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1 MCR3U Trigonometric Identities Use the above ratios to determine: (a) sin cos (b) sin2θ + cos2 θ A trigonometric identity is any mathematical equation with trigonometric expressions that is true for all values of the angle (variable) Fundamental Trigonometric Identities sin x cos x 1) Quotient Identity tan x 2) Pythagorean Identities sin2 x + cos2 x = 1 sin2 x = 1 – cos2 x cos2x = 1 – sin2 x 1 + tan2 x = sec2 x 1 + cot2 x = csc2 x 3) Reciprocal Identities csc x cos x 1 1 , sec x , cot x sin x cos x sin x 2 ** It is not always obvious that both sides of a trigonometric expression are equal. To prove that it is an identity, a proof that shows that both sides of the expression are equal is required. To prove that a given expression is an identity, follow the steps: Separate the two sides of the equation using L.S. R.S. method Simplify the more complicated side until it is identical to the other side or simplify both sides into the same expression Strategies to use: (i) express all tangent & reciprocal functions in terms of sine and cosine (ii) apply a Pythagorean identity if required (iii) factor or find a common denominator where necessary x y y , cos = , tan = to prove the r r x 1 + tan2 = sec 2 Example # 1 – Use the definitions sin = identity Left Side Right Side Example # 2 – Simplify each expression using the trigonometric identities a) (cos x) (tan x) b) cos2 x + sin2 x c) (sin x) (cos x) 1 – sin2 x 3 Example # 3 – Prove each identity b) tan x a) (tan x) (cos x) = sin x c) 1 1 tan x sin x cos x tan 2 x sin 2 x 2 1 tan x LS Example # 4 – Factor each expression a) 1 – sin2 b) sin – sin2 d) sin4 – cos4 c) sin2 – 2 sin + 1