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Haha, FUNdamental. Get it? Fundamental…FUN…nm.

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Our primary applications of trigonometry so far have been computational. We have not made full use of the properties of the functions to study the connections among the trigonometric functions themselves. We will now shift our emphasis more toward theory and proof, exploring where the properties of these special functions lead us, often with no immediate concern for real- world relevance at all. Hopefully in the process you will gain an appreciation for the rich and intricate tapestry of interlocking patterns that can be woven from the six basic trig functions – patterns that will take on even greater beauty later on when you can view them through the lens of calculus. Our primary applications of trigonometry so far have been computational. We have not made full use of the properties of the functions to study the connections among the trigonometric functions themselves. We will now shift our emphasis more toward theory and proof, exploring where the properties of these special functions lead us, often with no immediate concern for real- world relevance at all. Hopefully in the process you will gain an appreciation for the rich and intricate tapestry of interlocking patterns that can be woven from the six basic trig functions – patterns that will take on even greater beauty later on when you can view them through the lens of calculus.

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During today’s lesson you will learn: Definition of an identity Definition of an identity Basic Trigonometric Identities, Pythagorean Identities, Cofunction and Even-Odd Identities Basic Trigonometric Identities, Pythagorean Identities, Cofunction and Even-Odd Identities Simplify Trigonometric Expressions and Solve Trigonometric Equations Simplify Trigonometric Expressions and Solve Trigonometric Equations

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A.1 + 1 = 2 B.2(x – 3) = 2x – 6 C.x 2 + 3 = 7 D.(x 2 – 1)/(x+1) = x – 1 What are the similarities and differences in each of the equations above?

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Identities - statements which are true for all values of the variable for which both sides of the equation are defined Identities - statements which are true for all values of the variable for which both sides of the equation are defined In other words, an identity is an equation that is ALWAYS equal for values which are appropriate for its domain. In other words, an identity is an equation that is ALWAYS equal for values which are appropriate for its domain. Domain of validity – set of values for which an equation is defined. Domain of validity – set of values for which an equation is defined.

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Reciprocal Identities Reciprocal Identities Quotient Identities Quotient Identities

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Complete the exploration on p. 445 Complete the exploration on p. 445 6 min. 6 min. 1 st 4 min. – NO TALKING!! 1 st 4 min. – NO TALKING!! Last 2 min. – You can discuss with your neighbor Last 2 min. – You can discuss with your neighbor Write answers on your own paper!! Write answers on your own paper!!

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Use your calculator to evaluate each expression Write Value What conclusions can you draw about sin 2 + cos 2 ? (sin(25 )) 2 + (cos(25 )) 2 (sin(72 )) 2 + (cos(72 )) 2 (sin(90 )) 2 + (cos(90 )) 2 (sin(30 )) 2 + (cos(30 )) 2

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sin2 ++++ cos 2 = 1 1 + tan2 ==== sec 2 cot 2 ++++ 1 = csc 2 How can you change the first identity into the second? The third?

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Find sin & cos if tan = 8 & sin > 0.

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Type in as shown on your calculator CompareEvaluateWhat conclusions can you draw about sin (90 - ) & cos ? cos (90 - ) & sin ? tan (90 - ) & cot ? cot (90 - ) & tan ? sin (90 - 65 )cos (65 ) cos (90 - 71 )sin (71 ) tan (90 - 68 )cot (68 ) cot (90 - 47 )tan (47 )

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EvaluateCompare S or O EvaluateWhich functions are even? Which functions are odd? sin (-25 )sin (25 ) cos (-25 )cos (25 ) tan (-25 )tan (25 ) cot (-25 )cot (25 ) csc (-25 )csc (25 ) sec (-25 )sec (25 )

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Ex 3 If cot(- ) = 7.89, find tan ( - /2).

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Ex 4 Use basic identities to simplify the expression. a) 1 + tan 2 x b) sec 2 (-x) – tan 2 x csc 2 x csc 2 x

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a)(sin x) (tan x + cot x) b) tan x + tan x csc 2 xsec 2 x csc 2 xsec 2 x

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Ex 6 Find all values of x in the interval [0,2 ) that solve 2 cos x sin x – cos x = 0. a)Algebraically b)Verify by graphing

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a)4 cos 2 x – 4 cos x + 1 = 0 b)2 sin 2 x + 3 sin x = 2

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a)cos x = 0.75 b)sin 2 x = 0.4

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Mrs. Mullen asked the class to factor 1 – sin 2 x. Mrs. Mullen asked the class to factor 1 – sin 2 x. Anna wrote (1 – sin x)(1 + sin x). Alma wrote (cos x)(cos x). Who is correct? Explain how you made your choice. Who is correct? Explain how you made your choice.

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Using Fundamental Identities

Using Fundamental Identities

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