Download Trigonometry Cheat Sheet: Functions, Formulas, and Identities and more Exams Trigonometry in PDF only on Docsity! TRIGONOMETRY CHEAT SHEET 2 → → → © Paul Dawkins - https://tutorial.math.lamar.edu Right triangle definition Definition of the Trig Functions Unit Circle Definition For this definition we assume that 0 < θ < π or 0◦ < θ < 90◦. 2 For this definition θ is any angle. sin(θ) = opposite hypotenus e csc(θ) = hypotenuse opposite sin(θ) = y = y csc(θ) = 1 cos(θ) = adjacent sec(θ) = hypotenuse 1 y hypotenus e adjacen t cos(θ) = x = x sec(θ) = 1 tan(θ) = opposite adjace nt cot(θ) = adjacent opposit e 1 tan(θ) = y x x cot(θ) = x y Domain Facts and Properties Period The domain is all the values of θ that can be plugged into the function. sin(θ), θ can be any angle cos(θ), θ can be any angle tan(θ), θ n + 1 π, n = 0, ±1, ±2, . . . The period of a function is the number, T , such that f (θ + T ) = f (θ). So, if ω is a fixed number and θ is any angle we have the following periods. sin (ω θ) T = 2π ω cos (ω θ) T = 2π ω csc(θ), θ /= nπ, n = 0, ±1, ±2, . . . π sec(θ), θ n + 1 2 π, n = 0, ±1, ±2, . . . tan (ω θ) → T = ω 2π cot(θ), θ nπ, n = 0, ±1, ±2, . . . csc (ω θ) → T = ω2π sec (ω θ) → T = ω Range The range is all possible values to get out of the function. cot (ω θ) T = π ω TRIGONOMETRY CHEAT SHEET © Paul Dawkins - https://tutorial.math.lamar.edu −1 ≤ sin(θ) ≤ 1 −1 ≤ cos(θ) ≤ 1 −∞ < tan(θ) < ∞ −∞ < cot(θ) < ∞ sec(θ) ≥ 1 and sec(θ) ≤ −1 csc(θ) ≥ 1 and csc(θ) ≤ −1 3 2 3 2 TRIGONOMETRY CHEAT SHEET © Paul Dawkins - https://tutorial.math.lamar.edu For any ordered pair on the unit circle (x, y) : cos(θ) = x and sin(θ) = y Example cos 5π = 1 sin 5π = − √ 3 2 2 2 − 2 1sin cos (α − β) 2 2 = 1 2 2 TRIGONOMETRY CHEAT SHEET © Paul Dawkins - https://tutorial.math.lamar.edu Definition Inverse Trig Functions Inverse Properties y = sin−1(x) is equivalent to x = sin(y) y = cos−1(x) is equivalent to x = cos(y) y = tan−1(x) is equivalent to x = tan(y) Domain and Range Function Domain Range cos cos−1(x) = x cos−1 (cos(θ)) = θ sin sin−1(x) = x sin−1 (sin(θ)) = θ tan tan−1(x) = x tan−1 (tan(θ)) = θ Alternate Notation sin−1(x) = arcsin(x) y = sin−1(x) −1 ≤ x ≤ 1 π — 2 ≤ πy ≤ 2 cos−1(x) = arccos(x) y = cos−1(x) −1 ≤ x ≤ 1 0 ≤ y ≤ π y = tan−1(x) −∞ < x < ∞ − π < y < π tan−1(x) = arctan(x) Law of Sines, Cosines and Tangents Law of Sines Law of Tangents sin ( α ) sin ( β )= sin ( γ ) = a − b t an 1 (α − β) a b c Law of Cosines a2 = b2 + c2 − 2bc cos(α) a + b = tan 1 (α + β) b − c t an 1 (β − γ) b + c tan (β + γ) b2 = a2 + c2 2ac cos(β) a − c tan 1 (α − γ) c2 = a2 + b2 − 2ab cos(γ) Mollweide’s Formula a + c = tan 1 (α + γ) a + b 1 = 2 c 2
