Trigonometric Functions and Complex Numbers

Desheng Yang, Chunhui Shen;;;

• Publisher: WCPC
• Year: 2016
• ISBN:
  9781938134883
• Topic: Mathematik
• Subtopic: Trigonometrie

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Trigonometric Functions and Complex Numbers covers the followings areas in the International Mathematical Olympiad (IMO) and other mathematical competitions.

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• Trigonometric identity, graphs and properties of trigonometric equations, inverse trigonometric functions and trigonometric equations, solutions of triangles, trigonometric substitution and trigonometric inequality;
• The concept and operation of complex numbers, trigonometric form of a complex number, complex number and equation.

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The contents are essential for the IMO. A good help for students who want to improve in these areas.

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PART I

Trigonometric Function

Chapter 1

Trigonometric Identity

1.1. Trigonometric Ratios for Any Angle

Any angle and its measures

1. Positive angle, negative angle and zero angle

If a ray rotating around the endpoint counterclockwise formed as the positive angle, its measurement is positive, and if a ray rotating around the endpoint clockwise formed as the positive angle, its measurement is negative (see Fig. 1.1):

Fig. 1.1.

2. Coterminal angle

In the process of angle forming, we can find there are numerous angles whose initial and terminal sides coincide with a certain angle α. Their range is 360° integer times that of α in size. We put the set of the angles (including α itself) with the same edge as α:

{β|β= k · 360° + α, k ∈ Z}.

Namely, any angle with the same terminal side as angle α can be expressed as a total of angle α and integer number of perigons.

3. Quadrant angle

If the terminal ray of an angle in standard position lies in the first quadrant, then the angle is said to be a first-quadrant angle. Similarly, second-, third-, and fourth-quadrant angles are defined as follows:

First-quadrant angles: {α|k · 360° < α < k · 360° + 90°, k ∈...