Plane and Spherical Trigonometry

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D.C. Heath & Company, 1917 - 313 páginas
 

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Contenido

Linear and angular velocity
9
Examples
10
CHAPTER II
12
Values of the trigonometric functions of 30 45 and 60
13
Values of the trigonometric functions of 120 135 and 150
15
Signs of the trigonometric ratios
17
Trigonometric functions are single valued
18
A given value of a trigonometric function determines an infinite number of angles
19
Examples
20
CHAPTER III
22
Trigonometric tables
23
Formulas used in the solution of right triangles
25
Suggestions on solving a triangle
26
Examples
29
Oblique triangles
30
Applications
31
CHAPTER IV
34
Variation of the functions
35
Graphical representation
39
Periodicity of the trigonometric functions
40
Examples
41
ART PAGB 44 Use of formulas
42
Functions of 90 + a in terms of functions of a
43
Functions of 90 a in terms of functions of a
44
Functions of 1800 a in terms of functions of a
45
Laws of reduction
46
CHAPTER V
47
The use of exponents
48
Trigonometric identities
49
Examples
51
LINE VALUES 57 Representation of the trigonometric functions by lines
53
Variations of the trigonometric functions as shown by line values
55
Fundamental relations by line values
56
Examples
57
CHAPTER VI
61
The cosine of the sum of two acute angles
62
Generalization of formulas
63
Tangent of the sum of two angles
65
ART PAGE 69 Sine cosine tangent and cotangent of the difference of two angles
66
Double angles
67
Half angles
68
Sum and difference of two sines and of two cosines
69
Equations and identities
70
Examples
71
CHAPTER VII
74
Multiple values of an inverse function
75
Principal values 1
77
Application of the fundamental relations to angles expressed as inverse functions
78
The value of any function of an inverse function
79
ART PAGE 96 Area in terms of sides
90
Check formulas
91
Illustrative problems
92
The ambiguous case
95
Examples
98
MISCELLANEOUS EXERCISES
102
CHAPTER IX
113
Geometric interpretation
115
Applications of De Moivres theorem
116
Fifth roots of unity
117
Square root of a complex number
118
Any root of a complex number
119
Sin n a and cos n a expressed in terms of sin a and cos a
120
Comparison of the values of sin a a and tan a a being any acute angle
121
Sine and cosine series
123
Examples
124
SPHERICAL TRIGONOMETRY CHAPTER X
127
Law of sines
128
Law of cosines
129
Law of cosines extended
130
Relation between one side and three angles
131
ART PAGK 123 Relation between two sides and three angles
132
Formulas independent of the radius of the sphere
133
CHAPTER XI
134
Sufficiency of formulas
137
Napiers rules
138
Side and angle opposite terminate in same quadrant
139
The quadrant of any required part
140
Solution of a right triangle
141
Examples
143
Examples
144
CHAPTER XII
145
Sides found from the three angles
146
Delambres or Gausss formulas
147
Formulas collected
148
ART PAGE 150 Theorem to determine quadrant
149
Third theorem to determine quadrant
150
Two solutions
153
Area of spherical triangle
154
SOLUTION WHEN ONLY ONE PART IS REQUIRED 157 Statement of problem
156
Two parts required
158
Problems
159
Each unknown part found from two angles and the included side
160
Each unknown part found from two sides and an angle opposite one of them
161
Each unknown part found from two angles and a side opposite one of them
163
The general triangle
164
ANSWERS o
165
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Página 4 - The logarithm of a quotient is equal to the logarithm of the dividend minus the logarithm of the divisor. , M , ,• , . logi — = log
Página 130 - ... cos a = cos b cos с + sin b sin с cos A ; (2) cos b = cos a cos с + sin a sin с cos в ; ^ A. (3) cos с = cos a cos b + sin a sin b cos C.
Página 4 - The logarithm of the product of two numbers is equal to the sum of the logarithms of the numbers.
Página 111 - From the top of a hill the angles of depression of two successive milestones, on a straight level road leading to the hill, are observed to be 5° and 15°.
Página 3 - If the number is less than 1, make the characteristic of the logarithm negative, and one unit more than the number of zeros between the decimal point and the first significant figure of the given number.
Página 85 - B is negative, and BD — — a cos B. The substitution of this in (4) leads us again to (3). Thus we see that (3) is true in all cases. THE LAW OF COSINES. The square of any side .of a plane triangle is equal to the sum of the squares of the other two sides minus twice their product times the cosine of the included angle. This may be regarded as a generalization of the Pythagorean Theorem to which it reduces when the included angle is a right angle. These two laws are among the most important of...
Página 2 - If the number is greater than 1, the characteristic is one less than the number of places to the left of the decimal point.
Página 131 - By (150) and (152) we have cos a = cos b cos c -\- sin b sin c cos A, cos c = cos a cos b...
Página 31 - The product of all the lines, that can be drawn from one of the angles of a regular polygon of n sides, inscribed in a circle whose radius is a, to all the other angular points = no.
Página 111 - From the top of a cliff 150 ft. high the angles of depression of the top and bottom of a tower are 30° and 60°, respectively.

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