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]