Trigonometry Trigonometric Identities

 An identity is an equation which is true for all values of the variable.  There are many trig identities that are useful in changing the appearance of an expression.  The most important ones should be committed to memory.

Trigonometric Identities Reciprocal IdentitiesQuotient Identities

cos 2 θ + sin 2 θ = 1 θ (x, y) r y x By Pythagoras’ Theorem x 2 + y 2 = r 2 Divide both sides by r

Trigonometric Identities  Pythagorean Identities  The fundamental Pythagorean identity  Divide by sin 2 x  Divide by cos 2 x

Identities involving Cosine Rule  Using the usual notation for a triangle, prove that c(bcosA – acosB) = b 2 – a 2

 Using the usual notation for a triangle, prove that c(bcosA – acosB) = b 2 – a 2 Identities involving Cosine Rule

Trigonometric Formulas Page 9 of tables

Replace B with – B

Replace B with A

Replace A with – A

Solving Trig Equations  To solve trigonometric equations:  If there is more than one trigonometric function, use identitiesto simplify  Let a variable represent the remaining function  Solve the equation for this new variable  Reinsert the trigonometric function  Determine the argument which will produce the desired value

cos 2 A = (1 + cos 2A) 1212 (1 – cos 2 A) (i) Using cos 2A = cos 2 A – sin 2 A, or otherwise, prove 1212 cos 2 A = (1 + cos 2A) Paper 2 Q4 (b) cos 2A = cos 2 A –sin 2 A cos 2A = cos 2 A – 1 + cos 2 A cos 2A = 2cos 2 A– 11 +

= cos x (ii) Hence, or otherwise, solve the equation 1 + cos 2x = cos x, where 0º ≤ x ≤ 360º Paper 2 Q4 (b) 1 + cos 2x 2cos 2 x – cos x = 0 2cos 2 x cos x(2cos x – 1) = 0 From (i) 360º180º 1 –1 –1 cos x = 0

Expand Collect like terms Rearrange Factorise

Replace t with sin x 2π2ππ 1 –1 –1