One says that a coordinate system is introduced on an axis if there is a one-to-one corre-spondence between points of the axis and numbers.. The value of a segment −→ OA is called the c
Trang 13.3 SPHERICALTRIGONOMETRY 73
TABLE 3.5 Basic properties and relations characterizing spherical triangles
1 Triangle inequality
The sum of lengths of two sides is greater than the length of the third side The absolute value of the difference between the lengths of two sides is
less than the length of the third side,
a + b > c, |a – b| < c
2 Sum of two anglesof a triangle The sum of two angles of a triangle is greater thanthe third angle increased by π,
α + β < π + γ
3 The greatest side andthe greatest angle The greatest side is opposite the greatest angle,a < b if α < β;
a = b if α = β
4 Sum of anglesof a triangle The sum of the angles lies between π and π < α + β + γ <3π 3π,
5 Sum of sidesof a triangle The sum of sides lies between0< a + b + c <2 0and2π
π
sin α =
sin b sin β =
sin c sin γ
7 The law of cosinesof sides cos c = cos a cos b + sin a sin b cos γ
8 The law of cosinesof angles cos γ = – cos α cos β + sin α sin β cos c
9 Half-angle formulas
sinγ
2 =
sin(p – a) sin(p – b) sin a sin b , cos
γ
2 =
sin p sin(p – c) sin a sin b ,
tanγ2 =
sin(p – a) sin(p – b) sin p sin(p – c)
10 Half-side theorem
sin2c =
– sin P sin(P – γ) sin α sin β , cos
c
2 =
sin(P – α) sin(P – β) sin α sin β ,
tanc
2 =
– sin P sin(P – γ) sin(P – α) sin(P – β)
11 Neper’s analogs
tan c
2cos
α – β
2 = tan
a + b
2 cos
α + β
2 , tanc2sin
α – β
2 = tan
a – b
2 sin
α + β
2 , cot γ
2cos
a – b
2 = tan
α + β
2 cos
a + b
2 , cotγ
2sin
a – b
2 = tan
α – β
2 sin
a + b
2
12 D’Alembert (Gauss)formulas sin
γ
2sin
a + b
2 = sin
c
2cos
α – β
2 , sin
γ
2sin
a + b
2 = cos
c
2cos
α + β
2 , cosγ
2sin
a – b
2 = sin
c
2sin
α – β
2 , cos
γ
2cos
a – b
2 = cos
c
2sin
α + β
2
13 Product formulas sin a cos β = cos b sin c – cos α sin b cos c, sin a cos b = cos β sin c – cos a sin β cos γ
14 The “circumradius” R cot R = sin(P – α) sin(P – β) sin(P – γ)
α
2 sin(α – P )
15 The “inradius” r tan r = sin(p – α) sin(p – β) sin(p – γ)
α
2sin(p – α)
Trang 274 ELEMENTARY GEOMETRY
TABLE 3.5 (continued)
Basic properties and relations characterizing spherical triangles
16 the spherical excess εWillier’s formula for tan P
2 = tan
ε
4 = tan
p
2tan
p – a
2 tan
p – b
2 tan
p – c
2
2 –
ε
4
=
! tanp–a2 tanp–b2 tanp2 tanp–c2
TABLE 3.6
Solution of spherical triangles
No Three parts
1 Three sides
a , b, c
The angles α, β, and γ are determined by the half-angle formulas and the cyclic
permutation
Remark.0< a+b+c <2π The sum and difference of two sides are greater than the third
2 Three angles
α , β, γ
The sides a, b, and c are determined by the half-side theorems and the cyclic
permutation
Remark.π < α+β +γ <3 π The sum of two angles is less than π plus the third angle.
3 Two sides a, b
and the
included
angle γ
First method.
α + β and α – β are determined from Neper’s analogs, then α and β can be found; side a is determined from the law of cosines, sin c = sin γ sin a
sin α.
Second method.
The law of cosines of sides is applied, cos c = cos a cos b + sin a sin b cos γ, cos β = cos b – sin a sin c
sin a sin c , cos α =
cos a – sin b sin c sin b sin c .
Remark 1.If γ > β (γ < β), then c must be chosen so that c > b (c < b).
Remark 2.The quantities c, α, and β are determined uniquely.
4 A side c
and the two
angles α, β
adjacent to it
First method.
a + b and a – b are determined from Neper’s analogs, then a and b can be found; angle γ is determined from the law of sines, sin γ = sin c sin α
sin a.
Second method.
The law of cosines of angles is applied, cos γ = – cos α cos β + sin α sin β cos c, cos a = cos α + cos β cos γ
sin β sin γ , cos b =
cos β + cos α cos γ sin α sin γ .
Remark 1.If c > b (c < b), then γ must be chosen so that γ > β (γ < β).
Remark 2.The quantities γ, a, and b are determined uniquely.
5 Two sides a, b
and the angle α
opposite one
of them
β is determined by the law of sines, sin β = sin α sin b
sin a. The elements c and γ can be found from Neper’s analogs.
Remark 1.The problem has a solution for sin b sin α≤sin a.
Remark 2.Different cases are possible:
1 If sin a≥sin b, then the solution is determined uniquely.
2 If sin b sin α < sin a, then there are two solutions β1and β2, β1+ β2= π.
3 If sin b sin α = sin a, then the solution is unique: β = 12π
6 Two angles
α , β and the
side a opposite
one of them
b is determined by the law of sines, sin b = sin a sin β
sin α. The elements c and γ can be found from Neper’s analogs.
Remark 1.The problem has a solution for sin a sin β≤sin α.
Remark 2.Different cases are possible:
1 If sin α≥sin β, then the solution is determined uniquely.
2 If sin β sin α < sin a, then there are two solutions b1and b2, b1+ b2= π.
3 If sin β sin α = sin a, then the solution is unique: b = 12π
Trang 3REFERENCES FORCHAPTER3 75
The following basic relations hold for spherical triangles:
sin a = cos π2 – a
= sin α sin c = cot π2 – b
cot β = tan b cot β, sin b = cos π2 – b
= sin β sin c = cot π2 – a
cot α = tan a cot α, cos c = sin π2 – a
sin π2 – b
= cos a cos b = cot α cot β, cos α = sin π2 – a
sin β = cos a sin β = cot π2 – b
cot c = tan b cot c, cos β = sin π2 – b
sin α = cos b sin α = cot π2 – a
cot c = tan a cot c,
(3 3 2 3 )
which can be obtained from the Neper rules: if the five parts of a spherical triangle (the right
angle being omitted) are written in the form of a circle in the order in which they appear in
the triangle and the legs a and b are replaced by their complements to 12π (Fig 3.38b), then
the cosine of each part is equal to the product of sines of the two parts not adjacent to it, as well as to the product of the cotangents of the two parts adjacent to it.
References for Chapter 3
Alexander, D C and Koeberlein, G M., Elementary Geometry for College Students, 3rd Edition, Houghton
Mifflin Company, Boston, 2002
Alexandrov, A D., Verner, A L., and Ryzhik, B I., Solid Geometry [in Russian], Alpha, Moscow, 1998 Chauvenet, W., A Treatise on Elementary Geometry, Adamant Media Corporation, Boston, 2001.
Fogiel, M (Editor), High School Geometry Tutor, 2nd Edition, Research & Education Association, Englewood
Cliffs, New Jersey, 2003
Gustafson, R D and Frisk, P D., Elementary Geometry, 3rd Edition, Wiley, New York, 1991.
Hadamard, J., Lec¸ons de g´eom´etrie ´el´ementaire, Vols 1 and 2, Rep Edition, H.-C Hege and K Polthier
(Editors), Editions J Gabay, Paris, 1999
Hartshorne, R., Geometry: Euclid and Beyond, Springer, New York, 2005.
Jacobs, H R., Geometry, 2nd Edition, W H Freeman & Company, New York, 1987.
Jacobs, H R., Geometry: Seeing, Doing, Understanding, 3rd Edition, W H Freeman & Company, New York,
2003
Jurgensen, R and Brown, R G., Geometry, McDougal Littell/Houghton Mifflin, Boston, 2000.
Kay, D., College Geometry: A Discovery Approach, 2nd Edition, Addison Wesley, Boston, 2000.
Kiselev, A P., Plain and Solid Geometry [in Russian], Fizmatlit Publishers, Moscow, 2004.
Leff, L S., Geometry the Easy Way, 3rd Edition, Barron’s Educational Series, Hauppauge, New York, 1997 Moise, E., Elementary Geometry from an Advanced Standpoint, 3rd Edition, Addison Wesley, Boston, 1990 Musser, G L., Burger, W F., and Peterson, B E., Mathematics for Elementary Teachers: A Contemporary
Approach, 6th Edition, Wiley, New York, 2002.
Musser, G L., Burger, W F., and Peterson, B E., Essentials of Mathematics for Elementary Teachers: A
Contemporary Approach, 6th Edition, Wiley, New York, 2003.
Musser, G L and Trimpe, L E., College Geometry: A Problem Solving Approach with Applications, Prentice
Hall, Englewood Cliffs, New Jersey, 1994
Pogorelov, A V., Elementary Geometry [in Russian], Nauka Publishers, Moscow, 1977.
Pogorelov, A., Geometry, Mir Publishers, Moscow, 1987.
Prasolov, V V., Problems in Plane Geometry [in Russian], MTsNMO, Moscow, 2001.
Prasolov, V V and Tikhomirov, V M., Geometry, Translations of Mathematical Monographs, Vol 200,
American Mathematical Society, Providence, Rhode Island, 2001
Roe, J., Elementary Geometry, Oxford University Press, Oxford, 1993.
Schultze, A and Sevenoak, F L., Plane and Solid Geometry, Adamant Media Corporation, Boston, 2004 Tussy, A S., Basic Geometry for College Students: An Overview of the Fundamental Concepts of Geometry,
2nd Edition, Brooks Cole, Stamford, 2002.
Vygodskii, M Ya., Mathematical Handbook: Elementary Mathematics, Rev Edition, Mir Publishers, Moscow,
1972
Trang 5Chapter 4
Analytic Geometry
4.1 Points, Segments, and Coordinates
on Line and Plane
4.1.1 Coordinates on Line
4.1.1-1 Axis and segments on axis.
A straight line on which a sense is chosen is called an axis If an axis is given and a scale
segment, i.e., a linear unit used to measure any segment of the axis, is indicated, then the segment length is defined (see Fig 4.1).
Figure 4.1 Axis.
A segment bounded by points A and B is called a directed segment if its initial point and endpoint are chosen Such a segment with initial point A and endpoint B is denoted
by − AB −→ Directed segments are usually called simply “segments” for brevity.
The value of a segment −− AB → of some axis is defined as the number AB equal to its
length taken with the plus sign if the senses of the interval and the axis coincide, and with the minus sign if the senses are opposite Obviously, the length of a segment is its absolute value The segment length is usually denoted by the symbol | AB | It follows from the above that
AB = –BA, | AB | = | BA | (4 1 1 1 )
Main identity For any arbitrary arrangement of points A, B, and C on the axis, the
values of the segments − AB −→ , −−→ BC , and −→ AC satisfy the relation
4.1.1-2 Coordinates on line Number axis.
One says that a coordinate system is introduced on an axis if there is a one-to-one
corre-spondence between points of the axis and numbers.
Suppose that a sense, a scale segment, and a point O called the origin are chosen on a line The value of a segment −→ OA is called the coordinate of the point A on the axis It is usually denoted by the letter x The coordinates of different points are usually denoted by subscripts; for example, the coordinates of points A1, , Anare x1, , xn The point An with coordinate xnis denoted by An(xn) An axis with a coordinate system on it is called
a number axis.
77
Trang 678 ANALYTIC GEOMETRY
4.1.1-3 Distance between points on axis.
The value A1A2 of the segment −−→ A1A2 on an axis is equal to the difference between the
coordinate x2of the endpoint and the coordinate x1of the initial point:
A1A2= x2– x1. (4. 1 1 3 )
The distance d between two arbitrary points A1(x1) and A2(x2) on the line is given by the relation
d = | A1A2| = | x2– x1| (4 1 1 4 )
Remark If segments do not lie on some axis but are treated as arbitrary segments on the plane or in space, then there is no reason to assign any sign to their lengths In such cases, the symbol of absolute value is usually omitted in the notation of lengths of segments We adopt this convention in the sequel
4.1.2 Coordinates on Plane
4.1.2-1 Rectangular Cartesian coordinates on plane.
If a one-to-one correspondence between points on the plane and numbers (pairs of numbers)
is specified, then one says that a coordinate system is introduced on the plane.
A rectangular Cartesian coordinate system is determined by a scale segment for
mea-suring lengths and two mutually perpendicular axes The point of intersection of the axes is
usually denoted by the letter O and is called the origin, while the axes themselves are called the coordinate axes As a rule, one of the coordinate axes is horizontal and the right sense is positive This axis is called the abscissa axis and is denoted by the letter X or by OX On the vertical axis, which is called the ordinate axis and is denoted by Y or OY , the upward sense is usually positive (see Fig 4.2a) The coordinate system introduced above is often denoted by XY or OXY
( ) c
( ) a
( ) d
( ) b
O
O
O
O
A
A A
Left Right
Upper half-plane
Lower half-plane
half-plane half-plane
X Y
X
X
X
X
Y
Y
Y Y
Figure 4.2 A rectangular Cartesian coordinate system.
Trang 74.1 POINTS, SEGMENTS,ANDCOORDINATES ONLINE ANDPLANE 79
The abscissa axis divides the plane into the upper and lower half-planes (see Fig 4.2b), while the ordinate axis divides the plane into the right and left half-planes (see Fig 4.2c) The two coordinate axes divide the plane into four parts, which are called quadrants and numbered as shown in Fig 4.2d.
Take an arbitrary point A on the plane and project it onto the coordinate axes, i.e., draw perpendiculars to the axes OX and OY through A The points of intersection of the perpendiculars with the axes are denoted by AX and AY, respectively (see Fig 4.2a) The
numbers
x = OAX, y = OAY, (4 1 2 1 )
where OAX and OAY are the respective values of the segments −→ OAX and −→ OAY on the
abscissa and ordinate axes, are called the coordinates of the point A in the rectangular Cartesian coordinate system The number x is the first coordinate, or the abscissa, of the point A, and y is the second coordinate, or the ordinate, of the point A One says that the point A has the coordinates (x, y) and uses the notation A(x, y).
Example 1 Let A be an arbitrary point in the right half-plane Then the segment −→ OA Xhas the positive
sense on the axis OX, and hence the abscissa x = OA X of A is positive But if A lies in the left half-plane, then the segment A X has the negative sense on the axis OX, and the number x = OA X is negative If the
point A lies on the axis OY , then its projection on the axis OX coincides with the point O and x = OA X=0
Thus all points in the right half-plane have positive abscissas (x >0), all points in the left half-plane have
negative abscissas (x <0), and the abscissas of points lying on the axis OY are zero (x =0)
Similarly, all points in the upper half-plane have positive ordinates (y >0), all points in the lower half-plane
have negative ordinates (y <0), and the ordinates of points lying on the axis OX are zero (y =0)
Remark 1 Strictly speaking, the coordinate system introduced above is a right rectangular Cartesian
coordinate system A left rectangular Cartesian coordinate system can, for example, be obtained by changing
the sense of one of the axes There also exist right and left oblique Cartesian coordinate systems, where the
coordinate axes intersect at an arbitrary angle
Remark 2 A right rectangular Cartesian coordinate system is usually called simply a Cartesian coordinate system
4.1.2-2 Transformation of Cartesian coordinates under parallel translation of axes.
Suppose that two rectangular Cartesian coordinate systems OXY and $ O $ X $ Y are given and
the first system is taken to the second by the translation of the origin O of the first system to
the origin $ O of the second system Under this translation, the axes preserve their directions
(the respective axes of the systems are parallel), and the origin moves by x0in the direction
of the OX-axis and by y0in the direction of the OY -axis (see Fig 4.3a) Obviously, the
point $ O has the coordinates (x0, y0) in the coordinate system OXY
Let an arbitrary point A have coordinates (x, y) in the system OXY and coordi-nates ( ˆx, ˆy) in the system $ O $ X $ Y The transformation of rectangular Cartesian coordinates
by the parallel translation of the axes is given by the formulas
x = ˆx + x0,
y = ˆy + y0 or
ˆx = x – x0,
ˆy = y – y0 (4. 1 2 2 ) 4.1.2-3 Transformation of Cartesian coordinates under rotation of axes.
Suppose that two rectangular Cartesian coordinate systems OXY and O $ X $ Y are given and
the first system is taken to the second by the rotation around the point O by an angle α (see Fig 4.3b).