Algebra Index Laws Standard index laws Standard index laws (a m) n = a mn am × an = am + n a1 = a a n = a × a × a × ... × a n times m am ÷ an = a n a = am - n a0 = 1 (ab) n = a nb n for a ≠ 0 a n an = n b b ( ) for b ≠ 0 Negative Indices a-1 = 1 a a - n = 1n a 1 a-n = an a -1 b = b a ( ) for a ≠ 0 for a ≠ 0 a -n = b ( ) for a ≠ 0 = for a ≠ 0, b ≠ 0 b n a ( ) bn an for a ≠ 0, b ≠ 0 Fractional Indices 1 a2 = a a a a 1 n m 2 m n - 1 2 for a ≥ 0 a = a for a ≥ 0, n � 2 a n = am for a ≥ 0 a - for a ≥ 0, n � 2 a -n n n = a m Formula Sheet 1 m 2 m = = for a � 0 1 for a � 0, n � 2 1 for a � 0 1 for a � 0, n � 2 a n = = 1 a am n am Brought to you by www.mathletics.com © 3P Learning Ltd Algebra Operations Arithmetic rules Commutative rule Associative rule Distributive rule a+b=b+a (a + b) + c = a + (b + c) a(b + c) = ab + ac a×b=b×a (a × b) × c = a × (b × c) a(b - c) = ab - ac Expanding Basic expansions Binomial expansions (a + b) (c + d) = ac + ad + bc + bd a(b + c) = ab + ac a(b - c) = ab - ac (a + b)2 = a2 + 2ab + b2 a(b + c) + d (e + f ) = ab + ac + de + df (a - b)2 = a2 - 2ab + b2 (a + b) (a - b) = a2 - b2 Factorising Common factors Grouping in pairs ab + ac = a(b + c) ac + ad + bc + bd = a(c + d) + b(c + d) ab - ac = a(b - c) = (a + b) (c + d) Perfect squares Difference of two squares a2 + 2ab + b2 = (a + b)2 a2 - b2 = (a + b) (a - b) a2 - 2ab + b2 = (a - b)2 Formula Sheet Brought to you by www.mathletics.com © 3P Learning Ltd Measurement Perimeter and Area 1 (of 2) Square Rectangle s Perimeter = 4s Area = s 2 s where s = side length Triangle l Perimeter = 2l + 2w Area = lw w where l = length w = width Heron’s formula for area of a triangle a b h c b Area = 1 2 bh Area = s(s - a) (s - b) (s - c) where b = base h = perpendicular height from base where s = 1 Parallelogram Trapezium 2 (a + b + c) and a, b, c are the side lengths of triangle a h h b b 1 2 Area = bh Area = where b = base h = perpendicular height from base where a and b are the lengths of the parallel sides h = perpendicular height between the parallel sides Formula Sheet h(a + b ) Brought to you by www.mathletics.com © 3P Learning Ltd Measurement Perimeter and Area 2 (of 2) Rhombus 1 2 Area = Kite xy Area = x where x and y are the lengths of the diagonals 1 2 xy x where x and y are the lengths of the diagonals y y Annulus Circle r Circumference = πd or Circumference = 2 πr R d r Area = πr 2 where r = radius d = diameter Area = π (R 2 - r 2) where R = radius of larger circle r = radius of smaller circle Sector Pythagoras’ Theorem r θ Arc length = Area = where r θ θ 360° θ 360° × 2πr × πr 2 = radius = angle at the centre (in degrees) Formula Sheet c a c =a +b 2 2 2 b where c is the hypotenuse of a right-angled triangle and a, b are the shorter sides. Brought to you by www.mathletics.com © 3P Learning Ltd Measurement Surface Area and Volume Cube Rectangular Prism Surface area = 6s 2 Surface area = 2hl + 2hw + 2lw Volume = s 3 where s = side length Volume = lwh where s h w l = height = width = length h w l Prism Pyramid Volume = Ah Volume = where A = cross-sectional area h = perpendicular height where A = area of the base h = perpendicular height 1 3 Ah h A A h Cylinder Cone Sphere Surface area = 2πr 2 + 2πrh Surface area = πr 2 + πrs Surface area = 4πr 2 Volume = πr 2h Volume = where r = radius h = perpendicular height 1 3 πr 2h Volume = where r = radius h = perpendicular height s = slant height 4 3 πr 3 where r = radius r h s h r r Formula Sheet Brought to you by www.mathletics.com © 3P Learning Ltd Measurement Unit Conversions (Metric) Length Area Mass 1 cm = 10 mm 1 cm2 = 100 mm2 1 g = 1000 mg 1 m = 100 cm 1 m2 = 10 000 cm2 1 m = 1000 mm 1 ha = 10 000 m2 1 kg = 1000 g 1 t = 1000 kg 1 km2 = 1 000 000 m2 1 km = 1000 m 1 km2 = 100 ha Capacity 1 L = 1000 mL Volume Volume and capacity 1 cm3 = 1000 mm3 1 mL = 1 cm3 1 m3 = 1 000 000 cm3 1 kL = 1000 L 1 m3 = 1000 L 1 m3 = 1 kL 1 ML = 1000 kL 1 ML = 1000 m3 Time Digital information and file size 1 minute = 60 seconds 1 byte = 8 bits 1 kilobyte = 2 10 bytes = 1024 bytes 1 hour = 60 minutes 1 megabyte = 2 20 bytes = 1024 kilobytes 1 day = 24 hours 1 year = 365 days (in a non-leap year)* 1 gigabyte = 2 30 bytes = 1024 megabytes 1 year = 366 days (in a leap year) 1 terabyte = 2 40 bytes = 1024 gigabytes *Some mathematical calculations use: 1 1 year = 365 4 days Formula Sheet Brought to you by www.mathletics.com © 3P Learning Ltd Surds & Logarithms Surds Logarithms For a ≥ 0, b ≥ 0: Definitions For b � 0, b ≠ 1 and x � 0: 2 ( a) = a If log b x = y then x = b y a2 = a log b (bx) = x blog b x = x ab = a × b a = b a b (since x log b b = x × 1) (b ≠ 0) Change of base law For a � 0, b � 0, x � 0 and a ≠ 1, b ≠ 1: log x log a x = log b a b For a ≥ 0, n > 2: a=a n a= 1 2 1 Log laws For b � 0, b ≠ 1, x � 0 and y � 0: an am = a m log b b = 1 (since b1 = b) log b 1 = 0 (since b0 = 1) 2 log b (x a) = a log b x n am = a m n log b (xy) = log b x + log b y log b x = log b x - log b y (y) Formula Sheet Brought to you by www.mathletics.com © 3P Learning Ltd The Cartesian Plane Midpoint of an interval Distance between two points The midpoint, M, of an interval with endpoints ( x1, y1 ) and ( x2, y2 ) has coordinates: The distance, d units, between two points ( x1, y1 ) and ( x2, y2 ) is given by: M= ( x1 + x2 y1 + y2 , 2 2 ) d= (x2 - x1 )2 + ( y2 - y1 )2 Gradient of a straight line Gradient-intercept form of a straight line The gradient, m, of a straight line joining two points ( x1 , y1 ) and ( x2 , y2 ) is given by: The equation of a straight line with gradient m and y-intercept c is given by: rise m = run or y2 - y1 m = x -x 2 1 y = mx + c Point-gradient formula for the equation of a straight line The equation of a line with gradient m and passing through ( x1 , y1 ) is given by: y - y1 = m(x - x1) Two-point formula for the equation of a straight line The equation of a straight line joining two points ( x1 , y1 ) and ( x2 , y2 ) is given by: y - y1 y2 - y1 x - x1 = x2 - x1 Parallel lines Perpendicular lines If two lines with gradients m1 and m2 are parallel then: If two lines with gradients m1 and m2 are perpendicular then: m1 = m2 Formula Sheet m1 × m2 = -1 or m2 = - m1 1 Brought to you by www.mathletics.com © 3P Learning Ltd Trigonometry 1 Right-angled triangles In a right-angled triangle: opposite sin θ = hypotenuse cos θ = adjacent hypotenuse hypotenuse opposite θ opposite tan θ = adjacent adjacent Exact ratios 30° 1 sin 30° = 2 sin 45° = 1 2 3 cos 30° = 2 cos 45° = 1 2 tan 30° = 1 3 tan 45° = 1 2 3 3 sin 60° = 2 60° 1 1 cos 60° = 2 tan 60° = 45° 2 1 45° 3 1 Angles of elevation and depression line of sight angle of depression angle of elevation line of sight Formula Sheet Brought to you by www.mathletics.com © 3P Learning Ltd Trigonometry 2 All triangles For any triangle with vertices A, B and C and opposite sides a, b and c respectively: A b C c a B Sine rule a sin A sin A a = = b sin B sin B b Cosine rule = = c Area of triangle Area = c = a + b - 2ab cosC 2 sin C sin C 2 cos C = c 2 1 2 absinC a2 + b2 - c2 2ab Complementary angles Supplementary angles For complementary angles θ and (90° - θ): For supplementary angles θ and (180° - θ): sin θ = cos (90° - θ) sin (180° - θ) = sin θ cos θ = sin (90° - θ) cos (180° - θ) = - cos θ tan θ = cot (90° - θ) tan (180° - θ) = - tan θ Relationships tan θ = sin θ cos θ Formula Sheet cosec θ = 1 sin θ sec θ = 1 cos θ cot θ = 1 tan θ Brought to you by www.mathletics.com © 3P Learning Ltd