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Name: Block: Date: Pre Calculus 12 Chapter 6 Trigonometric Identities Sec 6.1 Reciprocal, Quotient, and Pythagorean Identities and Proving Trigonometric Identities A Trigonometric Identity is a trigonometric equation that is true for all permissible values of the variable in the expressions on both sides of the equation. A proof {trigonometric one} requires you to manipulate one side {or both} of an algebraic equation so that they ultimately match. What you need to do for a proof is to show me that the left hand side of the equation is exactly the same of the right hand side of the equation. This is done by: 1. Finding common denominators to add fractions 2. Adding\Subtracting like terms together in an expression 3. Distributing through a set of brackets 4. Factoring common terms 5. Changing Division of Two Fractions into Multiplication 6. Canceling Common Terms 7. Using the “conjugate” to write a fraction in a different form. 8. Substituting Existing Identities YOU ARE NOT ALLOWED TO DO – EVER – TO MOVE “STUFF” FROM ONE SIDE OF AN EQUATION TO THE OTHER! YOU ARE NOT ALLOWED TO “SOLVE” – You can only manipulate how “stuff” looks. An Identity is a statement that we have either: a. assumed it to be true (a definition) b. or have already proved it to be true Chapter 4/5 has some identities that we have used: 1 1 1 sec θ = csc θ = cot θ = cos θ sin θ tan θ tan θ = sin θ cos θ cot θ = The “two” main types of problems for this section are: a. Simplify: you attempt to change from the given expression to a much smaller one b. Prove: Show that one side is exactly the same as the “other side” Ex1 : Simplify: sec 2 x × cos x tan x sec x cos θ sin θ Ex 2: Prove: sin θ cot θ = cos θ cot θ = cos θ csc θ cos θ (sec θ − 1) = 1 − cos θ 1 − tan θ = − tan θ 1 − cot θ Pythagorean Identities Given a point P (x, y) on a terminal arm of an angle θ, in standard position. Other Form (using tan) Other form (using co-tan) Pythagorean Identities: sin 2 θ + cos 2 θ = 1 1 + tan 2 θ = sec 2 θ Notation, we’ve agreed that: sin 2 x = ( sin x ) Using the previous identities, prove: sin 2 θ cot 2 θ = 1 − sin 2 θ 1 − cos x sin x = sin x 1 + cos x 1 + cot 2 θ = csc 2 θ 2 cos θ cos θ + = 2sec θ 1 + sin θ 1 − sin θ tan x (1 − cos x ) sin 3 x = 1 + cos x sec x (1 − cos θ )(1 + tan θ ) = tan 2 2 2 θ sec 2 θ + csc 2 θ = ( tan θ + cot θ ) 2