Hsc 3u Maths Formulae 5f6322

MATHEMATICS REVISION OF FORMULAE AND RESULTS Surds

  

Co-ordinate Geometry

a × b = ab a b

b



Gradient formula: m =



Midpoint Formula: midpoint =



Perpendicular distance from a point to a line:

2

Absolute Value

ab = a . b a+b ≤ a + b



m1 − m2 1 + m1 m2

Equations of a Line gradient-intercept form: y = mx + b

x3 − y3 = x − y (x2 + xy + y2 )

point-gradient form:

x3 + y3 = x + y (x2 − xy + y2 )

two point formula:

Real Functions intercept formula:

A function is even if f −x = f(x). The graph is symmetrical about the y-axis. A function is odd if f −x = − f(x). The graph has point symmetry about the origin.

The Circle The equation of a circle with: Centre the origin (0, 0) and radius r units is:

x2 + y2 = r2 

x1 + x2 y1 + y2 , 2 2

Acute angle between two lines (or tangents)

tanθ =

Factorisation



or m = tanθ

a2 + b2

x is the distance of x from the origin on the number line x − y is the distance between x and y on the number line 



x2 − x1

2

ax1 + by1 + c

Geometrically:



y2 − y1

( a) = a

a = a if a ≥ 0 a = − a if a < 0

+ (y2 − y1 )2

Distance formula: d =

a

=

x2 − x1



Centre (a, b) and radius r units is:

(x − a)2 + (y − b)2 = r2

y − y1 = m(x − x1 ) y − y1 x − x1 x a

+

y b

=

y2 − y1 x2 − x1

=1

general equation: ax + by + c = 0 

Parallel lines:

m1 = m2



Perpendicular lines:

m1 .m2 = − 1

Trigonometric Results 



sinθ =

cosθ =

The Quadratic Polynomial

opposite

(SOH)

hypotenuse

adjacent



The general quadratics is: y = ax2 + bx + c



The quadratic formula is:



The discriminant is: Δ = b − 4ac

hypotenuse

opposite

tanθ =



Complementary ratios:

2

If Δ < 0 the roots are not real If Δ = 0 the roots are equal If Δ is a perfect square, the roots are rational

cos 90° − θ = sinθ tan 90° − θ = cotθ



If α and β are the roots of the quadratic equation

ax2 + bx + c = 0

sec 90° − θ = cosecθ

then:

cosec(90° − θ) = secθ

sin2 θ + cos2 θ = 1 1 + cot2 θ = cosec2 θ

tanθ =

sinθ cosθ

and cotθ =

The Sine Rule a

=

b sinB

=



If a quadratic function is positive for all values of x, it is positive definite i.e. Δ < 0 and a > 0



If a quadratic function is negative for all values of x, it is negative definite i.e. Δ < 0 and a < 0



If a function is sometimes positive and sometimes negative, it is indefinite i.e. Δ > 0

b2 + c2 − a2 2bc

The Area of a Triangle 1

sinθ

c sinC

a2 = b2 + c2 − 2bcCosA



b

x = − 2a

The axis of symmetry is:

The Parabola

The Cosine Rule

CosA =

cosθ

c a



tan2 θ + 1 = sec2 θ



b

α + β = − a and αβ =

Pythagorean Identities

sinA

2a

If Δ ≥ 0 the roots are real

(TOA)

adjacent

sin 90° − θ = cosθ



−b ± b2 − 4ac

(CAH)





x=

Area = 2 abSinC



The parabola x2 = 4ay has vertex (0,0), focus (0,a), focal length ‘a’ units and directrix y = − a



The parabola (x − h) = 4a(y − k) has vertex (h, k)

2

Differentiation

Geometrical Applications of Differentiation





Stationary points:



Increasing function:



Decreasing function:



Concave up:



Concave down:



Minimum turning point:



Maximum turning point:



Points of inflexion: about the point.



Horizontal points of inflexion: = 0 and dx2 = 0 and dx concavity changes about the point.

First Principles: f ' (x) = h lim →∞ f ' (c) = xlim →c

    

If y = xn then Chain Rule:

dx

dx

or

h f (x) – f (c) h

= nxn−1

Quotient Rule: If y =

u v

then

dx dy dx

dy dx

d dx d dx

=

v

du dv +u dx dx v2

sinx = cosx cosx = − sinx tanx = sec2 x

Exponential Functions:

d dx d dx

Logarithmic Functions:

dy

d dx

>0

dx dy

<0

dx d2 y dx2 d2 y dx2

<0 >0

du

= u dx + v dx

Trigonometric Functions:

dx



dv

=0

dx

du

f (u) = f ' (u)

Product Rule: If y = uv then

d



d

dy

f (x + h) – f (x)

dy

ef (x) = f ' (x)ef (x) ax = ax .lna loge f (x) =

f ' (x) f (x)

d2 y dx2

dy dx dy dx

= 0 and = 0 and

d2 y dx2 d2 y dx2

>0 <0

= 0 and concavity changes

dy

d2 y

Approximation Methods

Sequences and Series





The Trapezoidal Rule:

d = U2 − U1

b

h f x dx = y + y + 2 y1 + y2 + y3 + …+ yn−1 2 0 n 

Arithmetic Progression

a

Un = a + n − 1 d

Simpson’s Rule:

Sn = 2 [2a + n − 1 d]

n

b

n

h f x dx = y + y + 4 y1 + y3 + … + 2 y2 + y4 + … 3 0 n a

Sn = 2 [a + l] where l is the last term 

In both rules, h =

b−a where n

r=

n is the number of strips.

by 



If

dx dx dx

b a

f x

2

= xn then y =

xn+1 n+1



If



Trigonometric Functions:

dx

= ax + b

n

then y =

ax + b n

𝑟 −1

The Trigonometric Functions 

 radians = 180⁰



Length of an arc:

l = rθ



Area of a sector:

A = 2 r2 θ



Area of a segment:

A = 2 r2 (θ − sinθ)



Exponential Functions:

Logarithmic Functions: f ' (x) f (x)

dx = loge x + C

1

Small angle results:

lim sinx x→0 x

a

1

[In these formulae,  is measured in radians.]

sec2 x dx = tanx + C

+ C and

1−r

a

sinx → 0 cosx → 1 tanx → 0

eax

a 1 − rn

=

1 −r

cos x dx = sinx + C

eax dx = 

a rn − 1

a(n + 1)

sin x dx = − cosx + C



S∞ =

x dx.

The volume obtained by rotating the curve y = f (x) about the x-axis between x = a and x = b is given by

π

U1

Sn =

If f (x) ≥ 0 for a ≤ x ≤ b, the area bounded by the curve y = f (x), the x-axis and x = a and x = b is given b f a

U2

Un = arn−1

Integration 

Geometric Progression

ax dx =

1

.ax ln a

= x lim →0

tanx x

=1



For y = sin nx and y = cos nx the period is



For y = sin nx the period is

π n

2π n

Logarithmic and Exponential Functions 

The Index Laws:

ax × ay = ax+y ax ÷ ay = ax−y ax

y

a−x = x

= axy 1 ax

y

ay = ax a0 = 1 

The logarithmic Laws: If loga b = c then ac = b

loga x + loga y = loga xy loga x − loga y = loga

x y

loga 𝑥 n + nloga x loga a = 1 and loga 1 = 0 

The Change of Base Result:

loga b =

loge b loge a

log

b

= log10 a 10

EXTENSION 1 REVISION OF FORMULAE AND RESULTS 

Co-ordinate Geometry 

When solving asinθ + bcosθ = c we can solve by writing in the form Rsin(θ + α) = c where:

Dividing an interval in the ratio m:n

mx2 + nx1 my2 + ny1 , m+n m+n 

Acute angle between two lines (or tangents)

m1 − m2 tanθ = 1 + m1 m2 Trigonometric Ratios 

Subsidiary Angle Method (Rsin(θ + α))

R = a2 + b 2 and tan α =

tan(A + B) =

tanA + tanB

a

Parameters 

The parametric equations for the parabola x2 = 4ay are x = 2at and y = at2



All other formulae in this subject are not to be committed to memory but students must know how they are derived.

Sum and Difference Results sin(A + B) = sinA cosB + cosA sinB sin(A – B) = sinA cosB – cosA sinB cos(A + B) = cosA cosB – sinA sinB cos(A – B) = cosA cosB + sinA sinB

b

Polynomials 

A real polynomial is in the form:

1– tanAtanB

P(x) = pnx2 + pn-1xn-1 + ... p2x2 + p1x + p0 tan(A – B) =



tanA – tanB 1+ tanAtanB

Double Angle Results sin2A = 2sinA cosA cos2A = cos2A – sin2A cos2A = 1 – 2sin2A cos2A = 2cos2A – 1 2tanA

tan2A = 1 – tan2 A 

θ

The ‘t’ Formulae where t = tan2 sin x = cos x =



p1, p2, p3, ....., pn are coefficients and are real numbers, usually integers.



The degree of the polynomial is the highest power of x with non-zero coefficient.



A polynomial of degree n has at most n real roots but may have less.



The result of a long division can be written in the form P(x) = A(x) . Q(x) + R(x)



The remainder theorem states that when P(x) is divided by (x – a) the remainder is P(a).



The factor theorem states that if x = a is a factor of P(x) then P(a) = 0.



If , , , , ... are the roots of a polynomial then

2t 1+ t2 1 − t2 1+ t2 2t

tan x = 1 − t2

b

c

d

 = − a,  = a,  = − a,  =

e a

Numerical Estimation of the Roots of an Equation

Inverse Trigonometric Functions



Halving the Interval Method





Newton’s Method If x = x0 is an approximation to a root of P(x) = 0 then x1 = x0 –

P(x0 ) P' (x0 )

y = sin-1x

y = cos-1x

is generally a

better approximation.

y = tan-1x

Be familiar with the conditions under which this method fails. Mathematical Induction 

Step 1:

Prove result true for n = 1 (It is sometimes necessary to have a different first step.)



Step 2:

Assume it is true for n = k and then prove true for n = k + 1



Step 3:

Conclusion as given in class



Integration

1



1

cos  = 2 1 + cos2θ  

−2 ≤ y ≤ − 2

Domain:

–1  x  1

Range:

0≤ y≤ 

Domain:

all real x

Range:

−2 ≤ y ≤ − 2

π

π

π

π

Properties:

General Solutions of Trigonometric Equations:

Derivatives: d

2

Range:

if sin = q, then  = n + (–1)n sin-1q if cos = q, then  = 2n  cos-1q if tan = q, then  = n + tan-1q

sin2 θ dθ and cos2 θ dθ can be solve using the substitutions: sin2  = 2 1 − cos2θ

–1  x  1

sin-1(-x) = -sin-1x cos-1(-x) =  – cos-1x tan-1(-x) = –tan-1x π sin-1x + cos-1x = 2 sin(sin-1x) = x cos(cos-1x) = x tan(tan-1x) = x 



Domain:

dx d dx d

Integration by first making a substitution.

dx

sin-1 x = cos-1 x =

1 1− x2 −1 1− x2 1

tan-1 x = 1 + x2

Table of Standard Integrals as provided in HSC 

Integrals: 1

𝑑𝑥 = sin-1

a2 – x2 1 1 𝑑𝑥 = tan-1 2 2 a +x a

x x = −cos-1 a a x a