Specialist Mathematics 2019 v1.2

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2 Mensuration circumference of a circle area of a parallelogram CC = ππππ area of a circle AA = ππrr AA = h area of a trapezium AA = 1 ( + )h area of a triangle AA = 1 h total surface area of a cone SS = ππππππ + ππrr total surface area of a cylinder SS = ππππh + ππrr surface area of a sphere SS = 4ππrr volume of a cone VV = 1 3 ππrr h volume of a cylinder VV = ππrr h volume of a prism VV = AAh volume of a pyramid VV = 1 3 AAh volume of a sphere VV = 4 3 ππrr3 Calculus xxnn = nnxx nn 1 eexx = ee xx xx nn = xxnn+1 nn cc ee xx = ee xx + cc ln(xx) = 1 xx 1 xx = ln xx + cc sin(xx) = cos(xx) cos(xx) = sin(xx) tan(xx) = sec (xx) sin 1 xx = 1 xx sin(xx) = cos(xx) + cc cos(xx) = sin(xx) + cc sec (xx) = tan(xx) + cc 1 xx = sin 1 xx + cc cos 1 xx = 1 1 xx xx = cos 1 xx + cc tan 1 xx = + xx + xx = tan 1 xx + cc Page of 5

3 Calculus chain rule product rule quotient rule integration by parts volume of a solid of revolution If h(xx) = ffgg(xx) then h (xx) = ff gg(xx)gg (xx) If h(xx) = ff(xx)gg(xx) then h (xx) = ff(xx)gg (xx) + ff (xx)gg(xx) If h(xx) = ff(xx) gg(xx) then h (xx) = ff (xx) gg(xx) ff(xx)gg (xx) (gg(xx)) ff(xx)gg (xx) = ff(xx)gg(xx) ff (xx)gg(xx) about the xx-axis about the yy-axis If yy = ff(uu) and uu = gg(xx) then = (uuuu) = uu + vv uu vv vv = uu vv uu = uuuu vv VV = ππ [ff(xx)] VV = ππ [ff(yy)] Simpson s rule simple harmonic motion acceleration ff(xx) ww [ff(xx 3 0) + 4[ff(xx 1 ) + ff(xx 3 ) + ] + [ff(xx ) + ff(xx 4 ) + ] + ff(xx nn )] If xx tt = ω xx then xx = AA sin(ωtt + α) or xx = AA cos(ωtt + β) vv = ω (AA xx ) TT = ππ ω = = xx = vv tt = 1 vv ff = 1 TT Real and complex numbers complex number forms zz = xx + yyyy = rr(cos(θθ) + ii sin(θθ)) = rr cis(θθ) modulus zz = rr = xx + yy argument arg(zz) = θ, tan(θθ) = yy, ππ < θ ππ xx product zz 1 zz = rr 1 rr cis(θθ 1 + θθ ) quotient De Moivre s theorem zz 1 zz = rr 1 rr cis(θθ 1 θθ ) zz nn = rr nn cis(nnnn) Page 3 of 5

4 Statistics binomial theorem (xx + yy) nn = xx nn + nn 1 xxnn 1 yy + + nn rr xxnn rr yy rr + + yy nn permutation PP rr = nn combination CC rr = nn rr = nn! rr! (nn rr)! nn nn! = nn (nn 1) (nn ) (nn rr + 1) (nn rr)! sample means approximate confidence interval for μμ mean standard deviation xx zz ss ss, xx + zz nn nn μμ σσ nn Trigonometry Pythagorean identities sin (AA) + cos (AA) = 1 tan (AA) + 1 = sec (AA) cot (AA) + 1 = cosec (AA) angle sum and difference identities double-angle identities product identities sin(aa + BB) = sin (AA) cos(bb) + cos (AA) sin (BB) sin(aa BB) = sin (AA) cos(bb) cos (AA) sin (BB) cos(aa + BB) = cos (AA) cos(bb) sin (AA) sin (BB) cos(aa BB) = cos (AA) cos(bb) + sin (AA) sin (BB) sin(aa) = sin (AA) cos (AA) cos(aa) = cos (AA) sin (AA) = 1 sin (AA) = cos (AA) 1 sin(aa) sin(bb) = 1 (cos(aa BB) cos(aa + BB)) cos(aa) cos(bb) = 1 (cos(aa BB) + cos(aa + BB)) sin(aa) cos(bb) = 1 (sin(aa + BB) + sin(aa BB)) cos(aa) sin(bb) = 1 (sin(aa + BB) sin(aa BB)) Page 4 of 5

5 Vectors and matrices magnitude 1 = = = cos(θθ) scalar (dot) product vector equation of a line Cartesian equation of a line vector (cross) product vector projection vector equation of a plane Cartesian equation of a plane 1 1. =. = rr = + kk xx 1 1 = yy = zz 3 3 = sin(θθ) nn = = on = cos(θθ) =. rr. nn =. nn + + cccc + = 0 determinant If AA = then det(aa) = cc multiplicative inverse matrix linear transformations cc 1 = 1 det(aa), det(aa) 0 cc dilation 0 0 cos(θθ) sin(θθ) rotation sin(θθ) cos(θθ) reflection (in the line yy = xx tan(θθ)) cos(θθ) sin(θθ) sin(θθ) cos(θθ) Page 5 of 5