonicity is increasing function in [2kπ-, 2kπ+]; The decreasing function (k ∈ z) in [2kπ+, 2kπ+π] is increasing function in [2kπ-π, 2kπ]; In [2kπ, 2kπ+ π], it is a decreasing function (k∈Z), in (kπ-, kπ+) it is a increasing function, and in (kπ, kπ+ π) it is a decreasing function (k ∈ z).

Inverse trigonometric function graph

Properties of inverse trigonometric functions

Arcsine function, arccosine function, arctangent function, anti-cotangent function

The inverse function of y = sinx (x ∈ [-,]) is called arcsine function, and the inverse function of x = arsiny = cosx (x ∈ [0, π]) is called anti-cosine function, and it is called x = x = arccosyy = tanx (x ∈ (-x ∈).

Understand that arcsinx means belonging to [-,]

And the angle arccosx with sine value equal to x indicates that it belongs to [0, π], the angle arctanx with cosine value equal to x indicates that it belongs to (-,), the angle arccotx with tangent value equal to x indicates that it belongs to (0, π), and the cotangent value is equal to X.

Domain [- 1, 1] [- 1, 1] (-∞, +∞) (-∞, +∞)

Range [-,] [0, π] (-,) (0, π)

Monotonicity is the increasing function on [- 1, 1], the subtraction function on [- 1, 1], the increasing function on (-∞, +∞) and the subtraction function on (-∞, +∞).

Parity arcsin (-x) =-arcsinx arccos (-x) = π-arccosx arctan (-x) =-arctanx arccot (-x) = π-arccotx.

Periodicity is not a synchronous function.

The identity sin (arcsinx) = x (x ∈ [-1]) arcsin (sinx) = x (x ∈ [-,]) cos (arccosx) = x (x ∈ [-65438].

arccot(cotx)=x(x∈(0，π))

Reciprocal identity arcsinx+arccosx = (x ∈ [- 1]) Arctanx+Arccotx = (x ∈ r)

formulas of trigonometric functions

Two-angle sum formula

sin(A+B) = sinAcosB+cosAsinB

sin(A-B) = sinAcosB-cosAsinB

cos(A+B) = cosAcosB-sinAsinB

cos(A-B) = cosAcosB+sinAsinB

tan(A+B)= 1

Tan ()

cot(A+B)= 1

Kurt (A-B)

Double angle formula

tan2A =

Sin2A=2SinA？ Kosa

Cos2A = Cos2A-Sin2A = 2 Cos2A- 1 = 1-2 Sin2A

Triple angle formula

sin3A = 3sinA-4(sinA)3

cos3A = 4(cosA)3-3cosA

tan3a = tana tan(+a) tan(-a)

half-angle formula

sin()=

cos()=

Tan () =

cot()=

tan()==

Sum difference product

sina+sinb=2sincos

sina-sinb=2cossin

cosa+cosb = 2coscos

cosa-cosb =-2 sin

tana+tanb=

Sum and difference of products

sinasinb = -[cos(a+b)-cos(a-b)]

cosacosb = [cos(a+b)+cos(a-b)]

sinacosb = [sin(a+b)+sin(a-b)]

cosasinb = [sin(a+b)-sin(a-b)]

Inductive formula

Sin(-a)=- Sina

cos(-a) = cosa

sin(-a) = cosa

Cos(-a) = Sina

sin(+a) = cosa

Cos(+a)=- Sina

sin(π-a) = sina

cos(π-a) = -cosa

sin(π+a) = -sina

cos(π+a) = -cosa

tgA=tanA =

General formula of trigonometric function

Sina =

cosa=

tana=

Other formulas

Answer? Sina +b? Cosa =×sin(a+c)[ where tanc=]

Answer? Crime (A)-B? Cos(a)=×cos(a-c)[ where tan(c)=]

1+sin(a) =(sin+cos)2

1-sin(a) = (sin-cos)2

Other non-critical trigonometric functions

CSC(a)= 1

Seconds (a) = seconds

Hyperbolic function

sinh(a)= 1

cosh(a)= 1

TG h(a)= 1

Formula one

Let α be an arbitrary angle, and the values of the same trigonometric function with the same angle of the terminal edge are equal:

sin(2kπ+α)= sinα

cos(2kπ+α)= cosα

tan(2kπ+α)= tanα

cot(2kπ+α)= cotα

Formula 2

Let α be an arbitrary angle, and the relationship between the trigonometric function value of π+α and the trigonometric function value of α;

Sine (π+α) =-Sine α

cos(π+α)= -cosα

tan(π+α)= tanα

cot(π+α)= cotα

Formula 3

The relationship between arbitrary angle α and the value of-α trigonometric function;

Sine (-α) =-Sine α

cos(-α)= cosα

tan(-α)= -tanα

Kurt (-α) =-Kurt α

Formula 4

The relationship between π-α and the trigonometric function value of α can be obtained by Formula 2 and Formula 3:

Sine (π-α) = Sine α

cos(π-α)= -cosα

tan(π-α)= -tanα

Kurt (π-α) =-Kurt α

Formula 5

The relationship between the trigonometric function values of 2π-α and α can be obtained by Formula-and Formula 3:

Sine (2π-α)=- Sine α

cos(2π-α)= cosα

tan(2π-α)= -tanα

Kurt (2π-α)=- Kurt α

Formula 6

α and the relationship between α and the trigonometric function value of α;

sin(+α)= cosα

cos(+α)= -sinα

tan(+α)= -cotα

cot(+α)= -tanα

sin(-α)= cosα

cos(-α)= sinα

tan(-α)= cotα

cot(-α)= tanα

sin(+α)= -cosα

cos(+α)= sinα

tan(+α)= -cotα

cot(+α)= -tanα

sin(-α)= -cosα

cos(-α)= -sinα

tan(-α)= cotα

cot(-α)= tanα

(higher than k∈Z)

It took me a long time to input this common formula in physics, hoping it will be useful to everyone.

Answer? sin(ωt+θ)+ B？ sin(ωt+φ) =×sin

Formulas of trigonometric functions Certificate (All)

Formula expression

Multiplication and factorization

a2-B2 =(a+b)(a-b)a3+B3 =(a+b)(a2-a b+B2)a3-B3 =(a-b)(a2+a b+B2)

Triangle inequality

|a+b|≤|a|+|b|

|a-b|≤|a|+|b|

| a |≤b & lt； = & gt-b≤a≤b

|a-b|≥|a|-|b|

-|a|≤a≤|a|

Solution of quadratic equation in one variable

-b+√(B2-4ac)/2a-b-b+√(B2-4ac)/2a

Relationship between root and coefficient

X 1+X2=-b/a

X 1*X2=c/a

Note: Vieta theorem.

Discriminant b2-4a=0 Note: The equation has two equal real roots.

B2-4ac >0 Note: The equation has real roots.

B2-4ac & lt； 0 Note: The equation has multiple yokes.

formulas of trigonometric functions

Two-angle sum formula

sin(A+B)= Sina cosb+cosa sinb sin(A-B)= Sina cosb-sinb cosa

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)

Double angle formula

tan2A = 2 tana/( 1-tan2A)ctg2A =(ctg2A- 1)/2c TGA

cos2a = cos2a-sin2a = 2 cos2a- 1 = 1-2 sin2a

half-angle formula

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))tan(A/2)=-√(( 1-cosA)/(( 1+cosA))

ctg(A/2)=√(( 1+cosA)/(( 1-cosA))ctg(A/2)=-√(( 1+cosA)/(( 1-cosA))

Sum difference product

2 Sina cosb = sin(A+B)+sin(A-B)2 cosa sinb = sin(A+B)-sin(A-B)

2 cosa cosb = cos(A+B)-sin(A-B)-2 sinasinb = cos(A+B)-cos(A-B)

sinA+sinB = 2 sin((A+B)/2)cos((A-B)/2 cosA+cosB = 2 cos((A+B)/2)sin((A-B)/2)

tanA+tanB = sin(A+B)/cosa cosb tanA-tanB = sin(A-B)/cosa cosb

ctgA+ctgBsin(A+B)/Sina sinb-ctgA+ctgBsin(A+B)/Sina sinb

The sum of the first n terms of some series

1+2+3+4+5+6+7+8+9+…+n = n(n+ 1)/2

1+3+5+7+9+ 1 1+ 13+ 15+…+(2n- 1)= N2

2+4+6+8+ 10+ 12+ 14+…+(2n)= n(n+ 1)

12+22+32+42+52+62+72+82+…+N2 = n(n+ 1)(2n+ 1)/6

13+23+33+43+53+63+…n3 = N2(n+ 1)2/4

1 * 2+2 * 3+3 * 4+4 * 5+5 * 6+6 * 7+…+n(n+ 1)= n(n+ 1)(n+2)/3

sine law

a/sinA=b/sinB=c/sinC=2R

Note: where r represents the radius of the circumscribed circle of the triangle.

cosine theorem

B2 = a2+C2-2 acco b

Note: Angle B is the included angle between side A and side C..

Tangent law

[(a+b)/(a-b)]= {[Tan(a+b)/2]/[Tan(a-b)/2]}

the standard equation of the circle

(x-a)2+(y-b)2=r2 Note: (a, b) is the central coordinate.

Circular general equation

X2+y2+Dx+Ey+F=0 Note: D2+E2-4f > 0

Parabolic standard equation

y2=2px y2=-2px x2=2py x2=-2py

Transverse area of right prism

S=c*h

Oblique prism side area

S=c'*h

Side area of regular pyramid

S= 1/2c*h '

Transverse area of regular prism

S= 1/2(c+c')h '

Yuantai lateral area

s = 1/2(c+c’)l = pi(R+R)l

Surface area of ball

S=4pi*r2

Cylindrical side area

S=c*h=2pi*h

Cone lateral area

S= 1/2*c*l=pi*r*l

Arc length formula

l=a*r

A is the radian number r>0 of the central angle.

Sector area formula

s= 1/2*l*r

Cone volume formula

V= 1/3*S*H

Cone volume formula

V= 1/3*pi*r2h

Oblique prism volume

V=S'L

Note: where s' is the area of straight section and l is the length of side.

Cylinder volume formula

V=s*h

cylinder

V=pi*r2h

-

Trigonometric function? Product sum difference sum difference product formula

If you can't remember it, push it yourself, and use the sine and cosine of the sum of two angles:

cos(A+B)=cosAcosB-sinAsinB

cos(A-B)=cosAcosB+sinAsinB

By adding or subtracting these two formulas, the sum and difference of two sets of products can be obtained:

Addition: cosAcosB=[cos(A+B)+cos(A-B)]/2

Subtraction: sinasinb =-[cos (a+b)-cos (a-b)]/2

sin(A+B)=sinAcosB+sinBcosA

sin(A-B)=sinAcosB-sinBcosA

By adding or subtracting these two formulas, the sum and difference of two sets of products can be obtained:

Addition: sinAcosB=[sin(A+B)+sin(A-B)]/2.

Subtraction: sinBcosA=[sin(A+B)-sin(A-B)]/2

Such ***4 groups are sum-difference integrals, and then the sum-difference integrals are reversed.

I don't know if it's to make you remember the exam, but you can also infer it temporarily when you really can't remember the exam.

Jia Zhengqian

Positive subtraction and positive remainder are in front.

What's more, what's more.

There is nothing left to reduce, and it is still negative.

The positive surplus is positive and JUNG WOO is negative.

Plus, plus, plus, minus, plus, minus.

.

3. Some conclusions in the triangle: (no need to remember)

( 1)anA+tan b+ tanC = tanA tanB tanC

(2)sinA+tsin B+sinC = 4 cos(A/2)cos(B/2)cos(C/2)

(3)cosA+cosB+cosC = 4 sin(A/2)sin(B/2)sin(C/2)+ 1

(4)sin2A+sin2B+sin2C = 4 Sina sinB sinC

(5)cos2A+cos2B+cos2C =-4 cosacosbcosc- 1

...........................

It is known that sin α = m sin (α+2 β), | m | < 1, and verified that tan (α+β) = (1+m)/(kloc-0/-m) tan β.

Solution: sinα=m sin(α+2β)

sin(a+β-β)=msin(a+β+β)

sin(a+β)cosβ-cos(a+β)sinβ= msin(a+β)cosβ+mcos(a+β)sinβ

sin(a+β)cosβ( 1-m)= cos(a+β)sinβ(m+ 1)

tan(α+β)=( 1+m)/( 1-m)tanβ

There is a full version of word document. You need to add me Q7 1032 1030.