§2 elementary function and numerical calculus
First, function concept and classification
[function and inverse function] suppose several volumes which D is assigns. If has two variables x and y, when the variable x takes some specific value when D, variable y also has a definite value according to determination relational f, then said that y is x function, f is called on D functional relations, records is y=f(x), x is called the independent variable, y is called the dependent variable. When x takes in D various numbers, corresponding y constitutes several volume of R, D to be called the domain of definition either the independent variable territory, R is called the value territory or the dependant variable territory. In turn, if regards as the independent variable y, x regards as a dependent variable, writes x expression with y: x=j(y), then said that y=f(x) and x=j(y) are inverse functions mutually.
[real variable function and complex variable function] work as when independent variable territory for real number field, the function is called the real variable function. When independent variable territory for complex field, the function is called the complex variable function.
[a circular function and function of many variables] only then an independent variable's function is called a circular function. Two or two above independent variable's functions are called the function of many variables.
[explicit function and implicit function] the dependent variable may express directly by the independent variable with mathematics formula the function is called the explicit function. If the functional relations contain in an equation or group of equations, the independent variable and the dependent variable differentiate not obviously, is called the implicit function.
[simple function and composite function], if y is u function y=f(u), but u is also x function, u=j(x), then y is called x composite function, u is called the middle variable, records makes y=f[j(x)], not middle variable's function is called the simple function.
[limited function and unbounded function], if has two to count m, M(m£M), causes m£f(x)£M, x establishes willfully to the domain of definition, then said that f(x) is in the domain of definition limited function, m is its world of mortals, M is its upper boundary. If such number m and M have one not to exist at least, then said that f(x) is in the domain of definition unbounded function.
[monotone function with nonmonotone function], if regarding the sector [a, b] random x1>x2 has f(x1) ³ f(x2) [or f(x1)£f(x2)], then said that f(x) is [a, b] the increasing function (or decreasing function). The increasing function and the decreasing function are generally called for the monotone function. It is not the increasing (or decreasing progressively) the function is called the nonmonotone function.
[odd function and even function], if x has permanently willfully regarding the domain of definition, then said that f(x) is an odd function; If x has permanently willfully regarding the domain of definition, then said that f(x) is an even function.
[periodic function with nonperiodic function], if has a real number T ¹ 0, enables x to have permanently willfully to the domain of definition in f(x+T)=f(x), then f(x) is called take T as the cyclical periodic function; Otherwise said that f(x) is the nonperiodic function.
[monodrome function and multiplevalued function], if regarding an independent variable x value, dependent variable y has one moreover only then a value with its correspondence, then said that y is x monodrome function. If regarding an independent variable x value, continues with its correspondence's y value, then said that y is x multiplevalued function.
[elementary function] the power function, the exponential function, the logarithmic function, the trigonometric function, the inverse trigonometric function are generally called for “the basic elementary function”, everything has the time limit mathematical operations by the basic elementary function process as well as has the time limit compound step to constitute, and can use the function which a mathematics formula expressed to belong to the elementary function.
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Second, power function and rational function
[definition] the shape likefunction is called the power function, in the formula a for willfully the solid constant.
x multinomial
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(a0, a1, L, an is constant, n is natural number)
Is called the rational integral function.
Two multinomial business
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Is called the rational fraction function.
The rational integral function and the rational fraction function are generally called for the rational function, sometimes uses mark R(x) to indicate.
[power function graph and characteristic]
Equation and graph 
Especially Drafting 

Curve initial point (0,0) and (1,1); When x>1, a bigger curve rise is quicker. When a is an even number, the function is an even function, in the sector in (0,¥) is the increasing function, in sector ( ¥, 0) for decreasing function. When a is an odd number, function for odd function and increasing function. 

Curve initial point (1,1). When a is at bay counts, the function is an even function, in the sector ( ¥, 0) is the increasing function, in sector (0, ¥) for decreasing function. When a is negative odd number, function for odd function and decreasing function. 
Third, exponential function and logarithmic function
[definition] the shape likefunction is called the exponential function.
When a=e, is writing is convenient, sometimesrecords does expx, records makes exp{f(x)}, and so on.
In function relationship, if x regards as the independent variable, y regards as a dependent variable, then said that y is take a as the bottom x logarithmic function, x is called the logarithm, records does. The exponential function and the logarithmic function are inverse functions mutually.
[functional digraph and characteristic]
Equation and graph 
Especially Drafting 
Exponential function 
Curve and y axis intersection in A(0,1). The approach line is y=0. 

Curve and x axis intersection in A(1,0). The approach line is x=0. 
[index operation principle]
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simultaneously [logarithm nature and operation principle] in the following formula, supposition a>0, the function supposition which meets are discuss in the domain of definition.
Zero and the negative number do not have the logarithm
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Logarithm identical equation _{} Trades the bottom formula
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[common logarithm and natural logarithm]
1o common logarithm: Take 10 are called the common logarithm as the bottom logarithm, records does
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2o natural logarithm: As the bottom logarithm is called the natural logarithm take e=2.718281828459L, records does
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3o common logarithm and natural logarithm relations:
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In the formula M is called the modulus,
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the 4o common logarithm first number asks the law:
If the logarithm is bigger than 1, then logarithm first number for positive number or zero, its value compared to integer figure few 1.
If the logarithm is smaller than 1, then the logarithm first number is a negative number, its absolute value is equal to front of the logarithm first place significant digit “0” integer (before decimal point that “0”).
The logarithm mantissa finds out by the logarithmic table.
Fourth, plane trigonometry function and inverse trigonometric function
1. Angle measure and conversion
[angle system and radian system]
1o entire circumferencethe arc is called includes 1 degree arc, but 1 degree arc to the center of circle angle is called 1 degree angle .1 degrees to be equal to that 60 points (record do), 1 classification (records in 60 seconds does). This kind with measures the angle the method to be called the angle system.
2o was equal to that the radius long arc is called includes 1 radian arc, but 1 radian arc to the center of circle angle is called 1 radian the angle, this kind measures the angle with the radian the method to be called the radian system.
[and radian conversion] radian and the relations are
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In the formula q and a express the identical angle number of degree and the radian number separately.
And radian conversion tableⅠ
Radian (r) 
(°) 
Minute (¢) 
Second (²) 
1 
57.29577951 
3437.746771 
206264.8063 
0.017453293 
1 
60 
3600 
0.0002908882 
0.016666667 
1 
60 
0.0000048481 
0.000277778 
0.016666667 
1 
_{}. In the table the blackbody digit is the precise value.
And radian conversion tableⅡ
360° 
180° 
90° 
60° 
45° 
30°  
Radian 
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[the ancestor rate (girth quotient) p] the circle perimeter and diameter ratio is called the girth quotient, expressed with the p. Because our country ancient times Southern Dynasty's mathematician Zu Chongzhi achieved the glorious triumph in the computation girth quotient aspect, thus the girth quotient also often was called the ancestor rate.
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Zu Chongzhi figures out p the value is 3.1415926<p<3.1415927.
2. Trigonometric function definition
[trigonometric function definition and sign modification]
Name 
Sine 
Cosine 
Tangent 
Cotangent 
Secant 
Cosec  
Deciding Righteousness 
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Symbol Number With 
Increasing Reducing Changing Melting 
Ⅰ 
+↑ 
+↓ 
+↑ 
+↓ 
+↑ 
+↓ 
Ⅱ 
+↓ 
 ↓ 
 ↑ 
 ↓ 
 ↑ 
+↑  
Ⅲ 
 ↓ 
 ↑ 
+↑ 
+↓ 
 ↓ 
 ↑  
Ⅳ 
 ↑ 
+↑ 
 ↑ 
 ↓ 
+↓ 
 ↓ 
[trigonometric function graph and characteristic]
Standard sine curve
Cycle:
With x axis point of intersection (same inflection point):
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Extreme point (maximum point or minimum point):
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Cosine curve
Cycle:
With x axis point of intersection (same inflection point):
Extreme point:
Generally sine curve
_{} Cycle:
In the formula A>0 is an oscillation amplitude, is the angular frequency, is the first phase With x axis point of intersection (same inflection point):

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Extreme point:
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At the same time, also belongs to the general sine tune It is extends the standard sine curve in the y axis direction
The line (supposes, may change into) _{} Long A time, compresses time in x axis direction, and
Translatessection of distances to obtain toward left.
Tangent curve
y=tan x
Cycle:
With x axis point of intersection (same inflection point):
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This tangent slope is 1.
Approach line:
Cotangent curve

Cycle:
With x axis point of intersection (same inflection point):
_{},
This tangent slope is  1.
Approach line:
Secant curve

Cycle:
Maximum point:
Minimum point:
Approach line:
Cosec curve

Cycle:
Maximum point:
Minimum point:
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Approach line:
3. Special angle triangle function value
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Radian  
0 
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0 
1 
0 
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1 
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15 
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18 
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Radian  
22.5 
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30 
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2 
36 
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45 
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1 
1 
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54 
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60 
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2 
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67.5 
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72 
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75 
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90 
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1 
0 
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0 
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1 
120 
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135 
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150 
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2 
180 
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0 
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0 
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In the tableindicatedthat (i.e. left and right limit) 0.1 acute angle's coangle's triangle function value is equal to this angle odd triangle function value, for example.
4. Trigonometric function basic relations and formula
[induction formula]
Trigonometric function induction formula table
Function Angle 
sin 
cos 
tan 
cot 
sec 
csc 
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Function Angle 
sin 
cos 
tan 
cot 
sec 
csc 
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In the table n is an integer.
[basic relations]
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Trigonometric function reciprocity table

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For example, if, then
[addition formula]
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[and difference and accumulates melts formula mutually]
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[double angle formula]
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In the formula n is a positive integer.
[halfangle formulas]
In the following formula the root number institute takes the mark to be consistent with equal sign left side of mark.
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[descending powers formula]
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Above in the formula n is a positive integer.
[trigonometric function is limited and formula]
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5. Inverse trigonometric function definition
[inverse trigonometric function domain of definition and principal value scope]
Function 
Principal value symbol 
Domain of definition 
Principal value scope 
Antisine 
If, then 
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_{} 
Arc 
If, then 
_{} 
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Countertangent 
If, then 
_{} 
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Arc 
If, then 
_{} 
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Arcsec 
If, then 
_{} 
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Arc 
If, then 
_{} 
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Generally the inverse trigonometric function and the principal value relations are
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In the formula n is the random integer.
[inverse trigonometric function graph and characteristic]
Antisine curve Arc curve
Inflection point (same curve center of symmetry): Inflection point (same curve center of symmetry):
_{}, this tangent slope is 1 _{}, this tangent slope is  1
Countertangent curve Arc curve
Inflection point (same curve center of symmetry): Inflection point:
_{}, this tangent slope is 1 _{}, this tangent slope is  1
Gradually coil in:_{} Curve center of symmetry:
Approach line:
Arcsec curve Arc curve
Apex:_{} Apex:
Approach line:_{} Approach line:
6. Inverse trigonometric function reciprocity and fundamental formula
[inverse trigonometric function reciprocity]
arc sin x = 
arc cos x = 
arc tan x = 
arc cot x = 
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Has * the number when only x for just when is suitable.
[inverse trigonometric function fundamental formula]
arc sin x + arc sin y = 
arc sin x  arc sin y = 
_{} 
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arc cos x + arc cos y = 
arc cos x  arc cos y = 
_{} 
_{} 
arc tan x + arc tan y = 
arc tan x  arc tan y = 
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2 arc sin x = 
2 arc cos x = 
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2 arc tanx = 
cos (n arc cos x) = 
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7. Triangle fundamental theorem

[sine law]
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In the formula R isthe ABC external connection radius (Figure 1.3).
[law of cosines]

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[pythagorean theorem] in the right triangle (C is right angle), cancels Fang Jiagu the side to be equal to the string side (Figure 1.4), namely
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The pythagorean theorem also said that the business high theorem, in the foreign books and periodicals calls Pythagoras the theorem.
[tangent theorem]
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Or
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[halfangle and length of side relational formula]
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In the formula, r isthe ABC interior contact radius, and
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In the formula S isthe ABC area.
8. Oblique triangle solution
Known element 
Other elements ask the law  
At the same time a and two jiao B, C 
_{}  
Two side a, b and included angle C 
_{}  
Trilateral a, b, c 
_{}  
Two side a, b and one side opposite angle A 
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when b sin A<a, has two solutions when b sin A>a, non solution when b sin A=a, has a solution 
Fifth, spherical trigonometry

1. Spherical trigonometry related name and nature
[greatcircle] with one intercepts a pass through the center O plane, is called the greatcircle in the ball surface obtained transversal, its radius is equal to the ball radius R (Figure 1.5).
[greatcircle arc length] connects in the spherical surface two A, the B geodetic line is passes A, in the B greatcircle the short arc, its center of circle angle for a (by radian idea), thenarc length a = Ra.
[two great circle arc included angles] on two great circle arc's point of intersection A corresponding greatcircle's tangent (the AB', AC') included angle is called these two great circle arcs the included angle, its also available two plane OAB and OAC constitute the dihedral angle measures (Figure 1.6).

[spherical surface two angular area] the spherical surface two angular ABA'C area (Figure 1.6 dashed area)(A is two great circle arc included angles, unit is radian).
[spherical triangle spherical excess (or spherical angle surplus)] Three greatcircles may constitute several spherical triangles in the spherical surface, we only considered that which triangles trilateral and is smaller than p.

Supposes a, b, g is nearby three (i.e. three section of greatcircle arc lengths, take radius R as Unit of measurement), A, B, C is three angles (i.e. three section of great circle arc 22 included angles, Figure 1.7). The spherical triangle's sums of three angle are certainly bigger than 180°, its difference d = A + B + Cp calls the spherical excess (unit radian), d >0.
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In the formula.
[spherical triangle area] spherical triangle ABC (Figure 1.7 dashed area) area S = R2d.
2. Spherical triangle fundamental theorem and formula
[sine law]
_{}
[law of cosines]
:
Angle:
[cotangent theorem]
:
Angle:
[tangent theorem]
_{}
[five element formulas]
:
Angle:
[halfangle formulas]
_{}
In the formula.
[halfside formula]
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[Delane cloth  Gaussformula]
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[Napier formula]
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3. Spherical triangle solution
[general spherical triangle formula]
Known element 
Solution formula 
Trilateral: a, b, g 
_{} 
Triangle: A <> 