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Chapter 5: Analytic Trigonometry Section 5.1a: Fundamental Identities HW: p. 451-452 1-7 odd, 27-49 odd Is this statement true? x 1 x 1 x 1 2 This identity is a true sentence, but only with the qualification that x must be in the domain of both expressions. If either side of the equality is undefined (i.e., at x = –1), then the entire expression is meaningless!!! The statement tan sin cos is a trigonometric identity because it is true for all values of the variable for which both sides of the equation are defined. The set of all such values is called the domain of validity of the identity. Basic Trigonometric Identities Reciprocal Identities 1 1 1 csc sec cot sin cos tan 1 1 1 sin cos tan csc sec cot Quotient Identities sin tan cos 0 cos cot sin is in the domain of validity of exactly three of the basic identities. Which three? Basic Trigonometric Identities Reciprocal Identities 1 1 1 csc sec cot sin cos tan 1 1 1 sin cos tan csc sec cot Quotient Identities sin tan cos cos cot sin For exactly two of the basic identities, one side of the equation is defined at 0 and the other side is not. Which two? Basic Trigonometric Identities Reciprocal Identities 1 1 1 csc sec cot sin cos tan 1 1 1 sin cos tan csc sec cot Quotient Identities sin tan cos cos cot sin For exactly three of the basic identities, both sides of the equation are undefined at 0 . Which three? Pythagorean Identities Recall our unit circle: sint P cost What are the coordinates of P? (1,0) P cos t ,sin t So by the Pythagorean Theorem: cos t sin t 2 Divide by cos t 2 1 : cos t sin t 1 2 2 2 cos t cos t cos t 2 2 2 1 tan t sec t 2 2 Pythagorean Identities Recall our unit circle: sint P What are the coordinates of P? P cos t ,sin t (1,0) cost So by the Pythagorean Theorem: cos t sin t 2 Divide by 1 sin t : 2 cos t sin t 1 2 2 2 sin t sin t sin t 2 2 2 cot t 1 csc t 2 2 Pythagorean Identities cos sin 1 1 tan sec 2 2 cot 1 csc 2 Given 2 tan 5 2 and cos 0, find sin 2 and cos . sec2 1 tan 2 1 52 26 sec 26 1 cos 26 We only take the positive answer…why? sin tan 5 5 sin 5cos cos 5 1 sin 5 sin 26 26 Cofunction Identities sin cos 2 cos sin 2 tan cot 2 cot tan 2 sec csc 2 csc sec 2 Can you explain why each of these is true??? Odd-Even Identities If sin x sin x cos x cos x csc x csc x sec x sec x tan x tan x cot x cot x cos 0.34 , find sin 2. Sine is odd sin sin 2 2 Cofunction Identity cos 0.34 Simplifying Trigonometric Expressions Simplify the given expression. sin x sin x cos x sin x sin 2 x cos 2 x 3 2 sin x 1 sin x How can we support this answer graphically??? Simplifying Trigonometric Expressions Simplify the given expression. 2 2 sec x 1 sec x 1 sec x 1 tan x 2 2 2 sin x sin x sin x 2 sin x 1 2 2 cos x sin x 1 2 sec x 2 cos x Graphical support? Simplifying Trigonometric Expressions Simplify the given expressions to either a constant or a basic trigonometric function. Support your result graphically. sin x cos x sin x 1 1 tan x cos x cos x cos x sin x cos x 1 cot x 1 sin x sin x sin x cos x sin x sin x tan x cos x sin x cos x cos x Simplifying Trigonometric Expressions Simplify the given expressions to either a constant or a basic trigonometric function. Support your result graphically. sec2 u tan 2 u 1 1 2 2 cos v sin v 1 Simplifying Trigonometric Expressions Use the basic identities to change the given expressions to ones involving only sines and cosines. Then simplify to a basic trigonometric function. sec y tan y sec y tan y sec y 1 sin y 1 sin y cos y cos y cos y cos y 1 cos y 1 sin y 1 sin y cos y cos y cos y 1 Simplifying Trigonometric Expressions Use the basic identities to change the given expressions to ones involving only sines and cosines. Then simplify to a basic trigonometric function. 1 sin y 1 sin y cos y cos y cos y 1 1 sin y sin y sin y 1 sin y cos y cos y cos y cos y 2 2 2 cos y Simplifying Trigonometric Expressions Use the basic identities to change the given expressions to ones involving only sines and cosines. Then simplify to a basic trigonometric function. 1 1 2 sec2 x csc x cos x sin x 1 1 sec2 x csc2 x 2 2 cos x sin x 1 cos 2 x sin 2 x sin x 2 2 2 2 2 cos x sin x sin x cos x sin x cos x sin x 1 sin x Let’s start with a practice problem… Simplify the expression cos x sin x cos x cos x sin x 1 sin x 1 sin x cos x 1 sin x cos x cos x 1 sin x cos x cos x sin x 1 sin x 1 sin x cos x cos x sin x sin x How about some 1 sin x cos x graphical support? 1 sin x 1 sec x 1 sin x cos x cos x 2 2 Combine the fractions and simplify to a multiple of a power of a basic trigonometric function. 1 1 1 sin x 1 sin x 1 sin x 1 sin x 1 sin x 1 sin x 1 sin x 1 sin x 2 2 1 sin x 1 sin x 2 2 1 sin x cos x 1 sin x 1 sin x 2sec x 2 Combine the fractions and simplify to a multiple of a power of a basic trigonometric function. sec x 1 sec x 1 1 1 sec x 1 sec x 1 sec x 1 sec x 1 2 2 sec x 1 2 tan 2 x 2 cot x 2 Combine the fractions and simplify to a multiple of a power of a basic trigonometric function. sin x 1 cos x sin x 1 cos x 1 cos x sin x sin x 1 cos x 2 2 sin x cos x 2 cos x 1 sin x 1 cos x 2 1 cos x 2 2cos x sin x 1 cos x sin x 1 cos x 2 2 2 2csc x sin x Quick check of your algebra skills!!! Factor the following expression (without any guessing!!!) 12 x 8 x 15 a c 180 b 8 2 What two numbers have a product of –180 and a sum of 8? 10,18 12 x 10 x 18 x 15 Group terms and factor: 2 x 6 x 5 3 6 x 5 Rewrite middle term: Divide out common factor: 2 6x 5 2x 3 Write each expression in factored form as an algebraic expression of a single trigonometric function. e.g., sin x 2sin x 1 2 Substitute: Let 2sin x 3sin x 1 u sin x sin x 2sin x 1 u 2u 1 2 2 Factor: “Re”substitute for your answer: u 1 2 sin x 1 2 Write each expression in factored form as an algebraic expression of a single trigonometric function. e.g., 2sin x 3sin x 1 2 2 2 1 cos x 1 2 cos x cos x sec x 2 1 cos x Write each expression in factored form as an algebraic expression of a single trigonometric function. e.g., 2sin x 3sin x 1 sin x cos x 1 sin x 1 sin x 1 2 2 sin x sin x 2 2 Let u sin x u u 2 u 1 u 2 2 sin x 1 sin x 2 Write each expression in factored form as an algebraic expression of a single trigonometric function. e.g., 2sin x 3sin x 1 sec x sec x tan x 2 2 sec2 x sec x sec2 x 1 2sec x sec x 1 2 2sec x 1 sec x 1 Write each expression as an algebraic expression of a single trigonometric function. e.g., 2sin x 3 tan 1 tan 1 tan 1 tan 1 1 tan 1 tan 2 tan x sec x 1 sec x 1 sec x 1 sec x 1 sec x 1 sec x 1 2 2 sec x 1