M1410501

# M1410501 - (Section 5.1 Fundamental Trig Identities 5.01...

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(Section 5.1: Fundamental Trig Identities) 5.01 CHAPTER 5: ANALYTIC TRIG SECTION 5.1: FUNDAMENTAL TRIG IDENTITIES PART A: WHAT IS AN IDENTITY? An identity is an equation that is true for all real values of the variable(s) for which all expressions contained within the identity are defined. For example, 1 x 2 ( ) 3 = 1 x 6 is an identity, because it holds true for all real values of x for which both sides of the equation are defined (i.e., for all real nonzero values of x ). If you are given the expression 1 x 2 ( ) 3 , it may be simplified to form the equivalent expression 1 x 6 .

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(Section 5.1: Fundamental Trig Identities) 5.02 PART B: LISTS OF FUNDAMENTAL TRIG IDENTITIES Memorize these in both “directions” (i.e., from left-to-right and from right-to-left). Reciprocal Identities csc x = 1 sin x sec x = 1 cos x cot x = 1 tan x sin x = 1 csc x cos x = 1 sec x tan x = 1 cot x Warning: Remember that the reciprocal of sin x is csc x , not sec x . Note: We typically treat “0” and “undefined” as reciprocals when we are dealing with trig functions. Your algebra teacher will not want to hear this, though! Quotient Identities tan x = sin x cos x and cot x = cos x sin x Technical Note: See Notes 4.10 on why this is consistent with SOH-CAH-TOA. Pythagorean Identities sin 2 x + cos 2 x = 1 1 + cot 2 x = csc 2 x tan 2 x + 1 = sec 2 x See Notes 4.30 ; see how to derive the last two from the first. Tip: The squares of csc x and sec x , which have the “Up-U, Down-U” graphs, are all alone on the right sides of the last two identities. Maybe because they are too profane …. Or maybe because they can never be 0 in value. (Why is that? Look at the left sides.)
(Section 5.1: Fundamental Trig Identities) 5.03 Cofunction Identities If x

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M1410501 - (Section 5.1 Fundamental Trig Identities 5.01...

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