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Trigonometry. Trigonometric Identities. 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.
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Trigonometry Trigonometric Identities
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 Identities Quotient Identities
cos2θ + sin2θ = 1 By Pythagoras’ Theorem x2 + y2 = r2 (x, y) r y Divide both sides by r θ x
Trigonometric Identities Pythagorean Identities The fundamental Pythagorean identity Divide by sin2 x Divide by cos2 x
Identities involving Cosine Rule Using the usual notation for a triangle, prove that c(bcosA–acosB) = b2–a2
Identities involving Cosine Rule Using the usual notation for a triangle, prove that c(bcosA–acosB) = b2–a2
Trigonometric Formulas Page 9 of tables
Solving Trig Equations • To solve trigonometric equations: • If there is more than one trigonometric function, • use identities to 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
1 2 1 2 cos2A = (1 + cos 2A). cos2A = (1 + cos 2A) (i) Using cos 2A = cos2A – sin2A, or otherwise, prove cos 2A = cos2A – (1 – cos2A) sin2A cos 2A = cos2A – 1 + cos2A 1 + cos 2A = 2cos2A – 1 2005 Paper 2 Q4 (b)
1 –1 180º 360º (ii) Hence, or otherwise, solve the equation 1 + cos2x = cosx, where 0º ≤ x ≤ 360º. 1 + cos2x 2cos2x = cosx From (i) 2cos2x –cosx = 0 cos x(2cosx – 1) = 0 cos x = 0 2005 Paper 2 Q4 (b)
Expand Collect like terms Rearrange Factorise
1 π 2π –1 Replace t with sinx