Angles and Applications in .NET framework

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CHAPTER 1 Angles and Applications
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1.1 Introduction 1.2 Plane Angle 1.3 Measures of Angles 1.4 Arc Length 1.5 Lengths of Arcs on a Unit Circle 1.6 Area of a Sector 1.7 Linear and Angular Velocity
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Trigonometric Functions of a General Angle
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2.1 Coordinates on a Line 2.2 Coordinates in a Plane 2.3 Angles in Standard Position 2.4 Trigonometric Functions of a General Angle 2.5 Quadrant Signs of the Functions 2.6 Trigonometric Functions of Quadrantal Angles 2.7 Undefined Trigonometric Functions 2.8 Coordinates of Points on a Unit Circle 2.9 Circular Functions
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Trigonometric Functions of an Acute Angle
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3.1 Trigonometric Functions of an Acute Angle 3.2 Trigonometric Functions of Complementary Angles 3.3 Trigonometric Functions of 30 , 45 , and 60 3.4 Trigonometric Function Values 3.5 Accuracy of Results Using Approximations 3.6 Selecting the Function in Problem Solving 3.7 Angles of Depression and Elevation
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Solution of Right Triangles
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4.1 Introduction 4.2 Four-Place Tables of Trigonometric Functions 4.3 Tables of Values for Trigonometric Functions 4.4 Using Tables to Find an Angle Given a Function Value 4.5 Calculator Values of Trigonometric Functions 4.6 Find an Angle Given a Function Value Using a Calculator 4.7 Accuracy in Computed Results
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Practical Applications
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5.1 Bearing 5.2 Vectors 5.3 Vector Addition 5.4 Components of a Vector 5.5 Air Navigation 5.6 Inclined Plane
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Reduction to Functions of Positive Acute Angles
6.1 Coterminal Angles 6.2 Functions of a Negative Angle 6.3 Reference Angles 6.4 Angles with a Given Function Value
Variations and Graphs of the Trigonometric Functions
7.1 Line Representations of Trigonometric Functions 7.2 Variations of Trigonometric Functions 7.3 Graphs of Trigonometric Functions 7.4 Horizontal and Vertical Shifts 7.5 Periodic Functions 7.6 Sine Curves
viii
Contents Basic Relationships and Identities
8.1 Basic Relationships 8.2 Simplification of Trigonometric Expressions 8.3 Trigonometric Identities
Trigonometric Functions of Two Angles
9.1 Addition Formulas 9.2 Subtraction Formulas 9.3 Double-Angle Formulas 9.4 Half-Angle Formulas
CHAPTER 10 Sum, Difference, and Product Formulas
10.1 Products of Sines and Cosines 10.2 Sum and Difference of Sines and Cosines
CHAPTER 11 Oblique Triangles
11.1 Oblique Triangles 11.2 Law of Sines 11.3 Law of Cosines 11.4 Solution of Oblique Triangles
CHAPTER 12 Area of a Triangle
12.1 Area of a Triangle 12.2 Area Formulas
CHAPTER 13 Inverses of Trigonometric Functions
13.1 Inverse Trigonometric Relations 13.2 Graphs of the Inverse Trigonometric Relations 13.3 Inverse Trigonometric Functions 13.4 Principal-Value Range 13.5 General Values of Inverse Trigonometric Relations
CHAPTER 14 Trigonometric Equations
14.1 Trigonometric Equations 14.2 Solving Trigonometric Equations
CHAPTER 15 Complex Numbers
15.1 Imaginary Numbers 15.2 Complex Numbers 15.3 Algebraic Operations 15.4 Graphic Representation of Complex Numbers 15.5 Graphic Representation of Addition and Subtraction 15.6 Polar or Trigonometric Form of Complex Numbers 15.7 Multiplication and Division in Polar Form 15.8 De Moivre s Theorem 15.9 Roots of Complex Numbers
APPENDIX 1 Geometry
A1.1 Introduction A1.2 Angles A1.3 Lines A1.4 Triangles A1.5 Polygons A1.6 Circles
APPENDIX 2 Tables
Table 1 Trigonometric Functions Angle in 10-Minute Intervals Table 2 Trigonometric Functions Angle in Tenth of Degree Intervals Table 3 Trigonometric Functions Angle in Hundredth of Radian Intervals
INDEX
Trigonometry
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Angles and Applications
1.1 Introduction
Trigonometry is the branch of mathematics concerned with the measurement of the parts, sides, and angles of a triangle. Plane trigonometry, which is the topic of this book, is restricted to triangles lying in a plane. Trigonometry is based on certain ratios, called trigonometric functions, to be defined in the next chapter. The early applications of the trigonometric functions were to surveying, navigation, and engineering. These functions also play an important role in the study of all sorts of vibratory phenomena sound, light, electricity, etc. As a consequence, a considerable portion of the subject matter is concerned with a study of the properties of and relations among the trigonometric functions.
1.2 Plane Angle
The plane angle XOP, Fig. 1.1, is formed by the two rays OX and OP. The point O is called the vertex and the half lines are called the sides of the angle.
Fig. 1.1
More often, a plane angle is thought of as being generated by revolving a ray (in a plane) from the initial S S position OX to a terminal position OP. Then O is again the vertex, OX is called the initial side, and OP is called the terminal side of the angle. An angle generated in this manner is called positive if the direction of rotation (indicated by a curved arrow) is counterclockwise and negative if the direction of rotation is clockwise. The angle is positive in Fig. 1.2(a) and (c) and negative in Fig. 1.2(b).
Fig. 1.2
CHAPTER 1 Angles and Applications
1.3 Measures of Angles
When an arc of a circle is in the interior of an angle of the circle and the arc joins the points of intersection of the sides of the angle and the circle, the arc is said to subtend the angle. A degree ( ) is defined as the measure of the central angle subtended by an arc of a circle equal to 1/360 of the circumference of the circle. A minute ( ) is 1/60 of a degree; a second ( ) is 1/60 of a minute, or 1/3600 of a degree.
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