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

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