1_-_Introduction_and_Review_-_Trigonomet

1_-_Introduction_and_Review_-_Trigonomet - INTRODUCTION AND...

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Unformatted text preview: INTRODUCTION AND REVIEW REVIEW ME 221 STATICS ME CE 221 Statics ME 222 Mechanics of Deformable Solids CE 305 CE 312 CE 321 CE 337 ME 423 ME 425 ME 426 ME 477 ME 495 GENERAL REVIEW GENERAL Trigonometry GEOMETRY/TRIGONOMETRY GEOMETRY/TRIGONOMETRY a b a b b a b a TRIGONOMETRY TRIGONOMETRY If A = 90o and the lengths AB and AC are equal, then; C β Angle θ = Angle β = 45o Angle Angle θ > Angle β Angle Angle θ < Angle β Angle Cannot be determined A θ B TRIGONOMETRY TRIGONOMETRY If A = 90o, AB = 4” and AC = 3”, then; a) BC = 1” b) BC = 7” c) BC = 25” d) BC = 5” 3” C β BC = ? A θ 4” B BC = 3 + 4 = 5 2 2 TRIGONOMETRY TRIGONOMETRY Right Triangle (sides relative to angle b) A a Hypotenuse Opposite C c Adjacent b B TRIGONOMETRY TRIGONOMETRY Right Triangle (AC)2 + (BC)2 = (AB)2 (BC) (AB) a + b + c = 180o 180 A a C c bB TRIGONOMETRY TRIGONOMETRY A a Sin (b) = Opposite/hypotenuse = AC/AB Cos (b) = Adjacent/Hypotenuse = CB/AB Hypotenuse pposite C c Adjacent b B Tan (b) = Opposite/Adjacent = AC/CB = Sin(b)/Cos(b) Cot (b) = Adjacent/Opposite = CB/AC = 1/Tan(b) Cot TRIGONOMETRY TRIGONOMETRY If α = 90o, AB = 3”, and AC = 4 then If a) BC = 5” C o b) Angle θ = 48.6 b) β 4” α 3” 5” c) Angle θ = 53.13o d) Angle β = 36.87 d) -1 o A θ B 4 o θ = tan = 53.13 3 TRIGONOMETRY TRIGONOMETRY Sin Cot a Cos a a Sin a Tan a Cos Tan Cot Circle with a radius of unit length TRIGONOMETRY TRIGONOMETRY Circle with a unit diameter Diameter Sin a A a A TRIGONOMETRY TRIGONOMETRY Sin2a + Cos2a = 1 Sin (a + b) = (sin a)(cos b) + (cos a)(sin b) Cos (a + b) = (cos a)(cos b) - (sin a)(sin b) Sin(2a) = 2sin(a)cos(a) Cos(2a) = Cos2(a) – Sin2(a) TRIGONOMETRY/Cosine Law D A a a2 c b B C BC2 = BA2 + AC2 - 2(BA)(AC)(cos(a)) TRIGONOMETRY/Cosine Law TRIGONOMETRY/ BC2 = BD2 + DC2 = (BA + AD)2 + DC2 BD BC2 = BA2 + AD2 + DC2 +2(BA)(AD) BC BC2 = BA2 + AC2 – DC2 + DC2 +2(BA)(AD) A BC2 = BA2 + AC2 +2(BA)(AD) b B D a a2 c C BC2 = BA2 + AC2 +2(BA)(AC)cos(a2) BC2 = BA2 + AC2 + 2(BA)(AC)(-cos(a)) BA 2(BA)(AC)(-cos(a)) BC = BA + AC - 2(BA)(AC)(cos(a)) TRIGONOMETRY/Sin Law TRIGONOMETRY/ AB BC AC = = Sin c Sin a Sin b A a b B c C TRIANGLES TRIANGLES A To proof it, use the parallel lines theory A B A+B+C = 180 B C The sum of the interior angles of a triangle is 180o GEOMETRY/TRIGONOMETRY a b d c For any closed polygon the sum of the interior angles is equal to: 90o(2*S – 4), where S is the number of sides GEOMETRY/TRIGONOMETRY GEOMETRY/TRIGONOMETRY A a B F C E D The sum of the interior angles of a polygon (6 sides) a + b + c + d + e + f = 90o (2*6 - 4) GEOMETRY/TRIGONOMETRY GEOMETRY/TRIGONOMETRY A a B F C E D The sum of the interior angles of a polygon (6 sides) a + b + c + d + e + f = 90o (2*6 - 4) ...
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