FM - Unit 4

Question 1

General Solutions for Trigonometric Equations

Answer 1

cos, x = 2nπ ± PV
sin, x = nπ + (-1)^n PV
tan, x = nπ + PV

Question 2

Roots of Unity

Answer 2

Where x^n = 1

find modulus & argument where arg(x) = (O + 2mπ) / n

Question 3

Trigonometric Equations In Complex Forms

Answer 3

cos(nO) = (z^n + z^-n) / 2
sin(nO) = (z^n - z^-n) / 2i

Question 4

Conditions For Matrix Simultaneous Equation Solutions

Answer 4

If det(M) =/ 0, there is a unique solution to the equations, else there may be none or infinite solutions.

If the equations are consistent, there are infinite solutions, else there are none.

Question 5

Geometric Interpretations of Simultaneous Equations

Answer 5

Planes meet at a point for unique solution

Planes form a sheath for consistent equations, infinite solutions

Planes form a prism or parallel for inconsistent equations

Planes identical for consistent equations, infinte solutions

Question 6

Polar To Cartesian

Answer 6

r = x^2 + y^2
x = rcos0
y = rsin0

Question 7

Perpendicular & Parallel Tangents to Initial Line

Answer 7

dx/d0 = 0 for perpendicular

dy/d0 for parallel

Question 8

Hyperbolic Functions

Answer 8

sinhx = ex - e-x / 2
coshx = ex + e-x / 2
tanhx = e2x - 1 / e2x + 1

Question 9

Deriving Inverse Hyperbolic Function Logarithm Form

Answer 9

let y = hf-1(x)
x = expform(y)

rearrange and quadratic formula

Question 10

1st Order Differentiation Methods

Answer 10

Integrating Factors
Equation has to be in the form dy/dx + yP(x) = Q(x)

Multiply by e^integral(P(x))

Reverse product rule to define LHS as a single derivative

Integrate

Separating Variables
Get all x and all y on LHS and RHS respectively

Question 11

Complementary Function General Forms

Answer 11

Auxillary function has:

Two real roots: y = Ae^mx + Be^nx

Two repeated roots: y = (A + Bx)e^mx

Two complex roots (p +qi): y = e^px (Acosqx + Bsinqx)

Question 12

Particular Integral General Forms

Answer 12

f(x) =

k, PI = A
kx, PI = Ax + B
kx^2, PI = Ax^2 + Bx + C
ksinx or kcosx, PI = Asinwx + Bcoswx
e^kx, PI = Ae^kx

Question 13

Justification For Choosing +ive Root In Derivation Of Hyperbolic Trig. Functions In Log. Form

Answer 13

Derivative of sinhx is always positive coshx