HSC Maths Formulas
Expansions
(a + b)² = a² + 2ab + b²
Line through the intersection of two lines
(ax1 + by1 + c1) + l(ax2 + by2 + c2) = 0
Equation of a Circle
(x-a)²+(y-b)²=r²
Equation of a locus (parabola)
(x-h)²=4a(y-k) or: (y-k)²=4a(x-h) Where the focal length is a, and ther vertex is at (h,k) The length of the latsu rectum is 4a
Equation of a locus (equidistant from 2 points)
(x₂-x)²+(y₂-y)²=(x₁-x)²+(y₁-y)²
Two-point form
(y-y1) / (y2 - y1) = (x - x1) / (x2 - x1)
Equation of a locus (forming 2 perpendicular lines)
(y₂-y)/(x₂-x)×(y₁-y)/(x₁-x)=-1
α²+β²
(α+β)²−2αβ
α²+β²+γ²
(α+β+γ)²-2(αβ+αγ+βγ)
log[a, 1]
0
log[a, a]
1
Sum of exterior angles
360°
Sector area
A = (r²∅)/2 [in radians]
Area of a parallelogram
A = bh
Area of a rectangle
A = lb
Area of a square
A = l²
Area of a minor segment
A = ½r²(∅ - sin∅) [in radians]
Area of a rhombus
A = ½xy
Area of a circle
A = πr²
Area between the curve and the x-axis
A=|b∫a f(x) dx|
Area between a curve and the y-axis
A=|b∫a f(y) dy|
Area between two curves (with two common points)
A=|b∫a[f(x) - g(x)]|
Similar triangles tests
AA, SSS, SAS, RHS
Sum and Product of Roots: ax³+bx²+cx+d=0
If the roots are α, β and γ: α+β+γ=-b/a (∑α) αβ+αγ+βγ=c/a (∑αβ) αβγ=-d/a (∑αβγ)
Sum and Product of Roots: ax⁴+bx³+cx²+dx+e=0
If the roots are α, β, γ and δ: ∑α=-b/a ∑αβ=c/a ∑αβγ=-d/a αβγδ=e/a
The Factor Theorum
If x-α is a factor of P(x), then P(α)=0
Sum and Product of Roots: ax²+bx+c
If α and β are roots of ax²+bx+c, α+β=-b/a αβ=c/a
Points on a parabola
P=(2ap,ap²) Q=(2aq, aq²)
Congruent triangles tests
SAS, AAS, SSS, RHS
The Remainder Theorum
The remainder when P(x) is devided by x-α is P(α)
Volume of a revolution
V=π b∫a y² dx
Angle sum of a quadrilateral
a + b + c + d = 360°
Angle sum of a triangle
a + b + c = 180°
The Sine Rule
a/sinA = b/sinB = c/sinC
Equations of the type asin∅ + bcos∅ = c
asin∅ + bcos∅ = Rsin(∅ + α) asin∅ - bcos∅ = Rsin(∅ - α) acos∅ + bsin∅ = Rcos(∅ - α) acos∅ - bsin∅ = Rcos(∅ + α) Where R = √(a² + b²), R>0; tanα = b/a, 0°≤∅≤360°
General form
ax + by + c = 0
∫ax^ndx
ax^(n+1)/(n+1) + C
The Cosine Rule
a² = b² + c² - 2bc.cosA
Sums and Differences of Angles
cos(α - β) = cosαcosβ + sinαsinβ cos(α + β) = cosαcosβ - sinαsinβ sin(α - β) = sinαcosβ - cosαsinβ sin (α +β) = sinαcosβ + cosαsinβ tan(α - β) = (tanα - tanβ) / (1 + tanαtanβ) tan(α + β) = (tanα + tanβ) / (1 - tanαtanβ)
Exact Values (cos)
cos30° = √3/2 cos45° = 1/√2 cos60° = 1/2
Reciprocal Ratios
cosecα = 1/sinα = H/O secα = 1/cosα = H/A cotα = 1/tanα = A/O
Complimentary Angles
cos∅ = sin(90° - ∅) sin∅ = cos(90° - ∅) sec∅ = cosec(90° - ∅) cosec∅ = sec(90° - ∅) tan∅ = cot(90° - ∅) cot∅ = tan(90° - ∅)
Pythagoras' Theorem
c² = a² + b²
Perpendicular distance formula
d = |ax1 + by1 + c| / √(a² + b²)
The Chain Rule
dy/dx=dy/du×du/dx if y=[f(x)]ⁿ, dy/dx=n[f(x)]ⁿ⁻¹.f'(x)
Exterior angle of a triangle
exterior < = sum of int. opp. <s
Derrivative of xⁿ
if f(x)=xⁿ, f'(x)=nxⁿ⁻¹
The Quotient Rule
if y=u/v, dy/dx=(vu'-uv')/v²
The Product Rule
if y=u×v, dy/dx=uv'+vu'
Arc Length
l = r∅ [in radians]
Limits involving infinity
lim[m/x, x→∞] = 0
Differentiation from first principles
lim[{f(x+h)-f(x)}/h, h→0]
a^b=c
log[a, c]=b
log[a, x]-log[a, y]
log[a, x/y]
log[y, x]
log[a, x]/log[a, y]
log[a, x]+log[a, y]
log[a, xy]
Gradient of a straight line
m = (y2 - y1) / (x2 - x1)
The gradient of the normal at P
m=-1/p
The gradient of the tangent at P
m=p
If PQ is a focal chord...
pq=-1, tangents of P and Q intersect at 90⁰ on the directrix
Angle sum of a polygon
s = 180(n - 2)
Size of an angle in a regular polygon
s = 180(n - 2) / n
Ratios as even or odd functions
sin(-∅) = -sin∅ [odd function] cos(-∅) = cos∅ [even function] tan(-∅) = -tan∅ [odd function]
Double Angles
sin2∅ = 2sin∅cos∅ cos2∅ = cos²∅ - sin²∅ cos2∅ = 2cos²∅ - 1 cos2∅ = 1 - 2sin²∅ tan2∅ = 2tan∅ / (1 - tan²∅)
Exact Values (sin)
sin30° = ½ sin45° = 1/√2 sin60° = √3/2
Pythagorean Identities
sin²∅ + cos²∅ = 1 tan²∅ + 1 = sec²∅ cot²∅ + 1 = cosec²∅
Ratios in terms of t = tanα/2, "The t formulas"
sinα = 2t / (1 + t²) cosα = (1 - t²) / (1 + t²) tanα = 2t / (1 - t²)
Trigonometric Ratios
sinα = O/H cosα = A/H tanα = O/A
Exact Values (tan)
tan30° = 1/√3 tan45° = 1 tan60° = √3
Angle between two lines
tanα = |(m1 - m2) / (1 + m1m2)|
Trigonometric Tdentities
tan∅ = sin∅ / cos∅ cot∅ = cos∅ / sin∅
Division of an interval in the ratio k:l
x = (lx₁ + kx₂) / (k + l) y = (ly₁ + ky₂) / (k + l)
Axis of Symetry
x = -b / 2a
Roots of a Quadratic Function
x = [-b ± √(b² - 4ac)] / 2a
The Quadratic Formula
x = [-b ± √(b² - 4ac)] / 2a
Equation of a vertical line
x = a
The Equation of the Normal at P
x+py=ap³+2ap
Intercept form
x/a + y/b = 1
The Equation of the Chord of Contact from (x₀,y₀) to x²=4ay
xx₀=2a(y+y₀)
Cartesian and Parametric Equation of a parabola
x²=4ay --> x=2at, y=at² x²=-4ay-->x=2at, y=-at² y²=4ax --> x=at², y=2at y²=-4ax--> x=-at², y=2at
Sums and Differences of Cubcs
x³ + y³ = (x + y)(x² - xy + y²) x³ - y³ = (x-y)(x² + xy + y²)
The Cartesian Equation for the Normal at P
x₁y+2ax=x₁y₁+2ax₁
Point-gradient form
y - y1 = m(x - x1)
The Quadratic Function
y = ax²+ bx + c
Equation of a horizontal line
y = b
Equation of a Semi-Circle
y = b ± √(r² - (x - a)²)
Gradient-intercept form
y = mx + b
Equation of the Chord PQ
y-½(p+q)x+apq=0
The Equation of the Tangent at P
y=px-ap²
log[a, x^y]
ylog[a, x]
The Point of Intersection for the Normals at P and Q
{-apq(p+q), a(p²+pq+q²+2)}
The tangents at P and Q meet at...
{a(p+q), apq}
Gradient of the chord PQ
½(p+q)
Area of a Triangle
½ab.sinC ½bh
Combinations (order doesn't matter)
ⁿCr=n!/r!(n-r)!
Permutations (order does matter)
ⁿPr=n!/(n-r)! or, is some objects are similar: n!/p!q! or, in a circle: (n-1)!
Descriminant
∆=b²-4ac If ∆<0, no real roots If ∆=0 one real root If ∆>0, 2 real roots If ∆ is a perfect square, roots are rational
Radians
∏radians = 180° 1 radian = 180/∏ 1° = ∏/180
The Distance Formula
√[ (x₂ - x₁)² + (y₂ - y₁)²]
