1 - Introduction and Review - Trigonometry

1 - Introduction and Review - Trigonometry - INTRODUCTION...

Info iconThis preview shows page 1. Sign up to view the full content.

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: INTRODUCTION AND REVIEW ME 221 STATICS 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 Trigonometry & Geometry GEOMETRY/TRIGONOMETRY Lines at 90 Degrees GEOMETRY/TRIGONOMETRY Two Parallel Lines GEOMETRY/TRIGONOMETRY a b a b b a b a TRIGONOMETRY If A = 90o and the lengths AB and AC are equal, then; C Angle = Angle = 45o Angle > Angle Angle < Angle Cannot be determined A B TRIGONOMETRY TRIGONOMETRY Right Triangle (sides relative to angle b) A a Hypotenuse Opposite C c Adjacent b B TRIGONOMETRY If A = 90o, AB = 4" and AC = 3", then; a) BC = 1" b) BC = 7" c) BC = 25" d) BC = 5" 3" C BC = ? A 4" B BC = 3 + 4 = 5 2 2 TRIGONOMETRY Right Triangle (AC)2 + (BC)2 = (AB)2 a + b + c = 180o A a C c b B TRIGONOMETRY A a Sin (b) = Opposite/hypotenuse = AC/AB Cos (b) = Adjacent/Hypotenuse = CB/AB Hypotenuse Opposite C c Adjacent b B Tan (b) = Opposite/Adjacent = AC/CB = Sin(b)/Cos(b) Cot (b) = Adjacent/Opposite = CB/AC = 1/Tan(b) TRIGONOMETRY If = 90o, AB = 3", and AC = 4 then a) BC = 5" C o b) Angle = 48.6 4" 3" 5" c) Angle = 53.13o d) Angle = 36.87 -1 o A B 4 o = tan = 53.13 3 TRIGONOMETRY Sin Cot a Cos a a Sin a Tan a Cos Tan Cot Circle with a radius of unit length TRIGONOMETRY Circle with a unit diameter Diameter Sin a A a A 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) sin( a + b) = sin( a ) cos(b) + cos(a ) sin(b) B D E b a O F In the figure below, assume that the side OB is a unit length, hence, OB = 1.0 sin( a + b) = BF = BD + DF C OCD = a and BCD = 90 - a BD = BC sin(BCD ) = BC sin(90 - a ) = BC cos(a ) = sin(b) cos(a ) G DF = CG = OC sin(a ) = cos(b) sin( a ) sin(a + b) = BF = BD + DF = sin( a ) cos(b) + cos(a ) sin(b) TRIGONOMETRY/Cosine Law D A a a2 c b B C BC2 = BA2 + AC2 - 2(BA)(AC)(cos(a)) TRIGONOMETRY/Cosine Law BC2 = BD2 + DC2 = (BA + AD)2 + DC2 BC2 = BA2 + AD2 + DC2 +2(BA)(AD) 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)) BC2 = BA2 + AC2 - 2(BA)(AC)(cos(a)) TRIGONOMETRY/Cosine Law BC2 = BA2 + AC2 - 2(BA)(AC)(cos(a)) A D From the right triangle BDC: BC2 = DC2 + BD2 From the right triangle DAC: DC2 = AC2 - DA2 (2) (1) a b B Draw CD at 90 degrees to BA c Substituting (2) in (1) C BC2 = AC2 DA2 + BD2 (3) BC2 = AC2 DA2 + (BA DA)2 BC2 = AC2 DA2 + BA2 + DA2 2(BA) (DA) BC2 = AC2 + BD2 2(BA)(CA)(cos(a)) TRIGONOMETRY/Sin Law AB BC AC = = Sin c Sin a Sin b A a b B c C TRIANGLES A To proof it, use the parallel lines theory A B A+B+C = 180o 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 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 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) TRIGONOMETRIC EQUATION Solve Asin + Bcos = D (1) A Take an angle such that tan = B -1 A = tan B From the right triangle 2 2 b A B a C c A B C = A + B , sin = and cos = C C TRIGONOMETRIC EQUATION Asin + Bcos = D (1) Multiply the left hand side of equation 1 by C/C = 1 b A B C sin + C cos = D (2) C C Substituting yields A B Csinsin + Ccoscos = D (2) a -1 D C[ cos( - ) ] = D; ( - ) = cos C -1 D -1 A = cos + tan C B C c TRIGONOMETRIC EQUATION Solve 25.5sin + 12cos = 10 (1) 25.5 Take an angle such that tan = 12 -1 25.5 = tan = 64.8o 12 b 28 .18 25.5 From the right triangle 2 2 a 12 c 25.5 12 C = 25.5 + 12 = 28.18, sin = & cos = 28.18 28.18 TRIGONOMETRIC EQUATION 25.5sin + 12cos = 10 (1) b Multiply the left hand side of equation 1 by C/C = 28.18 25.5 12 28.18 sin + 28.18 cos = 10 (2) 28.18 28.18 28 .18 25.5 Substituting yields 12 a -1 28.18sin64.8sin + 28.18cos64.8cos = 10 (2) c 10 28.18[ cos( - 64.8) ] = 10; ( - ) = cos = 69.22o 28.18 o = 69.22 + 64.8 = 134.02 GRAPHICAL SOLUTION 35 30 25 20 15 10 5 0 -5 -10 -15 0 50 100 Alpha 150 200 D ...
View Full Document

This note was uploaded on 02/03/2012 for the course CE 221 taught by Professor Buch during the Summer '08 term at Michigan State University.

Ask a homework question - tutors are online