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MTH 112 Elementary Functions Chapter 6 Trigonometric Identities, Inverse Functions, and Equations Section 1 Identities: Pythagorean and Sum and Difference Statements in Mathematics Conditional – May be true or false, depending on the values of the variables. – Example: 2x + 3y = 12 Fallacy – Never true, regardless of the values of the variables. – Example: x = x + 1 Identity – Always true, regardless of the values of the variables. – Example: x2 ≥ 0 Identities from Chapter 5 Reciprocal Relationships Tangent & Cotangent in terms of Sine and Cosine Cofunction Relationships – Note: For degrees, replace /2 with 90 Even & Odd Functions sec x 1 cos x tan x cos x sin x 2 cos( x) cos x sec( x) sec x csc x sin x cos x 1 sin x cot x cot x cot x tan x 2 sin( x) sin x csc( x) csc x 1 tan x cos x sin x csc x sec x 2 tan( x) tan x cot( x) cot x Pythagorean Identities What is known about the relationship between x, y and ? • x = cos • y = sin (x, y) 1 Unit Circle: x2 + y2 = 1 Pythagorean Identities What does this imply about the relationship between sin and cos ? (cos , sin ) 1 cos2 + sin2 = 1 Unit Circle: x2 + y2 = 1 Note: cos2 = [cos ]2 and cos 2 = cos (2) Pythagorean Identities cos2x + sin2x = 1 Dividing by cos2x gives … • 1 + tan2x = sec2x Dividing by sin2x gives … • cot2x + 1 = csc2x You should also recognize any variation of these. • example: sin2x = 1 - cos2x Sum & Difference Formulas 7/12 = 9/12 - 2/12 = 3/4 - /6 How can we use the known values of the trig functions of 3/4 and /6 to determine the trig values of 7/12? – Example: • cos(7/12) = cos(3/4 - /6) = ??? Sum & Difference Formulas Find cos s in terms of u and v. (note that s = u – v) (cos s, sin s) s (cos v, sin v) B (cos u, sin u) A u A s v B (1, 0) AB 2 2(cos u cos v sin u sin v) AB 2 2 cos s Sum & Difference Formulas The two expressions for AB gives … cos(u v) cos u cos v sin u sin v Substituting –v for v gives … cos(u v) cos u cos v sin u sin v Using the cofunctions identities gives … sin( u v) sin u cos v cos u sin v Substituting –v for v gives … sin( u v) sin u cos v cos u sin v Sum & Difference Formulas Back to our original example … cos(7/12) = cos(9/12 - 2/12) = cos(3/4 - /6) = cos(3/4) cos(/6) – sin(3/4) sin(/6) = -(√2)/2 • (√3)/2 – (√2)/2 • 1/2 = -(√6)/4 – (√2)/4 = -[(√6) + (√2)]/4 Sum & Difference Formulas Using the sum & difference formulas for sine and cosine, similar formulas for tangent can also be established. tan u tan v tan( u v) 1 tan u tan v tan u tan v tan( u v) 1 tan u tan v Sum & Difference Formulas Summarized sin( u v) sin u cos v cos u sin v cos(u v) cos u cos v sin u sin v tan u tan v tan( u v) 1 tan u tan v Simplifying Trigonometric Expressions No general procedure! But the following will help. – – – – – – – Know the basic identities. Multiply to remove parenthesis. Factor. Change all functions to sine and/or cosine. Combine or split fractions: (a+b)/c = a/c + b/c Other algebraic manipulations Know the basic identities. Try something and see where it takes you. If you seem to be getting nowhere, try something else!