Formulas of trigonometric functions table

Basic relations of trigonometric functions with the same angle

Reciprocal relation: quotient relation: square relation:

tanα？ cotα= 1

sinα？ cscα= 1

cosα？ secα= 1 sinα/cosα= tanα= secα/CSCα

cosα/sinα= cotα= CSCα/secαsin 2α+cos 2α= 1

1+tan2α=sec2α

1+cot2α=csc2α

(Hexagon mnemonic method: the graphic structure is "upper chord cut, Zuo Zheng middle cut,1"; The product of two functions on the diagonal is1; The sum of squares of trigonometric function values of two vertices on the shadow triangle is equal to the square of trigonometric function value of the next vertex; The trigonometric function value of any vertex is equal to the product of the trigonometric function values of two adjacent vertices. " )

Inductive formula (formula: odd variable couple, sign according to quadrant. )

Sine (-α) =-Sine α

cos(-α)=cosα tan(-α)=-tanα

Kurt (-α) =-Kurt α

sin(π/2-α)=cosα

cos(π/2-α)=sinα

tan(π/2-α)=cotα

cot(π/2-α)=tanα

sin(π/2+α)=cosα

cos(π/2+α)=-sinα

tan(π/2+α)=-cotα

cot(π/2+α)=-tanα

Sine (π-α) = Sine α

cos(π-α)=-cosα

tan(π-α)=-tanα

cot(π-α)=-coα

Sine (π+α) =-Sine α

cos(π+α)=-cosα

tan(π+α)=tanα

cot(π+α)=cotα

sin(3π/2-α)=-cosα

cos(3π/2-α)=-sinα

tan(3π/2-α)=cotα

cot(3π/2-α)=tanα

sin(3π/2+α)=-cosα

cos(3π/2+α)=sinα

tan(3π/2+α)=-cotα

cot(3π/2+α)=-tanα

Sine (2π-α)=- Sine α

cos(2π-α)=cosα

tan(2π-α)=-tanα

Kurt (2π-α)=- Kurt α

sin(2kπ+α)=sinα

cos(2kπ+α)=cosα

tan(2kπ+α)=tanα

cot(2kπ+α)=cotα

(where k∈Z)

General formula for sum and difference of formulas of trigonometric functions's two angles

sin(α+β)=sinαcosβ+cosαsinβ

sin(α-β)=sinαcosβ-cosαsinβ

cos(α+β)=cosαcosβ-sinαsinβ

cos(α-β)=cosαcosβ+sinαsinβ

tanα+tanβ

tan(α+β)=———

1-tanα？ tanβ

tanα-tanβ

tan(α-β)=———

1+tanα？ tanβ

2 tons (α/2)

sinα=————

1+tan2(α/2)

1-tan2(α/2)

cosα=————

1+tan2(α/2)

2 tons (α/2)

tanα=————

1-tan2(α/2)

Sine, cosine and tangent formulas of half angle; Power reduction formula of trigonometric function

Sine, cosine and tangent formulas of double angles Sine, cosine and tangent formulas of triangle

sin2α=2sinαcosα

cos 2α= cos 2α-sin 2α= 2 cos 2α- 1 = 1-2 sin 2α

2tanα

tan2α=———

1-tan2α

sin3α=3sinα-4sin3α

cos3α=4cos3α-3cosα

3tanα-tan3α

tan3α=————

1-3tan2α

Sum and difference product formula of trigonometric function

α+β α-β

sinα+sinβ=2sin——？ Because-

2 2

α+β α-β

sinα-sinβ=2cos——？ Sin-

2 2

α+β α-β

cosα+cosβ=2cos——？ Because-

2 2

α+β α-β

cosα-cosβ=-2sin——？ Sin-

2 2 1

sinα？ cosβ=-[sin(α+β)+sin(α-β)]

2

1

cosα？ sinβ=-[sin(α+β)-sin(α-β)]

2

1

cosα？ cosβ=-[cos(α+β)+cos(α-β)]

2

1

sinα？ sinβ=--[cos(α+β)-cos(α-β)]

2

Convert asinα bcosα into trigonometric function form of angle (formulas of trigonometric functions of auxiliary angle

Set, function

Set simple logic

Any x∈A x∈B is marked as a B.

A B，B A A=B

A b = {x | x ∈ a, and x∈B}

A b = {x | x ∈ a, or x∈B}

Card (A B)= Card (A)+ Card (B)- Card (A B)

(1) proposition

If the original proposition is p, then q

If q is the inverse proposition of p

If p is q, there is no proposition.

If the negative proposition is q, then p.

(2) the relationship between the four propositions

(3)A B, A is a sufficient condition for B to be established.

B A, a is a necessary condition for B.

A B, a is the necessary and sufficient condition of b.

Property Exponent and Logarithm of Function

(1) domain, range, corresponding rules

(2) Monotonicity

For any x 1, x2∈D

If X 1 < X2F (X 1) < F (X2), then F (X) is called increasing function on D.

If x 1 < x2 f(x 1) > f (x2), then f(x) is said to be a decreasing function on d.

(3) Parity

If f (-x) = f(x), f(x) is called an even function for any X in the domain of function F (x).

If f (-x) =-f(x), then f(x) is called odd function.

(4) periodicity

For any x in the definition domain of function f(x), if there is a constant t that makes f(x+t) = f (x), it is said that f(x) is a fractional exponential power of periodic function (1).

The significance of positive fractional exponential power is

The significance of negative fractional exponential power is

(2) The nature and algorithm of logarithm.

loga(MN)=logaM+logaN

logaMn=nlogaM(n∈R)

Exponential function logarithmic function

(1) y = ax (a > 0, a≠ 1) is called exponential function.

(2)x∈R，y>0

Image transfer (0, 1)

When a > 1, x > 0, y > 1; When x 1, y = ax is an increasing function.

0 < a < 1, y = ax is a decreasing function (1), and y = logax (a > 0, a≠ 1) is a logarithmic function.

(2)x>0，y∈R

Image transfer (1, 0)

When a > 1, x > 1, y > 0; 0 1, y = logax is increasing function.

0 < a < 1, y = logax is a decreasing function.

Exponential equation and logarithmic equation

fundamental form

logaf(x)=b f(x)=ab(a>0，a≠ 1)

Same bottom type

logaf(x)= logag(x)f(x)= g(x)> 0(a > 0，a≠ 1)

Substitution type f (ax) = 0 or f (logax) = 0.

Ordered sequence

Basic concept of arithmetic progression of sequence.

The general formula of (1) series an = f (n)

(2) Recursive formula of sequence

(3) The relationship between the general term formula of the sequence and the sum of the first n terms.

an+ 1-an=d

an=a 1+(n- 1)d

A, a and b are equal. 2A = A+B

m+n=k+l am+an=ak+al

Common summation formulas of geometric series

an=a 1qn_ 1

The proportion of A, G and B is equal G2 = AB.

M+n=k+l Oman = Aka

inequality

Basic properties of inequalities Important inequalities

a>b bb，b>c

a>b a+c>b+c

a+b>c a>c-b

a>b，c>d

a>b，c>0 ac>bc

a>b，c0，c>d>0 acb>0 dn>bn(n∈Z，n> 1)

a>b>0 > (n∈Z，n> 1)

(a-b)2≥0

a，b∈R a2+b2≥2ab

|a|-|b|≤|a b|≤|a|+|b|

Basic methods of proving inequality

comparative law

(1) To prove the inequality A > B (or A < B), just prove it.

A-b > 0 (or a-b < 0 =)

(2) if B > 0, to prove that A > B, just prove that,

To prove a < b, prove it.

Synthesis method is a method to deduce the inequality to be proved (from cause to effect) from the known or proved inequality according to the nature of inequality.

Analytical method is to seek the sufficient conditions for the conclusion to be established, and gradually seek the sufficient conditions for the required conditions to be established until the required conditions are known to be correct, which is obviously manifested as "holding the fruit"

plural

Algebraic form triangular form

A+bi =c+ Adi = c and B = D.

(a+bi)+(c+di)=(a+c)+(b+d)i

(a+bi)-(c+di)=(a-c)+(b-d)i

(a+bi)(c+di )=(ac-bd)+(bc+ad)i

a+bi = r(cosθ+isθ)

r 1 =(cosθ 1+isθ 1)？ R2(cosθ2+isθ2)

=r 1？ R2〔cos(θ 1+θ2)+isin(θ 1+θ2)〕

〔r(cosθ+sinθ)〕n = rn(cosnθ+isinnθ)

k=0， 1，…，n- 1

Analytic geometry

1, straight line

Linear equation of the distance between two points and a fixed fractional point

|AB|=| |

|P 1P2|=

y-y 1=k(x-x 1)

y=kx+b

The positional relationship between two straight lines, including angle and distance.

Or k 1 = k2, and b 1≠b2.

L 1 coincides with l2.

Or k 1 = k2 and b 1 = B2.

L 1 intersects with l2.

Or k 1≠k2

l2⊥l2

Or k 1k2 =- 1L 1 to l2.

The angle between l 1 and l2.

Distance from point to straight line

2. Conic curve

Circular ellipse

The standard equation (x-a) 2+(y-b) 2 = R2.

The center of the circle is (a, b) and the radius is r.

The general equation x2+y2+dx+ey+f = 0.

Where the center of the circle is (),

Radius r

(1) Use the distance d from the center of the circle to the straight line and the radius r of the circle to judge or use the discriminant to judge the positional relationship between the straight line and the circle.

(2) Use the sum and difference of center distance d and radius to judge the positional relationship between two circles.

Focus F 1 (-c, 0), F2(c, 0)

(b2=a2-c2)

weird

collinearity equation

The focal radius | mf 1 | = a+ex0, | mf2 | = a-ex0.

Hyperbolic parabola

hyperbola

Focus F 1 (-c, 0), F2(c, 0)

(a，b>0，b2=c2-a2)

weird

collinearity equation

The focal radius | mf 1 | = ex0+a, | mf2 | = ex0-a parabola y2 = 2px (p > 0).

Focus f

collinearity equation

translation of axes

Here (h, k) is the coordinate of the origin of the new coordinate system in the original coordinate system.

1. Set elements have ① certainty ② mutual difference ③ disorder.

2. Set representation ① enumeration ② description

③ Wayne diagram ④ number axis method

3. Set operation

⑴A∩(B∪C)=(A∪B)∩(A∪C)

⑵ Cu(A∩B)=CuA∪CuB

Cu(A∪B)= CuA∪CuB

4. The nature of the set

⑴ Number of subsets of n-tuple set: 2n.

Proper subset number: 2n-1; Nonempty proper subset number: 2n-2.

A summary of senior high school mathematics concepts

I. Functions

1. If there are n elements in set A, the number of all different subsets of set A is, and the number of all non-empty proper subset is.

The image symmetry axis equation of quadratic function is and the vertex coordinates are. When using the undetermined coefficient method to find the analytic expression of quadratic function, there are three methods to find the analytic expression, namely sum (vertex).

2. Power function, when n is positive odd number, m is positive even number, m

3. The approximate image of the function is

From the image, the range of the function is, the monotonic increasing interval is, and the monotonic decreasing interval is.

Second, trigonometric functions

1. Establish a rectangular coordinate system with the vertex of the angle as the coordinate origin and the starting edge as the positive semi-axis of the X axis. Take any point different from the origin on the last side of the angle, and record the distance from point P to the origin as sin =, cos =, tg =, ctg =, sec =, csc =.

2. In the relation of trigonometric functions with the same angle, the square relation is:,,;

Reciprocal relationships are:,,;

The division relation is:,.

3. The inductive formula can be summarized in ten words: odd even, and the sign looks at the quadrant. For example =,.

4. The maximum value of the function is, the minimum value is, the period is, the frequency is, the phase is, and the initial phase is; The symmetry axis of the image is a straight line, and the intersection of the image and the straight line is the symmetry center of the image.

5, the monotone interval of trigonometric function:

The increasing interval of is, and the decreasing interval is; The increasing interval of is, the decreasing interval is, the increasing interval is and the decreasing interval is.

6、

7. The double angle formula is: sin2 =

cos2 = = =

tg2 = .

8. The triple angle formula is: sin3 = cos3 =

9. The half-angle formula is: sin = cos =

tg = = = .

10, and the formula for raising power is:

1 1, and the formula of decreasing power is:

12, general formula: sin = cos = tg =

13、sin( )sin( )=，

cos( )cos( )= = .

14、 = ;

= ;

= 。

15、 = 。

16、sin 180= .

17, trigonometric function value of special angle:

sin 0 1 0

cos 1 0 0

Tg 0 1 does not exist 0 does not exist.

Ctg does not exist 1 0 0 does not exist.

18, the sine theorem is (where r represents the radius of the circumscribed circle of a triangle):

19, from the first form of cosine theorem, =

From the second form of cosine theorem, cosB=

20. the area of △ ABC is represented by s, the radius of circumscribed circle is represented by r, the radius of inscribed circle is represented by r, and the half circumference is represented by p:

① ; ② ;

③ ; ④ ;

⑤ ; ⑥

2 1, the projective theorem in trigonometry: in △ABC, …

22. In △ABC, …

23. In △ABC:

24, product and difference formula:

① ,

② ,

③ ,

④ 。

25. Sum-difference product formula:

① ,

② ,

③ ,

④ 。

Third, the inverse trigonometric function.

The domain of 1 is [- 1, 1], and the range of values is, odd function and increasing function;

The domain of is [- 1, 1], and the range of values is, odd or even, negative function;

The domain of is R, and the scope is odd function and increasing function;

The definition domain of is r, the value domain is odd or even, and the subtraction function.

2. when;

For anyone, there are:

When?

3. Solution set of the simplest trigonometric equation:

Fourth, inequality.

1. If n is a positive odd number, can it be deduced from? (yes)

What if n is a positive even number (only if all the numbers are non-negative)?

2. Can the same inequality be subtracted or divided (no)?

Can you add it up? (yes)

Can you multiply? (Yes, but with conditions)

3, the average inequality of two positive numbers is:

The average inequality of three positive numbers is:

The average inequality of n positive numbers is:

4. The relationship between harmonic mean, geometric mean, arithmetic mean and root mean square of two positive numbers is

6, two-way inequality is:

At that time, the left side got an equal sign, at that time, the right side got an equal sign.

Verb (abbreviation for verb) order

1, the general formula of arithmetic progression is, and the summation formula of the first n terms is: =.

2. The general formula of geometric series is,

The first n terms and formulas are:

3. When the common ratio Q of geometric series satisfies < At 1, =S=. Generally speaking, if the limit of the sum of the first n terms of an infinite series exists, it is called the sum of the terms of this series (or the sum of all terms), which is expressed by S, that is, S=.

4. If m, n, p, q∈N, and, then: if the series is arithmetic progression, there is; When the series is geometric series, there are.

5. In arithmetic progression, if Sn= 10 and S2n=30, then S3n = 60；;

6. In geometric series, if Sn= 10 and S2n=30, then S3n = 70；;

Plural intransitive verb

1, how to calculate? (Find the remainder obtained by dividing n by 4 first. )

2. Are two imaginary cube roots of 1, and:

3. The triangle inequality in the complex set is: where the left side is equal to the vector * * * line and the opposite direction (same direction) corresponding to the complex number z 1 and z2, and the right side is equal to the vector * * * line and the opposite direction (opposite direction) corresponding to the complex number z 1 and z2.

4, Dimov theorem is:

5. If it is not a zero complex number, then the n power of z has n roots, namely:

What is the special relationship between their corresponding points on the complex plane?

Is located on a circle with the center of the circle at the origin and the radius of, and divides this circle into n equal parts.

6. If the points corresponding to the complex number z 1 and z2 are A and B, then the area of △AOB(O is the origin of coordinates) is.

7、 = 。

8. Several basic trajectories of points corresponding to complex number z in the complex plane:

(1) The trajectory is a ray.

(2) the trajectory is a ray.

③ The trajectory is circular.

(4) The trajectory is a straight line.

⑤ There are three possible cases of trajectory: a) When the trajectory is elliptical in time; B) When the trajectory is a line segment; C) If, the trajectory does not exist.

⑥ There are three possible cases of trajectory: a) In time, the trajectory is hyperbolic; B) When the trajectory is two rays; C) If, the trajectory does not exist.

Seven, permutation and combination, binomial theorem

1, addition principle, what is the application of multiplication principle? What are the characteristics?

Additive classification, class independence; Multiplication is step by step, step by step correlation.

2. The formula of permutation number is: = =;

The relationship between permutation number and combination number is:

The formula of combination number is: = =;

Combined number attribute: =+=

= =

3. Binomial Theorem: General formula of binomial expansion:

Eight, analytic geometry

1, Schell formula:

2. The distance formula between two points on the number axis:

3, the distance between two points in the rectangular coordinate plane formula:

4. If point P divides the directed line segment into constant ratio λ, λ =

5. If the directed line segment between point and point P is within the constant ratio λ, λ = =;

=

=

If so, the coordinates of the center of gravity g of △ABC are.

6. The definition of finding the slope of a straight line is k=, and the two-point formula is k=.

7, several forms of linear equation:

Point tilt:, tilt truncation:

Two-point type: intercept type:

General formula:

The equation of the line system passing through the intersection of two straight lines is:

8. For straight lines, the angle θ from straight line to straight line satisfies:

The included angle θ between the sum of straight lines satisfies:

Straight line, then the included angle θ between straight lines satisfies:

The included angle θ between the sum of straight lines satisfies:

9. Distance from point to straight line:

10, and the distance between two parallel straight lines is

1 1, the standard equation of a circle is:

The general equation of a circle is:

Where the radius is and the center coordinate is.

Thinking: What kind of figure does the equation represent when summing?

12, if, then the equation of a circle with line segment AB as its diameter is

Cross two circles.

,

The equation of the circle system at the intersection is:

The equation of the circle system passing through the intersection of a straight line and a circle is:

13, the tangent equation with the circle as the tangent point is

Generally speaking, the tangent equation with a curve as the tangent point is: For example, the tangent equation of a parabola with a point as the tangent point is:, that is:.

Note: this conclusion can only be used to do multiple-choice questions or fill-in-the-blank questions. If it is a solution, it can only be done according to the routine process of finding the tangent equation.

14, there are two most commonly used methods to study the positional relationship between a circle and a straight line, namely:

① discriminant method: δ > 0, =0,<0, which is equivalent to the intersection, tangency and separation of straight lines and circles;

② Investigate the relationship between the distance from the center of the circle to the straight line and the radius: the distance is greater than the radius, equal to the radius, and less than the radius, which is equivalent to the separation, tangency and intersection of the straight line and the circle.

15, the four forms of parabolic standard equation are:

16, the focal coordinate of parabola is:, and the alignment equation is:.

If a point is a point on a parabola, the distance from the point to the focus of the parabola (called focal radius) is:, and the length of the chord (called path) passing through the focus of the parabola and perpendicular to the axis of symmetry of the parabola is:.

17, the two forms of elliptic standard equation are: and.

18, the focal coordinate of the ellipse is, the directrix equation is, the eccentricity is and the path length is. One of them is.

19. If a point is a point on an ellipse and is its left and right focal points, then the length of the focal radius of point P is its sum.

20. The two forms of hyperbolic standard equation are: and.

2 1, focal coordinate of hyperbola is, directrix equation is, eccentricity is, path length is, and asymptote equation is. One of them is.

22. The hyperbolic system equation with hyperbolic asymptotes is. Hyperbolic equations with hyperbola as the focus are.

23. If the straight line intersects the conic curve at two points A(x 1, y 1) and B(x2, y2), the chord length is;

If a straight line intersects a conic curve at two points A(x 1, y 1) and B(x2, y2), the chord length is.

24. The geometric meaning of the focal point parameter p of conic curve is the distance from the focal point to the directrix, which is true for both ellipse and hyperbola.

25. Translate the coordinate axis so that the origin of the new coordinate system is (h, k) in the original coordinate system. If the coordinates of point P in the original coordinate system are in the new coordinate system, then =, =.

Nine, polar coordinates, parameter equation

1. The general form of the linear parameter equation passing through this point is:

2. If the straight line passes through a point, the standard form of the straight line parameter equation is: the geometric meaning of the parameter t corresponding to point P is: the number of directed line segments.

If points P 1, P2 and P are points on a straight line, their corresponding parameters in the above parameter equation are: when point P is divided into directed line segments,; When the point p is the midpoint of the line segment p+0p 2,

3. The parameter equation of a circle with a point as the center and a radius is:

3. If the origin of the rectangular coordinate system is the pole, the positive semi-axis of the X axis is the polar axis, and the polar coordinates of point P are the rectangular coordinates, then ….

4. The polar coordinate equation of a straight line passing through the pole at an inclination angle is:

The polar coordinate equation of a straight line passing through a point and perpendicular to the polar axis is:

The polar coordinate equation of a straight line passing through a point and parallel to the polar axis is:

The polar coordinate equation of a straight line passing through a point with an inclination angle is:

5. The polar coordinate equation of a circle with the center at the pole and the radius r is:

The polar coordinate equation of a circle whose center is at point is:

The polar coordinate equation of a circle whose center is at point is:

Polar coordinate equation of a circle whose center is at point and radius is.

6, if m, n, then.

Solid geometry

1, the projective formula of dihedral angle is, where the meanings of each symbol are: the area of figure f on one plane of dihedral angle, the projection of figure f on another plane of dihedral angle, and the size of dihedral angle.

2. If the projection of a straight line on the plane is a straight line, and the straight line M is a straight line passing through the inclined foot on the plane, the angle it forms is, and the angle it forms is θ, then the relationship between these three angles is.

3. Volume formula:

Cylinder:, cylinder:.

Oblique prism volume: (where, it is the straight section area and side length);

Cone:, cone:.

Table body: frustum:

Sphere:.