Midterm_2_reference_sheet[1]

Midterm_2_reference_sheet[1] - Math 4C Fall 2007 Midterm 2...

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Unformatted text preview: Math 4C Fall 2007 Midterm 2 Reference Sheet Trigonometric functions sin " = opp hyp, cos " = adj hyp, tan " = opp adj Of acute angles sin " = y r , cos " = x r , tan " = y x ( x # 0) (circle radius r) Of any angle sin t = y, cos t = x, tan t = y x ( x " 0) Of real numbers Radians and degrees 180 = " (radians) ! sin A sin B sinC ! = = Solving general triangles Law of Sines a b c ! 2 2 2 2 2 Law of Cosines a = b + c " 2bc cos A, cos A = (b + c " a 2 ) /(2bc) ! 1 1 A = absin " , A = (a + b + c)(a + b # c)(a + c # b)(b + c # a) Area of a triangle 2 ! 4 Unit Circle ! ! Translations, reflections, and dilations y = Asin[B(x " C)] + D, y = Acos[B(x " C)] + D C units Horizontal translation (phase shift) Vertical translation (vertical shift) D units Horizontal dilation By factor of 1 B (period becomes 2 " B ) Vertical dilation (amplitude) ! By factor of A ! " f ( x) reflect across x-axis; f ("x) across y-axis Reflections ! ! ! ! Principal trigonometric identities 1 1 1 sin " cos " csc " = , sec " = , cot " = , tan " = , cot " = Definitions sin " cos " tan " cos " sin " 2 2 2 2 2 2 Pythagorean sin " + cos " =1, tan " +1 = sec " , 1+ cot " = csc " Opposite angle sin("# ) = "sin # , cos("# ) = cos # , tan("# ) = "tan # sin(" + 2 #k) = sin " , cos(" + 2 #k) = cos " , tan(" + #k) = tan " Periodicity ! " " sin( # $ ) = cos $ , cos( # $ ) = sin $ , sin(90 # $ ) = $ , cos(90 # $ ) = $ Complements ! 2 2 ! 2 cos " = (1+ cos 2" ) 2, sin 2 " = (1# cos 2" ) 2 Reduction ! sin(s + t) = sin s cos t + cos s sin t, sin(s " t) = sin s cos t " cos s sin t Addition formulas cos(s + t) = cos s cos t " sin s sin t, cos(s " t) = cos s cos t + sin s sin t ! tan s + tan t tan s " tan t tan(s + t) = , tan(s " t) = ! 1" tan s tan t 1+ tan s tan t ! Double-angle ! sin2" = 2sin " cos " , cos 2" = cos 2 " # sin 2 " = 2cos 2 " #1 =1# 2sin 2 " , tan2" = 2tan " (1# tan 2 " ) ! " 1# cos " " 1+ cos " " sin " Half-angle sin = , cos = , tan = ! 2 2 2 2 2 1+ cos " 1 ! sin A sin B = [cos( A " B) " cos( A + B)] Product-to-sum 2 1 ! sin A cos B = [sin( A + B) + sin( A " B)] 2 1 ! cos A cos B = [cos( A + B) + cos( A " B)] 2 A+B A"B A+B A"B ! sin A + sin B = 2sin cos sin Sum-to-product , sin A " sin B = 2cos 2 2 2 2 A+B A"B A+B A"B ! cos A + cos B = 2cos cos sin , cos A " cos B = "2sin 2 2 2 2 # # # # ! ! Inverse trigonometric function Domain restrictions sin [" , ], cos [0, # ], tan [" , ] 2 2 2 2 Vectors (2-D) ! ! Position (radius) Given P( x1 , y1 ) and Q( x 2 , y 2 ) , PQ = x 2 " x1 , y 2 " y1 Unit components Magnitude (length) If v = x, y , then v = xi + yj ! u= If v = x, y , then v = x 2 + y 2 Unit vector ! ! ! Scalar multiplication! If v = x, y , then kv = kx,ky Addition If u = a,b and v = c,d , then u + v = a + c,b + d ! ! ! Dot product If u = a,b and v = c,d , then u " v = ac + bd v v %u#v ( ! u#v ! , " = cos$1' Angle between u and v cos " = 'uv* ! * ! u v! & ) ! ! ! Vectors (3-D) Position (radius) Unit components ! Given P( x1 , y1 ,z1 ) and Q( x 2 , y 2 ,z 2 ) , PQ = x 2 " x1 , y 2 " y1 ,z 2 " z1 If v = x, y,z , then v = xi + yj + zk ! ! ! ! Magnitude (length) Scalar multiplication Addition Dot product If v = x, y,z , then v = x 2 + y 2 + z 2 If v = x, y,z , then kv = kx,ky.kz Unit vector u= v v ! ! Angle between u and v ! Direction cosines ! ! Magnitude/direction cosine form Cross-product !If v = ai + bj + ck and w = di + ej + fk, then ! v " w = (bf # ce)i + (cd # af ) j+ (ae # bd )k ! If u = a,b,c and v = d,e, f , then u + v = a + d,b + e,c + f ! ! If u = a,b,c and v = d,e, f , then u " v = ad + be + cf %u#v ( ! u#v cos " = , " = cos$1' 'uv* * ! uv ! & ) ! If v = ai + bj! ck, and " = angle between v and i, and " = angle + a between v and j, and " = angle between v and k, then cos " = , v b ! c ! cos " = and cos " = . v v ! # If v = ai + bj + ck, then v = v cos " i + v cos! j+ v cos $ k ! ...
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This note was uploaded on 04/18/2008 for the course MATH 4C taught by Professor Arnold during the Fall '08 term at UCSD.

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