Trigonometry: Math for Gifted Students

Official site with more information and practice: www.mathallstar.org. Trigonometry is an important subject in mathematics. It relates to many other subjects such as geometry, coordinate geometry, complex number, and so on. Therefore, trigonometric problems appear in almost every AMC12 or above comp [...]
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Official site with more information and practice: www.mathallstar.org. Trigonometry is an important subject in mathematics. It relates to many other subjects such as geometry, coordinate geometry, complex number, and so on. Therefore, trigonometric problems appear in almost every AMC12 or above competition either explicitly or implicitly. In addition, students attending lower level competition may find trigonometry can offer valuable alternative solutions to some geometry problems. In order to be proficient in trigonometry, it is necessary to memorize some formulas. However, there are hundreds, if not thousands, of trigonometric formulas. It is practically impossible and often unnecessary to remember all of them. Therefore, it is critical to know what formulas are essential and thus have to be remembered. Accordingly, the first objective of this book is to help students understand and remember those essential formulas. Remembering a sufficient number of formulas may help students achieve high scores in school tests. However, it is not sufficient to win math competitions. Students will have to master relevant techniques and be able to choose the most appropriate formula to solve particular problems. Let's take the following expression as an example: \begin{equation}\label{eq_ex} \cos 20 DEGREES\circ \cos 40 DEGREES\circ \cos 80 DEGREES\circ \end{equation} \indent The value of this expression can be calculated in multiple ways. A classic technique is to multiply it by $\sin 20 DEGREES\circ$. The result can be obtained by applying the double angle formula a few times. An alternative, relatively less known, solution is to apply the triple angle formula. This solution can produce the result immediately. Both approaches are workable in this case. Each of them can be used to tackle some generalized formed of \myJustRef{eq_ex}. As such, it is important for students to know all the relevant techniques and which one to choose in a particular case. Accordingly, the second objective of this book is to illustrate important techniques and to explain when to use them. In order to achieve this, some sample problems will appear repeatedly when different techniques are discussed. This will help students understand the pros and cons of different techniques when tacking specific problems. Upon completing this book, students should have the necessary basis for solving trigonometry problems in math competitions. In order to maximize learning results, students should attempt all the examples and practice problems once again after finishing the whole book. This will be helpful to re-enforce those techniques discussed and also offer a chance for students to reflect appropriateness of different techniques to so