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 = 12Fallacy –Never true, regardless of the values of the variables. –Example: x = x + 1Identity –Always true, regardless of the values of the variables. –Example: x 2 ≥ 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

Pythagorean Identities What is known about the relationship between x, y and  ? x = cos  y = sin  Unit Circle: x 2 + y 2 = 1  (x, y) 1

Pythagorean Identities What does this imply about the relationship between sin  and cos  ? Unit Circle: x 2 + y 2 = 1  (cos , sin  ) 1 cos 2  + sin 2  = 1 Note: cos 2  = [cos  ] 2 and cos  2 = cos (  2 )

Pythagorean Identities cos 2 x + sin 2 x = 1 Dividing by cos 2 x gives … 1 + tan 2 x = sec 2 x Dividing by sin 2 x gives … cot 2 x + 1 = csc 2 x You should also recognize any variation of these. example: sin 2 x = 1 - cos 2 x

Sum & Difference Formulas 7  /12 = 9  /  /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 s A B Find cos s in terms of u and v. (note that s = u – v) v u (cos v, sin v) (cos u, sin u) s A B (cos s, sin s) (1, 0)

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

Sum & Difference Formulas Back to our original example … cos(7  /12) = cos(9  /  /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.

Sum & Difference Formulas Summarized

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!