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Week 8 Sum and Difference Formulas for Cosine Presentation

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Week 8 9.1 Summary of Trig Identities Pythagorean Quotient Even Odd Reciprocal cos2x + sin2x = 1 = sin x / cos x tan (-x) = - tan x sin x = 1 / csc x tan x 1 + cot2x = css2x = cos x / sin x cot (-x) = - cot x cos x = 1 / sec x cot x 1 + tan2x = sec2x sin (-x) = - sin x tan x = 1 / cot x csc (-x) = - csc x x csc x = 1 / sin Example 1 Graph both sides of the identity cot (x) = 1 / tan x. In other words, on the graphing calculator graph y = cot x and y = 1 / tan x Using the reciprocal identity on the previous slides, we know these are the same line because cot x = 1 / tan x so any graphs of both of these would look the same. If any equations are the same graph, they are likely identities Example 2. Verifying tan (x) * cos (x) = sin (x) Step 1. Work on one side of equation, easiest to simplify the more complex side tan (x) * cos (x) is the complex side Step 2. Look for opportunities to simplify (not any) Step 3. Look for identity opportunities and make substitutions tan (x) = sin (x) / cos (x) ← Substitute tan(x) * cos (x) = ( sin(x) / cos (x) ) * cos ← eliminate = sin (x) Example 3. Verify using even-odd identities ( 1 + sin x ) [ 1 + sin (-x) ] = cos2 x ( 1 + sin x) ( 1 - sin (x) ) ← sin (-x) = - sin (x) 1 - sin (x) + sin (x) - sin2 x ← FOIL 1 - sin2 x = cos2 x Theorem ← cos2 x + sin2 x = 1 Pythagorean Example 4. Vertify (sec2 x - 1 / sec2 x) = sin2 x Remember, start with the complicated side! sec2 x - 1 / sec 2 x = (tan2 x + 1) ...
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