By the end of this module you will be expected to have learnt the following formulae:
Dividing and Factoring Polynomials
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If you have a polynomial f(x) divided by x - c, the remainder is equal to f(c). Note if the equation is x + c then you need to negate c: f(-c).
A polynomial f(x) has a factor x - c if and only if f(c) = 0. Note if the equation is x + c then you need to negate c: f(-c).
The Laws of Exponents
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b
x
b
y
=
b
x
+
y
{\displaystyle b^{x}b^{y}=b^{x+y}\,}
b
x
b
y
=
b
x
−
y
{\displaystyle {\frac {b^{x}}{b^{y}}}=b^{x-y}}
(
b
x
)
y
=
b
x
y
{\displaystyle \left(b^{x}\right)^{y}=b^{xy}}
a
n
b
n
=
(
a
b
)
n
{\displaystyle a^{n}b^{n}=\left(ab\right)^{n}\,}
(
a
b
)
n
=
a
n
b
n
{\displaystyle \left({\frac {a}{b}}\right)^{n}={\frac {a^{n}}{b^{n}}}}
b
−
n
=
1
b
n
{\displaystyle b^{-n}={\frac {1}{b^{n}}}}
b
c
x
=
(
b
x
)
c
{\displaystyle b^{\frac {c}{x}}=\left({\sqrt[{x}]{b}}\right)^{c}}
b
1
=
b
{\displaystyle b^{1}=b\,}
b
0
=
1
{\displaystyle b^{0}=1\,}
Logarithmic Function
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The inverse of
y
=
b
x
{\displaystyle y=b^{x}\,}
x
=
b
y
{\displaystyle x=b^{y}\,}
y
=
log
b
x
{\displaystyle y=\log _{b}x\,}
Change of Base Rule:
log
a
x
{\displaystyle \log _{a}x\,}
log
b
x
log
b
a
{\displaystyle {\frac {\log _{b}x}{\log _{b}a}}}
Laws of Logarithmic Functions
edit
When X and Y are positive.
log
b
X
Y
=
log
b
X
+
log
b
Y
{\displaystyle \log _{b}XY=\log _{b}X+\log _{b}Y\,}
log
b
X
Y
=
log
b
X
−
log
b
Y
{\displaystyle \log _{b}{\frac {X}{Y}}=\log _{b}X-\log _{b}Y\,}
log
b
X
k
=
k
log
b
X
{\displaystyle \log _{b}X^{k}=k\log _{b}X\,}
Conversion of Degree Minutes and Seconds to a Decimal
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X
+
Y
60
+
Z
3600
{\displaystyle X+{\frac {Y}{60}}+{\frac {Z}{3600}}}
s
=
θ
r
{\displaystyle s=\theta r\,}
A
r
e
a
=
1
2
r
2
θ
{\displaystyle Area={\frac {1}{2}}r^{2}\theta }
The Trigonometric Ratios Of An Angle
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Function
Written
Defined
Inverse Function
Written
Equivalent to
Cosine
cos
θ
{\displaystyle \cos \theta \,}
A
d
j
a
c
e
n
t
H
y
p
o
t
e
n
u
s
e
{\displaystyle {\frac {Adjacent}{Hypotenuse}}}
arccos
θ
{\displaystyle \arccos \theta \,}
cos
−
1
θ
{\displaystyle \cos ^{-1}\theta \,}
x
=
cos
y
{\displaystyle x=\cos \ y\,}
Sine
sin
θ
{\displaystyle \sin \theta \,}
O
p
p
o
s
i
t
e
H
y
p
o
t
e
n
u
s
e
{\displaystyle {\frac {Opposite}{Hypotenuse}}}
arcsin
θ
{\displaystyle \arcsin \theta \,}
sin
−
1
θ
{\displaystyle \sin ^{-1}\theta \,}
x
=
sin
y
{\displaystyle x=\sin \ y\,}
Tangent
tan
θ
{\displaystyle \tan \theta \,}
O
p
p
o
s
i
t
e
A
d
j
a
c
e
n
t
{\displaystyle {\frac {Opposite}{Adjacent}}}
arctan
θ
{\displaystyle \arctan \theta \,}
tan
−
1
θ
{\displaystyle \tan ^{-1}\theta \,}
x
=
tan
y
{\displaystyle x=\tan \ y\,}
Important Trigonometric Values
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You need to have these values memorized.
θ
{\displaystyle \theta \,}
r
a
d
{\displaystyle rad\,}
sin
θ
{\displaystyle \sin \theta \,}
cos
θ
{\displaystyle \cos \theta \,}
tan
θ
{\displaystyle \tan \theta \,}
0
∘
{\displaystyle 0^{\circ }}
0
0
1
0
30
∘
{\displaystyle 30^{\circ }}
π
6
{\displaystyle {\frac {\pi }{6}}}
1
2
{\displaystyle {\frac {1}{2}}}
3
2
{\displaystyle {\frac {\sqrt {3}}{2}}}
1
3
{\displaystyle {\frac {1}{\sqrt {3}}}}
45
∘
{\displaystyle 45^{\circ }}
π
4
{\displaystyle {\frac {\pi }{4}}}
2
2
{\displaystyle {\frac {\sqrt {2}}{2}}}
2
2
{\displaystyle {\frac {\sqrt {2}}{2}}}
1
{\displaystyle 1\,}
60
∘
{\displaystyle 60^{\circ }}
π
3
{\displaystyle {\frac {\pi }{3}}}
3
2
{\displaystyle {\frac {\sqrt {3}}{2}}}
1
2
{\displaystyle {\frac {\sqrt {1}}{2}}}
3
{\displaystyle {\sqrt {3}}}
90
∘
{\displaystyle 90^{\circ }}
π
2
{\displaystyle {\frac {\pi }{2}}}
1
0
-
a
2
=
b
2
+
c
2
−
2
b
c
cos
α
{\displaystyle a^{2}=b^{2}+c^{2}-2bc\cos \alpha \,}
b
2
=
a
2
+
c
2
−
2
a
c
cos
β
{\displaystyle b^{2}=a^{2}+c^{2}-2ac\cos \beta \,}
c
2
=
a
2
+
b
2
−
2
a
b
cos
γ
{\displaystyle c^{2}=a^{2}+b^{2}-2ab\cos \gamma \,}
a
sin
α
=
b
sin
β
=
c
sin
γ
{\displaystyle {\frac {a}{\sin \alpha }}={\frac {b}{\sin \beta }}={\frac {c}{\sin \gamma }}}
A
r
e
a
=
1
2
b
c
sin
α
{\displaystyle Area={\frac {1}{2}}bc\sin \alpha \,}
A
r
e
a
=
1
2
a
c
sin
β
{\displaystyle Area={\frac {1}{2}}ac\sin \beta \,}
A
r
e
a
=
1
2
a
b
sin
γ
{\displaystyle Area={\frac {1}{2}}ab\sin \gamma \,}
Trigonometric Identities
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sin
2
θ
+
cos
2
θ
=
1
{\displaystyle \sin ^{2}\theta +\cos ^{2}\theta =1\,}
t
a
n
θ
=
sin
θ
cos
θ
{\displaystyle tan\theta ={\frac {\sin \theta }{\cos \theta }}}
The reason that we add a + C when we compute the integral is because the derivative of a constant is zero, therefore we have an unknown constant when we compute the integral.
∫
x
n
d
x
=
1
n
+
1
x
n
+
1
+
C
,
(
n
≠
−
1
)
{\displaystyle \int x^{n}\,dx={\frac {1}{n+1}}x^{n+1}+C,\ (n\neq -1)}
∫
k
x
n
d
x
=
k
∫
x
n
d
x
{\displaystyle \int kx^{n}\,dx=k\int x^{n}\,dx}
∫
{
f
′
(
x
)
+
g
′
(
x
)
}
d
x
=
f
(
x
)
+
g
(
x
)
+
C
{\displaystyle \int \left\{f^{'}(x)+g^{'}(x)\right\}\,dx=f(x)+g(x)+C}
∫
{
f
′
(
x
)
−
g
′
(
x
)
}
d
x
=
f
(
x
)
−
g
(
x
)
+
C
{\displaystyle \int \left\{f^{'}(x)-g^{'}(x)\right\}\,dx=f(x)-g(x)+C}
Rules of Definite Integrals
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∫
a
b
f
(
x
)
d
x
=
F
(
b
)
−
F
(
a
)
{\displaystyle \int _{a}^{b}f\left(x\right)\ dx=F\left(b\right)-F\left(a\right)}
∫
a
b
f
(
x
)
d
x
=
−
∫
b
a
f
(
x
)
d
x
{\displaystyle \int _{a}^{b}f\left(x\right)\ dx=-\int _{b}^{a}f\left(x\right)\ dx}
∫
a
a
f
(
x
)
d
x
=
0
{\displaystyle \int _{a}^{a}f\left(x\right)\ dx=0}
Area between a curve and the x-axis is
∫
a
b
y
d
x
(
for
y
≥
0
)
{\displaystyle \int _{a}^{b}y\,dx\ ({\mbox{for}}\ y\geq 0)}
Area between a curve and the y-axis is
∫
a
b
x
d
y
(
for
x
≥
0
)
{\displaystyle \int _{a}^{b}x\,dy\ ({\mbox{for}}\ x\geq 0)}
Area between curves is
∫
a
b
|
f
(
x
)
−
g
(
x
)
|
d
x
{\displaystyle \int _{a}^{b}{\begin{vmatrix}f\left(x\right)-g\left(x\right)\end{vmatrix}}dx}
∫
a
b
y
d
x
≈
1
2
h
{
(
y
0
+
y
n
)
+
2
(
y
1
+
y
2
+
…
+
y
n
−
1
)
}
{\displaystyle \int _{a}^{b}y\,dx\approx {\frac {1}{2}}h\left\{\left(y_{0}+y_{n}\right)+2\left(y_{1}+y_{2}+\ldots +y_{n-1}\right)\right\}}
Where:
h
=
b
−
a
n
{\displaystyle h={\frac {b-a}{n}}}
∫
a
b
f
(
x
)
d
x
≈=
h
[
f
(
x
1
)
+
f
(
x
2
)
+
…
+
f
(
x
n
)
]
{\displaystyle \int _{a}^{b}f\left(x\right)\,dx\approx =h\left[f\left(x_{1}\right)+f\left(x_{2}\right)+\ldots +f\left(x_{n}\right)\right]}
Where:
h
=
b
−
a
n
{\displaystyle h={\frac {b-a}{n}}}
and
x
i
=
1
2
[
(
a
+
{
i
−
1
}
h
)
+
(
a
+
i
h
)
]
{\displaystyle x_{i}={\frac {1}{2}}\left[\left(a+\left\{i-1\right\}h\right)+\left(a+ih\right)\right]}
![{\displaystyle b^{\frac {c}{x}}=\left({\sqrt[{x}]{b}}\right)^{c}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/260916fb4cd9fa2865d3d3129e5233c2b9485af5)
![{\displaystyle \int _{a}^{b}f\left(x\right)\,dx\approx =h\left[f\left(x_{1}\right)+f\left(x_{2}\right)+\ldots +f\left(x_{n}\right)\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c8892220f2181ca22226fdf533fd77b93b9ac525)
![{\displaystyle x_{i}={\frac {1}{2}}\left[\left(a+\left\{i-1\right\}h\right)+\left(a+ih\right)\right]}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5259213f3fabd4478a2a51ec53b732043773d266)