Chapter 5 Analytic Trigonometry Pre-Calculus OHHS Mr. J. Focht

5.2 Proving Trig Identities What You'll Learn A Proof Strategy Proving Identities Disproving Non-Identities 5-2

Vocabulary A Trigonometric Proof is a sequence of expressions showing how to proceed from one given expression and ending at a specified expression. Each step of the proof will have to be explained. We usually start with the more complicated expression and work toward the simpler. 5-2

Example Show that is an identity. Factor a2-b2=(a-b)(a+b) Cancel/Reduce Subtraction 5-2

Now You Try Prove this identity 5-2

Rules to Prove By Do not add, multiply, subtract, divide, use a square root, or square both sides of an identity. These rules apply only to an equation – we are not dealing with an equation. Proceed by simple, easily explained steps. Try to stay on one side of the identity – but there will be exceptions. 5-2

Example Prove that LOVE can change to MATH. Rules: Change only 1 letter each step. Each step must be a word. LOVE MOVE MOTE MOTH MATH 5-2

Now You Try Change CAT to DOG 5-2

Proving an Identity Prove: tan x + cot x = sec x csc x. Quotient ID Finding LCD Fraction Addition 5-2 Continued on the next page

Proving an Identity Prove: tan x + cot x = sec x csc x. Fraction Addition Def of Fraction Multiplication Reciprocal ID 5-2

Now You Try Prove: (1-tan x)2 = sec2x – 2 tan x 5-2

Example: Using Difference of Squares Prove: Multiply by 1 Fraction Multiplication Cancel/Reduce Pythagorean ID 5-2

Now You Try Prove: 5-2

Example Working on Both Sides Prove: Pythagorean ID Cancel/Reduce Factor 5-2 Continued on the next page

Example Working on Both Sides Prove: Factor Cancel/Reduce Both sides reduced to the same expression 5-2

Now You Try Prove: sin2x cos3x = (sin2x – sin4x)(cos x) 5-2

Disproving Non-Identities Disproving a non-identity is easy. Just find 1 value that makes the wannabe identity false. Example: Prove this to be false. sin(2x) = 2 sin x Try x = 30: sin(2∙30) = 2 sin 30 sin 60= 2 sin 30 = 2 (½) is not true 5-2

Now You Try Show that is not an identity. 5-2

5.2 Home Work P. 460, #2, 7, 12, 16, 20, 24, 28, 34, 36, 46, 50, 58, 59, 62 5-2