Trigonometry Triangle ABC Sine Rule = = can be used upside down if trying to find an angle Important that side a is opposite angle A and side b opposite angle B and side c opposite angle C Cosine Rule is given in formulae sheets but you need to be able to rearrange to get as the subject = ( ) Area of Triangle important that the angle C is the angle between the two sides a and b + = Radians =ᵒ = Sector Area Arc length Ɵ Ɵ Basic Trigonometry SOHCAHTOA = = = www.chattertontuition.co.uk 0775 950 1629 Page 1
=( ) =( ) =(+) =( ) ( )=(+) = ( ) ( )= Useful Trigonometric Triangles Just remember that sin30ᵒ= and you can work out the rest using Pythagoras Theorem and basic trigonometry Just remember that tan45ᵒ=1 and you can work out the rest using Pythagoras Theorem and basic trigonometry 1 60ᵒ 2 1 45ᵒ 30ᵒ 45ᵒ ᵒ= ᵒ= ᵒ= 1 ᵒ=ᵒ= ᵒ= www.chattertontuition.co.uk 0775 950 1629 Page 2
Solving quadratics factorise, complete the square or quadratic equation Quadratic Equation = ± ++=0 Completing the square we can use this to work out where the turning points are and to help sketch the curve or to solve quadratic equations. Eg 4 16+5=4( 4)+5=4(( 2) 2 ) 5 4(( 2) 4) 5= 4( 2) 21 Turning point (2, -21) Line of symmetry =2 Hidden Quadratics By using substitution we can change an equation into a quadratic that can be solved. eg +6 +8=0 Let = or +2 8=0 let = both of which can be solved so we have +6+8=0 so we have +2 8=0 www.chattertontuition.co.uk 0775 950 1629 Page 3
Laws of logarithms = = = = + = = = Change of base = Solving equations with logs Eg =5 take logs of both sides = (5 )= 5 = could also have taken logs to base 5 and then would get the same answer = = www.chattertontuition.co.uk 0775 950 1629 Page 4
Calculus Differentiation multiply by power and reduce power by 1 = = Integration = increase power by 1 and then divide by new power + = + Area under curve Area under curve ()= between curve and axis between curve and y axis ()= Stationary Points Minimum, maximums and points of inflection Differentiate and set =0 and solve to find To investigate whether a minimum or maximum differentiate again to get, substitute in the value and if positive then minimum, if negative then maximum, if 0 then a point of inflection Increasing function Where the gradient is positive (sloping upwards) Decreasing function Where the gradient is negative (sloping downwards) www.chattertontuition.co.uk 0775 950 1629 Page 5
Coordinate Geometry Equation of a line =( ) If have gradient m and a point (, ) that the line goes through Gradient of line If have two points on the line (, ) and (, ) Equation of a circle ( ) +( ) = Centre (a, b) and radius r Midpoint If have two points on the line (, ) and (, ) (, ) Length of a line ( ) +( ) If have two points on the line (, ) and (, )By Pythagoras Theorem Normal is perpendicular to Tangent The gradients are the negative reciprocal of each other eg gradient tangent then gradient normal is www.chattertontuition.co.uk 0775 950 1629 Page 6
Sequences Formulae given in booklet but do need to recognise if a geometric sequence or arithmetic Geometric each term is a constant multiple of the previous term Arithmetic each term is a constant addition to the previous term Convergent eg,,,, Divergent eg 7, 15, 23, 31, each term gets closer and closer to a number each term does not converge Periodic the terms start repeating eg 3, 2, 4, 5, 3, 2, 4, 5, 3, 2, 4, 5, has period 4 Period is number of terms before the sequence repeats () sum of terms from r = 1 up to r = n eg (4 1)=3+7+11+15+19=55 (sum of all the terms from r = 1 up to r = 5) www.chattertontuition.co.uk 0775 950 1629 Page 7
Indices = ( ) = = = = = = = Polynomials Factor Theorem If ()= then = is a root and ( ) is a factor Remainder Theorem When () is divided by ( ) the remainder is () To sketch a cubic polynomial try to factorise ( )( )( )=0 will cross the axis at a, b and c Surds = = ( ) ( )( ) www.chattertontuition.co.uk 0775 950 1629 Page 8
Quadratic inequalities solve ( )( )<0 ( )( ) >0 solve where this equals 0 (when = =) and then see whether we want the bit between a and b or the bits either side A positive quadratic has a shape so will be below the axis between a and b ( )( )< << ( )( ) > < > Transformations ()+ translation of a units in the positive y direction (+) translation of a units in the negative direction () stretch of scale factor a parallel to the y axis () stretch of scale factor parallel to the axis Discriminant Part of the quadratic equation. Tells us how many real roots there are. > = (or repeated roots) < Circle Theorems A tangent meets a radius at right angles The perpendicular distance from the centre of a circle to a chord bisects that chord Angles in a semicircle are right angles www.chattertontuition.co.uk 0775 950 1629 Page 9
Curves The angle gets repeated at π - The angle gets repeated at 2π - The angle gets repeated at π + www.chattertontuition.co.uk 0775 950 1629 Page 10
= If you found these helpful and would like to see some more then visit our website http://www.chattertontuition.co.uk/a-level-maths-papers Chatterton Tuition offer tuition in all subjects as well as A level maths. We are local to North Yorkshire but also offer online tutoring. www.chattertontuition.co.uk 0775 950 1629 Page 11