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Trigonometric Identities Reciprocals Quotients Pythagorean When simplifying expressions, or factoring expressions, look for common factors. If none exist, look for related functions from the identities that will assist in the simplification. Examples: 1. Substitute: 2. Factor difference of perfect squares. Substitute: 3. Version used: Substitute: 4. Multiply and express as a monomial: Version used: 1) Simplify the expression (1-cos2x)(cscx) to a single trigonometric function. 2) Simplify 1 cos2 x sec 2 x 1 Ans: cos2x 3) Simplify: cos2x + cos2x tan2x Ans: cos2x + cos2x = 1 sin cos cos sin 4) Simplify: 1 sin Ans: 5) secθ (cosθ) – cos2θ 1 cos Ans: 1 – cos2θ = sin2θ Ans: Sin x More Examples 2. Simplify the expression (1 – cos2x)(cscx) to a single trig functions 3. Simplify: cos2x + (cos2x)(tan2x) 4. Write this expression as a monomial with a single trig function: (secθ)(cosθ) – cos2θ 6. Write as a monomial with a single trig function a) sinθcotθ b) sinθsecθ c) secθcotθsinθ d) secθsinθcscθ e) cscθ(1- cos2θ) f) sinθ(cot2θ + 1) g) secθcosθ – cos2θ Using Identities in Equation Solving IF there is more than one trig function in the equation, identities are needed to reduce the equation to a single trig function 1) Solve: 2cos2θ + 3sinθ – 3 = 0 2(1 – sin2 θ) + 3sin θ – 3 = 0 2 – 2sin2 θ + 3sin θ – 3 = 0 -2sin2 θ + 3sin θ – 1 = 0 replace using sin2 θ + cos2 θ= 1 distribute combine like terms 0 = 2sin2 θ - 3sin θ + 1 move to other side to make a >0 0 = (2sinθ - 1)(sin θ – 1) factor sin θ = ½ sin θ = 1 t-chart and solve for sin θ 1) RA = 30 Quadrantal do the steps to solve for θ 2) positive θ = 90° 3) QI Θ = RA QII θ = 180 – RA Θ = 30° θ = 180 – 30 Θ = 150° Θ = {30°, 90°, 150°} 2) 2cos2θ – sinθ = 1 3) sec2θ – tanθ – 1 = 0 4) cosθ = secθ 5) 2sinθ = cscθ