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Trigonometric Functions and Complex Numbers

Desheng Yang
World Century
Publication Date: 
Number of Pages: 
World Century Mathematical Olympiad Series I
Problem Book
[Reviewed by
Allen Stenger
, on

This is an Olympiad-level problem book, with complete solutions, in the two related subject areas of trigonometric functions (2/3 of the book) and complex numbers (1/3 of the book). The body of the book consists of worked examples. There is also an Exercises section at the end of each chapter, with solutions in the back of the book. The book is a translation of a Chinese-language book published in 2010.

As the title indicates, the book deals primarily with trigonometric functions (not trigonometry) and complex numbers (not complex variables). There are a few geometric problems, and a few trigonometric identities, but most of the trigonometric questions ask for the values of particular trigonometric functions. Typical examples are (p. 53) is “Find \(\sin 10^\circ \sin 50^\circ \sin 70^\circ\)” and (p. 113) to solve \(2 \sin(3x+\tfrac{\pi}{4}) = \sqrt{3}\). Most of the complex number problems are actually college algebra problems that use complex numbers. A typical example (p. 262) is “If \(i\) is a root of equation \(x^3 + 2x + k = 0\), find two roots and \(k\).”

The English is very shaky and in many cases incomprehensible. Some of the language is reminiscent of the assembly instructions we used to get with imported gadgets. For example (pp. 134–135):

Convert complex geometric transformation and complex deductive reasoning for operation of trigonometric function. This method is simple and clear thinking. Communication triangle and the geometry relationship, in addition to the direct use of trigonometric function definition and the triangle formula, mainly by means of sine law and cosine law and area formula.

For another example (p. 21):

Emphasis should be put on the classified discussion. This makes full use of principle “the big horn is transformed into the small angle” and the formula “when the odd element changes, the even element does not change.”

These are especially bad examples, but many of the sentences are unidiomatic. Some of the problem statements are garbled, for example on p. 259 we are given that \(p\) is an even prime number (which to me means \(p=2\)), but from the solution it appears that \(p\) is meant to be an odd prime number. There are references on pp. 135 and 145 to the Mailer Laws, but these are not explained; I have not been able to track down these laws, or determine from the context what law is being used.

Bottom line: Accurate mathematically, but limited, and the language makes it too difficult to follow, especially for the high-school students who are its intended audience. Even people who already understand the mathematics will have difficulty following the statements. There are other, better books, such as Andreescu & Andrica’s Complex Numbers from A to ... Z and Andreescu & Feng’s 103 Trigonometry Problems: From the Training of the USA IMO Team.

Allen Stenger is a math hobbyist and retired software developer. He is an editor of the Missouri Journal of Mathematical Sciences. His personal web page is His mathematical interests are number theory and classical analysis.

  • Trigonometric Function:
    • Trigonometric Identity
    • Trigonometric Identity
    • Trigonometric Functions
    • Inverse Trigonometric Functions
    • Solutions of Triangles
    • Trigonometric Substitution and Trigonometric Inequality
  • Complex Number:
    • Concept of Complex Number
    • Operations with Complex Number
    • Trigonometric Form of a Complex Number
    • Complex Number and Equation