Trigonometry (cheatsheet)

Trigonometry (cheatsheet) - Trigonometry CheatSheet 1 How...

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Trigonometry CheatSheet 1 How to use this document This document is not meant to be a list of formulas to be learned by heart. The first few formulas are very basic (they descend from the definition and/or Pythagoras’ theorem) and you might want to memorize them, but you should be able to retrieve everything else from these two/three basic formulas, and you should be able to do it quickly (during an exam you don’t want to invest ten minutes in recalculating a formula that you might or might not need!!!). These formulas are arranged in a logical order, starting from the most basic, so that each formula can be retrieved using formulas that come first only. Every formula is accompanied by a short explanation about how you can retrieve it (there might be more than one method). 2 Formulas that come from Pythagoras’ theorem and/or the def- inition cos 2 x + sin 2 x = 1 (1) It’s an immediate consequence of the definition and Pythagoras’ theorem tan 2 x + 1 = sec 2 x (2) It follows from the definition of tangent, secant and the previous formula cot 2 x + 1 = csc 2 x (3) Same as before cos ( - x ) = cos x (4) Cosine is even. It follows from the definition. sin ( - x ) = - sin x (5) tan ( - x ) = - tan x (6) cot ( - x ) = - cot x (7) Sine, tangent and cotangent are all odd. It follows from the definition. The period of sine, cosine secant and cosecant is 2 π , and the period of tangent and cotangent is π : sin ( x + 2 ) = sin x (8) cos ( x + 2 ) = cos x (9) tan ( x + ) = tan x (10) 1
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3 Addition of angles You might want to take the first two formulas as black boxes and memorize them. However, if you know about the complex numbers you can retrieve them both very quickly. sin ( x + y ) = sin x cos y + cos x sin y (11) cos ( x + y ) = cos x cos y - sin x sin y (12) These two formulas can be derived using the property of exponentials e i ( x + y ) = e ix e iy Plug the Euler’s identity e ix = cos x + i
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This note was uploaded on 11/03/2011 for the course MATH 1090 taught by Professor Greenwood during the Spring '08 term at MIT.

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Trigonometry (cheatsheet) - Trigonometry CheatSheet 1 How...

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