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Formulas from Trigonometry 1 5 Radians and Degrees π − θ = cos θ sin 2 If θ is an angle expressed in radians, and α is the same angle measured in degrees, then θ α = . π 180 tan 6 Definition of Trigonometric Functions sin θ = opposite hypotenuse cos θ = adjacent hypotenuse tan θ = opposite adjacent cot θ = adjacent opposite sec θ = hypotenuse adjacent csc θ = hypotenuse opposite π − θ = cot θ 2 cot π − θ = tan θ 2 π csc − θ = sec θ 2 Symmetry, Periodicity sin(−θ) = − sin θ sin(θ + 2π) = sin θ cos(θ + 2π) = cos θ tan(θ + π) = tan θ 7 π cos − θ = sin θ 2 π sec − θ = csc θ 2 The length s of an arc on a circle of radius r opposite the angle θ, measured in radians, is given by the formula s = rθ . 2 Co-Functions cos(−θ) = cos θ sin(θ + π) = − sin θ cos(θ + π) = − cos θ tan(−θ) = − tan θ Pythagorean Identities sin2 θ + cos2 θ = 1 tan2 θ + 1 = sec2 θ 3 Values for Specific Angles cot2 θ + 1 = csc2 θ θ 0o sin θ cos θ tan θ 0 0 1 0 30o 45o 60o 90o π/6 √π/4 √π/3 π/2 3/2 1 √1/2 √2/2 2/2 1/2 0 √3/2 √ 3/3 1 3 ±∞ 8 Addition, Subtraction sin(α + β) = sin α cos β + cos α sin β sin(α − β) = sin α cos β − cos α sin β 4 Fundamental Identities sin θ cos θ 1 sec θ = cos θ 1 cot θ = tan θ tan θ = cos(α + β) = cos α cos β − sin α sin β cos θ sin θ 1 csc θ = sin θ cos(α − β) = cos α cos β + sin α sin β cot θ = 1 tan(α + β) = tan α + tan β 1 − tan α tan β tan(α − β) = tan α − tan β 1 + tan α tan β 9 π 2 arcsin(−x) = − arcsin x arccos(−x) = π − arccos x arctan(−x) = − arctan x Double Angles, Half Angles arcsin x + arccos x = sin 2θ = 2 sin θ cos θ cos 2θ = cos2 θ − sin2 θ = 1 − 2 sin2 θ = 2 cos2 θ − 1 2 tan θ tan 2θ = 1 − tan2 θ 1 − cos θ 1 + cos θ 2 θ 2 θ sin = cos = 2 2 2 2 θ sin θ 1 − cos θ tan = = 2 1 + cos θ sin θ 10 13 sin α sin β sin γ = = a b c 14 c2 = a2 + b2 − 2ab cos γ 15 1 A = ab sin γ 2 Heron’s Formula: 1 With s = (a + b + c) (= semiperimeter) the 2 area A of a triangle is given by Sinusoids A= y = a sin(bx + c) + d Amplitude Period Phase shift Maximum value Minimum value Miscellaneous Formulas Area A of a triangle: Sinusoids are functions of the form 12 Law of Cosines Products 1 sin α sin β = (cos(α − β) − cos(α + β)) 2 1 cos α cos β = (cos(α − β) + cos(α + β)) 2 1 sin α cos β = (sin(α − β) + sin(α + β)) 2 11 Law of Sines |a| 2π/|b| −c/b d + |a| d − |a| Inverse Functions For −1 ≤ x ≤ 1 we define: y = sin−1 x = arcsin x if and only if −π/2 ≤ y ≤ π/2 and sin y = x. For −1 ≤ x ≤ 1 we define: y = cos−1 x = arccos x if and only if 0 ≤ y ≤ π and cos y = x. For any real number x we define: y = tan−1 x = arctan x if and only if −π/2 < y < π/2 and tan y = x 2 p s(s − a)(s − b)(s − c)