rules for surds
-simplify surds as much as possible
-always factorize surds if possible by making them a square number x something, then put the root number| surd
-if denominator has a surd, rationalize by multiplying fraction with the denominator surd/ denominator surd
-you can only multiply & divide surds with surds
-if you have a surd x number = number| surd
-when rationalizing:
2 terms denominator - flip signs ( - ) to ( + ), ( + ) to ( - )
eg. 7 2 - √ 3
———— X ———— = 14 - 7√ 3
2 + √ 3 2 - √ 3
0/x & x/0
0/x = 0
x/0 = undefined
indices laws
- 1^ a = 1
- a^ 0 = 1
- a^ m x a^ n = a^ m+n
- a^ m / a^ n = a^ m-n
- (a^ m) ^n = a^ mn
- a^-m = 1/ a^m
- a^ m/n = n root/—-^m (m- power, n-root)
how to change reoccurring decimals to fractions
-put over 9’s depending on the no. of reoccurring digits
-eg. if 0.27 with dash, 27/99
how to change reoccurring decimals without a reoccurring one in front to fractions
-make an equations
-times both sides by how many places needed to put . in front of reoccurring digit
-minus litter side by one and no. side by the og no.
-get the letter by itself
eg. 0.683 with dash = x
100x = 68.3 with dash
99x = 68.3 - 0.683
x = 67.65/ 99
x = 41/ 60
how to find gradient of perpendicular straight line (has to intersect at 90)
-m for perpendicular is inverse
- flip fraction & make ( - )
How to find equation of parallel straight line
-m is same
-change c only
How to find intercept knowing the other of straight line
-if don’t know y = auto - (0, __)
-if don’t know x = auto - (__, 0)
-use the equation and make the known intercept = 0
how to find θ for right angled triangles
-SOH CAH TOA
sin θ = O/ H
cos θ = A/ H
tan θ = O/ A
-O is side opposite θ
-H is longest side
unit circle:
-45, 45, 90 (isosceles triangle)
-hypotenuses is √2 ,rest is 1
SOH CAH TOA
-sin 45 = √2/ 2
-cos 45 = √2/ 2
-tan 45 = 1/1
unit circle:
-30, 60, 90 (equilateral triangle)
-hypotenuse is 2, rest is 1 or √3
SOH CAH TOA
-sin 30 = 1/2
-cos 30 = √3/ 2
-tan 30 = √3/ 3
-sin 60 = √3/ 2
-cos 60 = 1/2
-tan 60 = √3
functions for sine, cosine, tan
1) sine
x< 180 (x = +) __ sin x = sin (180 - x)
x> 180 (x = -) __ x - 180 = y , sin x = sin (360- y)
2) cosine
x< 90/ x> 270 (x = +) __ cos x = -cos (180 - x)
90< x< 270 (x = -) __ cos x = cos (360 - x)
3) tan
tan x = tan (180 + x)
sine rule
A, B, C - angle
a, b, c - side (side has to be opposite same letter angle)
a b
——– = ——–
sin A sin B
-cross multiply
-2 sides, 1 angle/ 1 side, 2 angles
cosine rule
A, B, C - angle
a, b, c - side (side has to be opposite same letter angle)
-c^2 = a^2 + b^2 - 2ab cos C
-cos C = a^2 + b^2 - c^2
————————-
2ab
-3 sides/ 2 sides, 1 angle
area of non right angle triangle
1/2 x ab sincC
how to find length & angle for 3D triangle in a square/ rectangle
length:
-find hypothenuse of base
-find hypothenuse of 3D triangle
angle:
-SOH CAH TOA with values
rules for differentiation
-multiply coefficient of x with power, minus power by 1
-if power = 0, take only coefficient if there is or just 1
-if number without x, remove it
Proofing statements for circle geometry
1) The center angle is twice the circumference angle on the same arc
2) A diameter subtends a 90 angle at the circumference
3) Angles from the same arc are equal
4) Opposite angles in a cyclic quadrilateral add up to 180
5) The radius perpendicular to the radius at the point of contact forms a 90 degree
6) A tangent is perpendicular to the radius at the point of contact forms a 90 degree
7) Alternate segment: the angle between a tangent & chord equals the angle in the alternate segment
finding inverse functions
-write function as y = mx + c
-swap y & x positions
-solve for y
how to find vectors
1) break down the lines & add them
-takes the shortest path
-take only paths with known vectors
-be careful of direction
