This document provides definitions and theorems related to allied mathematics II, including: - The definition of a root of a polynomial as a number that makes the polynomial equal to zero. - The fundamental theorem of algebra, which states that every polynomial of degree greater than 0 has at least one real or complex root. - Theorems regarding the expansions of trigonometric functions like sin, cos, and tan in terms of powers of the angle. - Problems applying the theorems, like finding the value of an angle given a trigonometric ratio or expanding a trigonometric function. The document also introduces hyperbolic functions like sinh, cosh, and tanh and provides their

1. Allied Mathematics II
SWATHI SUNDARI.S
Assistant Professor of Mathematics
DEPARTMENT OF MATHEMATICSC(SF)
V.V.VANNIAPERUMAL COLLEGE FOR WOMEN
VIRUDHUNAGAR - 626001.
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2. Formation of Equations
Definition
Let f(x) and g(x) be two polynomials. We say that g(x) divides f(x) if there
exists another polynomial h(x) such that f(x) = h(x)g(x).
Definition
A real or complex number a is called a root of the polynomial f(x) if f(a) = 0.
Remark
a is a root of f(x) if and only if x − a divides f(x) and we say that x − a is a
factor of f(x).
Definition
If r is a positive integer such that (x − a)r
divides f(x) and (x − a)r+1
does not
divide f(x), then we say α is a root of multiplicity r.
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3. Theorem
Fundamental Theorem of Algebra: Every polynomial of degree n > 0 has
atleast one real or imaginary root.
Theorem
Every polynomial of degree n > 0 has exactly n roots where the roots are
counted according to multiplicity r.
Theorem
For any equation with real coefficients the complex roots occur in conjugate
pairs. (i.e) If α + iβ is a root then α − iβ is also a root.
Theorem
Let f(x) be a polynomial with rational coefficients . Let α, β be rational
numbers where β > 0 and β is not a perfect square. Then if α +
√
β is a root
of f(x) = 0 then α −
√
β is also a root.
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4. Expansion of Sin θ,Cos θ and Tan θ
Theorem
When θ is expressed in radians
(i) sin θ = θ -
θ3
3! +
θ5
5! - ....
(ii) cos θ = 1 -
θ2
2! +
θ4
4! - ....
(iii) tan θ = θ +
θ3
3 +
2θ15
15 + ....
Proof :
(i) Let f(θ)= sin θ . By Taylor’s expansion of f about the origin is given by
f(θ) = f(0) + f (0) θ
1! + f (0)θ2
2! + f (0)θ3
3! + ... (1)
Now, f(θ)= sin θ ⇒f(0)= 0; f’(θ)= cos θ ⇒f’(0)= 1;
f (θ)= -sin θ ⇒f”(0)= 0; f”’(θ)= -cos θ ⇒f”’(0)= −1.
Therefore, From (1), sin θ = θ -
θ3
3! +
θ5
5! - ....
(ii) Proof is similar to (i) .
(iii) Proof is similar to (i) .
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5. Problem
If
sinθ
θ = 863
864, then the value of θ is
1
12 radians (approaximately).
Problem
If
tanθ
θ = 2524
2523, then the value of θ is 1◦
58 radians (approaximately).
Problem
If x is small cos(α + x) = cos α − x sin α−
x2
2 cos α +
x3
6 sin α
Problem
If cos (
π
3 + θ) = 0.49 then the value of θ is 40 minutes (approaximately).
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6. Problem
If x =
2
1! -
4
3! +
6
5! -
8
7! +.... and y = 1 +
2
1! -
23
3! +
25
5! -.... then prove that
x2
= y.
Problem
If θ is small then
(i) θ cot θ = 1 -
θ2
3 -
θ4
45 approximately.
(ii)
1
6 sin3
θ =
θ3
3! - (1 + 32
)
θ5
5! + (1 + 32
+ 34
)
θ7
7! -...
Problem
(i) limx→0
3 sin x−sin 3x
x−sin x = 24
(ii) limx→0
cos2
ax−cos2
bx
1−cos cx =
2(b2
−a2
)
c2 .
(ii) limx→π/2 (
sin x+cos 2x
cos2x
) =
3
2.
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7. Hyperbolic Functions
Definition
The hyperbolic functions are defined by
(i) sinh x =
ex
−e−x
2
(ii) cosh x =
ex
+e−x
2
(iii) tanh x =
sinh x
cosh x
(iii) coth x =
cosh x
sinh x
(iii) cosech x =
1
sinh x
(iii) sech x =
1
cosh x
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8. Theorem
(i) sinh−1
x = loge(x + x2 + 1)
(ii) cosh−1
x = loge(x + x2 − 1)
(iii) tanh−1
x = 1
2 log (1+x
1−x ).
Problem
(i) (cosh x + sinh x)n
= cosh nx + sinh nx
(ii) (1+tanh x
1−tanh x ) = cosh 2x + sinh 2x
(iii) If tan(ai + b) = x + iy then
x
y =
sin 2a
sinh 2b
(iv) If tan(Ai + B) = x + iy then x2
+ y2
+ 2x cot 2A = 1
(v) If sin(Ai + B) = x + iy then
x2
sin2A
-
y2
cos2A
=1
(vi) If cos(x + iy) = cos θ + i sin θ then cos 2x + cosh 2y = 2
(vii) If sin(θ + iψ) = tan α + i sin α then cos 2θ cosh 2ψ = 3
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9. Problem
(viii) If cos(x + iy) = r(cos α + i sin α) then y = 1
2 log [sin(x−α)
sin(x+α)]
(ix) If tan(x
2 ) = tanh(x
2 ) then cos x cosh x = 1
(x) Let u = logetan(π
4 + θ
2 ) iff cosh u = sec θ
(xi) Let cos α + i sin α = logetan(π
4 + θ
2 ) iff cosh u = sec θ
(xii) If cos(θ + iψ) = cos(θ + iψ) then sin2
θ = ±sin α
(xii) If tan(θ + iψ) = (cos α + i sin α) then θ = 1
2 nπ + 1
4 π and
ψ = 1
2 log tann(1
4 π + 1
2 α)
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10. Thank You
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