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3.9 Proving Trig Identities

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Using fundamental identities (from 3-8), we can prove other identities Best way is to start with one side and manipulate algebraically or use fundamental identities to get it to be equivalent to other side Note: There are no hard fast rules to adhere to… sometimes we just have to try something. Hints: – try writing in its component parts or reciprocal – add fractions (common denominator) – multiply fraction by conjugate of bottom – utilize Pythagorean identities If it asks for a counterexample you need to find just one value that it will not work for. (it might actually work for some values – so think about your choice) (all sines & cosines)

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Lets try some!!! There are 6 problems around the classroom. Work your way around the room doing as many problems as you can Use your friends for help

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Homework #309 Pg 174 #1–49 odd

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Ex 1) Prove: Work more complicated side

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Ex 2) Prove: Work this side Write in terms of sin ϕ & cos ϕ Pythag Ident (cos 2 θ + sin 2 θ = 1)

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Ex 3) Prove: Work this side Mult by conj of bottom secθ & cosθ are reciprocals! 1 Pythag identity (cos 2 θ + sin 2 θ = 1)

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Ex 4) Prove: Work this side Write tanθ in terms of sinθ & cosθ cscθ & sinθ are reciprocals! Use Pythag identity (cos 2 θ + sin 2 θ = 1) cosθ & secθ are reciprocals!

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Ex 5) Show that sin(β + θ) = sinβ + sinθ is not an identity. You just need 1 counterexample. There are lots of answers!! Here is just one: LHS: RHS:

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Ex 6) Prove: Pythag identity (cos 2 θ + sin 2 θ = 1) Work this side Diff of 2 squares 1

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Verifying Trigonometric Identities What is an Identity? An identity is a statement that two expressions are equal for every value of the variable. Examples:

Verifying Trigonometric Identities What is an Identity? An identity is a statement that two expressions are equal for every value of the variable. Examples:

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