Trigonometry without tears

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Trig without Tears (Printer-Friendly Version)

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Trig without Tears
or, How to Remember Trigonometric Identities
revised 7 Mar 2009 1997– Copyright © 1997 – 2009 Stan Brown, Oak Road Systems Summary: Faced with the large number of trigonometric identities, students tend to try to memorize them all Thatway lies disaster When you all. disaster. memorize a formula by rote, you have no way to know whether you’re remembering it correctly. I believe it is much more effective (and, in the long run, much easier to understand thoroughly how the trig easier) functions work, memorize half a dozen formulas and work out the formulas, rest as needed. That’s what these pages show you how to do. This is theprinter-friendly version. For reading on line, you probably want the browser-friendly version. You’re welcome to print copies of this page for your own use, and to link from your own Web pages to this page. But please don’t make any electronic copies and publish them on your Web page or elsewhere.

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Contents

Trig without Tears Introduction About Trigonometry About Trigwithout Tears Bonus Topics What You Won’t Find Here Notation Interval Notation Degrees and Radians The Six Functions The Basic Two: Sine and Cosine Expressions for Lengths of Sides The Other Four: Tangent, Cotangent, Secant, Cosecant Six Functions in One Picture

http://oakroadsystems.com/twt/pftwt.htm

23/04/2009

Trig without Tears (Printer-Friendly Version)

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Why Call ItSine? Functions of Special Angles Functions of 45° Functions of 30° and 60° Mnemonic for All Special Angles Functions of Any Angle Not Just Triangles Any More Why Bother? Reference Angles Signs of Function Values Examples: Function Values Identities for Related Angles Periodic Functions Solving Triangles Law of Sines Law of Cosines Detective Work: Solving All Types of Triangles The Cases Special Note:Side-Side-Angle Solving Triangles on the TI-83 or TI-84 The “Squared” Identities Sum and Difference Formulas Sine and Cosine of A±B Euler’s Formula Sine and Cosine of a Sum Sine and Cosine of a Difference Some Geometric Proofs Tangent of A±B Product-Sum Formulas Product to Sum Sum to Product Double Angle and Half Angle Formulas Sine or Cosine of a Double Angle Tangent of a Double Angle Sine orCosine of a Half Angle Tangent of a Half Angle Inverse Functions Principal Values Functions of Arcfunctions Example 1: cos(Arctan x) Example 2: cos(Arcsin x) Example 3: cos(Arctan 1/x)) Arcfunctions of Functions Example 4: Arccos(sin u) Example 5: Arcsec(cos u) Example 6: Arctan(sin u) Notes and Digressions

http://oakroadsystems.com/twt/pftwt.htm

23/04/2009

Trig without Tears(Printer-Friendly Version)

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The Problem with Memorizing On the Other Hand ... Proof of Euler’s Formula Polar Form of a Complex Number Powers and Roots of a Complex Number Square Root of i Principal Root of Any Number Multiple Roots Logarithm of a Negative Number Cool Proof of Double-Angle Formulas Great Book on Problem Solving What’s New

http://oakroadsystems.com/twt/pftwt.htm23/04/2009

Trig without Tears (Printer-Friendly Version)

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Trig without Tears Part 1:

Introduction

About Trigonometry

Trigonometry is fascinating! It started as the measurement (Greek metron) of triangles (Greek trigonon), but now it has been formalized under the influence of algebra and analytic geometry and we talk of trigonometric functions not just functions. sides andangles of triangles. triangles Trig is almost the ideal math subject. Big and complex enough to have all sorts of interesting odd corners, it is still small and regular enough to be taught thoroughly in a semester. (You can master the essential points in a week or so.) It has lots of obvious practical uses, some of which are actually taught in the usual trig course. And trig extends plenty of...
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