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MTH 112 Elementary Functions Chapter 6 Trigonometric Identities, Inverse Functions, and Equations

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

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MTH 112 Elementary Functions Chapter 6 Trigonometric Identities, Inverse Functions, and Equations

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  1. MTH 112Elementary FunctionsChapter 6Trigonometric Identities, Inverse Functions, and Equations Section 1 Identities: Pythagorean and Sum and Difference

  2. 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

  3. 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

  4. (x, y)  1 Unit Circle: x2 + y2 = 1 Pythagorean Identities • What is known about the relationship between x, y and ? • x = cos  • y = sin 

  5. 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)

  6. 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

  7. 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) = ???

  8. (cos s, sin s) (cos v, sin v) s A (cos u, sin u) B (1, 0) Sum & Difference Formulas Find cos s in terms of u and v. (note that s = u – v) s B u A v

  9. Sum & Difference Formulas The two expressions for AB gives … Substituting –v for v gives … Using the cofunctions identities gives … Substituting –v for v gives …

  10. 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

  11. Sum & Difference Formulas • Using the sum & difference formulas for sine and cosine, similar formulas for tangent can also be established.

  12. Sum & Difference FormulasSummarized

  13. Simplifying TrigonometricExpressions • 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!

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