Trigonometry/Basic Calculus Formulae Sheet

Every single formula you need for M2 in one place.

## MI Template

P
(
n
)

{\displaystyle P(n)}

 be the following proposition:

When

# n
1

{\displaystyle n=1}

,
LHS =
RHS =

∴
P
(
1
)
 is true.

{\displaystyle \therefore P(1)\text{ is true.}}

Assume that

P
(
k
)

{\displaystyle P(k)}

is true for a positive integer

k

{\displaystyle k}

.
That is,

When

# n
k
1

{\displaystyle n=k+1}

,

(proof that LHS is equal to RHS)

L
H
# S
R
H
S

{\displaystyle LHS = RHS}

.

∴
P
(
k
1
)
 is true if
P
(
k
)
 is true.

{\displaystyle \therefore P(k+1)\text{ is true if }P(k)\text{ is true.}}

By the principle of mathematical induction,

P
(
n
)

{\displaystyle P(n)}

is true for all positive integers

n

{\displaystyle n}

.

## Trigonometry

1. sin
2

⁡
x

cos
2

⁡
# x
1

{\displaystyle \sin^2x+\cos^2x=1}
2. 1

tan
2

⁡
# x

sec
2

⁡
x

{\displaystyle 1+\tan^2x=\sec^2x}
3. 1

cot
2

⁡
# x

csc
2

⁡
x

{\displaystyle 1+\cot^2x=\csc^2x}
4. sin
⁡
(
A
±
B
# )
sin
⁡
A
cos
⁡
B
±
cos
⁡
A
sin
⁡
B

{\displaystyle \sin(A\plusmn B)=\sin A\cos B\plusmn \cos A \sin B}
5. cos
⁡
(
A
±
B
# )
cos
⁡
A
cos
⁡
B
∓
sin
⁡
A
sin
⁡
B

{\displaystyle \cos(A\plusmn B)=\cos A\cos B\mp \sin A \sin B}
6. tan
⁡
(
A
B
# )

tan
⁡
A
±
tan
⁡
B

1
∓
tan
⁡
A
tan
⁡
B

{\displaystyle \tan(A+B)=\frac{\tan A \plusmn \tan B}{1\mp \tan A \tan B}}
7. sin
⁡
2
# x
2
sin
⁡
x
cos
⁡
x

{\displaystyle \sin 2x = 2\sin x \cos x}
8. cos
⁡
2
x

=

cos
2

⁡
x
−

sin
2

⁡
x

=
1
−
2

sin
2

⁡
x

=
2

cos
2

⁡
x
−
1

{\displaystyle \begin{align} \cos 2x & = \cos^2x-\sin^2x \\ & = 1-2\sin^2x \\ &= 2\cos^2x-1\end{align}
}
9. tan
⁡
2
# x

2
tan
⁡
x

1
−

tan
2

⁡
x

{\displaystyle \tan 2x = \frac{2 \tan x}{1-\tan^2 x}}
10. sin
2

⁡
# x

1
2

(
1
−
cos
⁡
2
x
)

{\displaystyle \sin^2 x = \frac{1}{2}(1-\cos 2x)
}
11. cos
2

⁡
# x

1
2

(
1
cos
⁡
2
x
)

{\displaystyle \cos^2 x = \frac{1}{2}(1+\cos 2x)
}
12. tan
2

⁡
# x

1
−
cos
⁡
2
x

1
cos
⁡
2
x

{\displaystyle \tan^2 x = \frac{1-\cos 2x}{1+\cos 2x}
}
13. 2
sin
⁡
A
cos
⁡
B

=
sin
⁡
(
A
B
)
sin
⁡
(
A
−
B
)

=
sin
⁡
(
A
B
)
−
sin
⁡
(
B
−
A
)

{\displaystyle \begin{align} 2 \sin A \cos B & = \sin (A+B)+\sin(A-B) \\ & = \sin (A+B)-\sin(B-A) \\ \end{align}
}
14. 2
cos
⁡
A
cos
⁡
# B
cos
⁡
(
A
B
)
cos
⁡
(
A
−
B
)

{\displaystyle 2 \cos A \cos B = \cos (A+B)+\cos(A-B)
}
15. 2
sin
⁡
A
sin
⁡
# B
cos
⁡
(
A
−
B
)
−
cos
⁡
(
A
B
)

{\displaystyle 2\sin A\sin B=\cos(A-B)-\cos(A+B)}
16. sin
⁡
x
sin
⁡
# y
2
sin
⁡
(

x
y

2

)
cos
⁡
(

x
−
y

2

)

{\displaystyle \sin x+\sin y=2\sin(\frac{x+y}{2})\cos(\frac{x-y}{2})}
17. sin
⁡
x
−
sin
⁡
# y
2
cos
⁡
(

x
y

2

)
sin
⁡
(

x
−
y

2

)

{\displaystyle \sin x-\sin y=2\cos(\frac{x+y}{2})\sin(\frac{x-y}{2})}
18. cos
⁡
x
cos
⁡
# y
2
cos
⁡
(

x
y

2

)
cos
⁡
(

x
−
y

2

)

{\displaystyle \cos x+\cos y=2\cos(\frac{x+y}{2})\cos(\frac{x-y}{2})}
19. cos
⁡
x
−
cos
⁡
# y
−
2
sin
⁡
(

x
y

2

)
sin
⁡
(

x
−
y

2

)

{\displaystyle \cos x-\cos y=-2\sin(\frac{x+y}{2})\sin(\frac{x-y}{2})}

## Limits

1. lim

x
→
0

sin
⁡
θ

θ

=
1

,
w
h
e
r
e

θ

i
s

m
e
a
s
u
r
e
d

i
n

r
a
d
i
a
n
s

{\displaystyle \lim\_{x \to 0} \frac{\sin \theta}{\theta}=1\mathrm{,where\ \theta\ is\ measured\ in\ radians}}
# 2. e

lim

x
→
∞

(
1

1
x

)
x

{\displaystyle e=\lim\_{x \to \infin} (1+\frac{1}{x})^x}
3. e
x

=

∑

# n
0

∞

x
n

n
!

,
o
r

i
n

o
t
h
e
r

w
o
r
d
s
,

e
x

=
1

x

1
!

x
2

2
!

x
3

3
!

x
4

4
!

.
.
.

{\displaystyle e^x=\sum\_{n=0}^\infin \frac{x^n}{n!}\mathrm{, or\ in\ other\ words,\ }e^x=1+\frac{x}{1!}+\frac{x^2}{2!}+\frac{x^3}{3!}+\frac{x^4}{4!}+\ ...}
4. e
x

=

lim

n
→
∞

(
1

x
n

)
n

{\displaystyle e^x=\lim\_{n \to \infin} (1+\frac{x}{n})^n}
5. e
x

=

lim

n
→
∞

(
1
n
x

)

1
n

{\displaystyle e^x=\lim\_{n \to \infin} (1+nx)^{\frac{1}{n}}}
6. lim

x
→
∞

e
x

−
1

x

=
1

{\displaystyle \lim\_{x \to \infin} \frac{e^x-1}{x}=1}

## Differentiation

1. First Principle:

d
y

d
x

=

lim

Δ
x
→
0

Δ
y

Δ
x

=

lim

Δ
x
→
0

f
(
x
Δ
x
)
−
f
(
x
)

Δ
x

{\displaystyle \frac{dy}{dx}=\lim\_{\Delta x \to 0} \frac{\Delta y}{\Delta x} = \lim\_{\Delta x \to 0} \frac{f(x+\Delta x)-f(x)}{\Delta x}}
2. d

d
x

# k
0

,
w
h
e
r
e

k

i
s

a

c
o
n
s
t
a
n
t

{\displaystyle \frac{d}{dx}k=0\mathrm{, where\ }k\mathrm{\ is\ a\ constant}}
3. d

d
x

(

x
n

# )
n

x

n
−
1

{\displaystyle \frac{d}{dx}(x^n)=nx^{n-1}}
4. d

d
x

(
k
u
# )
k
(

d

d
x

u
)

{\displaystyle \frac{d}{dx}(ku)=k(\frac{d}{dx}u)}
5. d

d
x

(
u
v
# )
u

d
v

d
x

v

d
u

d
x

{\displaystyle \frac{d}{dx}(uv)=u\frac{dv}{dx}+v\frac{du}{dx}}
6. d

d
x

(

u
v

# )

v

d
u

d
x

−
u

d
v

d
x

v
2

{\displaystyle \frac{d}{dx}(\frac{u}{v})=\frac{v\frac{du}{dx}-u\frac{dv}{dx}}{v^2}}
7. d
y

d
x

=

d
y

d
u

⋅

d
u

d
x

{\displaystyle \frac{dy}{dx}=\frac{dy}{du}\cdot\frac{du}{dx}}
8. d

d
x

e

k
x

=
k

e

k
x

,
w
h
e
r
e

k

i
s

a

c
o
n
s
t
a
n
t

{\displaystyle \frac{d}{dx}e^{kx}=ke^{kx}\mathrm{, where\ }k\mathrm{\ is\ a\ constant}}
9. d

d
x

ln
⁡
# x

1
x

{\displaystyle \frac{d}{dx}\ln x=\frac{1}{x}}
10. d
sin
⁡
x

x

=
cos
⁡
x

{\displaystyle \frac{d \sin x}{x}=\cos x}
11. d
cos
⁡
x

x

=
−
sin
⁡
x

{\displaystyle \frac{d \cos x}{x}=-\sin x}
12. d
tan
⁡
x

x

=

sec
2

⁡
x

{\displaystyle \frac{d \tan x}{x}=\sec^2 x}
13. d
csc
⁡
x

x

=
−
csc
⁡
x
cot
⁡
x

{\displaystyle \frac{d \csc x}{x}=-\csc x \cot x}
14. d
sec
⁡
x

x

=
sec
⁡
x
tan
⁡
x

{\displaystyle \frac{d \sec x}{x}=\sec x\tan x}
15. d
cot
⁡
x

x

=
−

csc
2

⁡
x

{\displaystyle \frac{d \cot x}{x}=-\csc^2 x}
16. d

d
x

f
(
x
)

=

f
′

(
x
)

2

f
(
x
)

{\displaystyle \frac{d}{dx}\sqrt{f(x)}=\frac{f^\prime (x)}{2\sqrt{f(x)}}}

## Integration

1. ∫

e
x

d
# x

e
x

C

{\displaystyle \int e^x dx = e^x + C}
2. ∫
sin
⁡
x

d
# x
−
cos
⁡
x
C

{\displaystyle \int \sin x\ dx = -\cos x + C}
3. ∫
cos
⁡
x

d
# x
sin
⁡
x
C

{\displaystyle \int \cos x\ dx = \sin x + C}
4. ∫
tan
⁡
x

d
# x
ln
⁡

sec
⁡
x

C

{\displaystyle \int \tan x\ dx = \ln |\sec x| + C}
5. ∫

sec
2

⁡
x

d
# x
tan
⁡
x
C

{\displaystyle \int \sec^2 x\ dx = \tan x + C}
6. ∫

csc
2

⁡
x

d
# x
−
cot
⁡
x
C

{\displaystyle \int \csc^2 x\ dx = -\cot x + C}
7. ∫
sec
⁡
x
tan
⁡
x

d
# x
sec
⁡
x
C

{\displaystyle \int \sec x \tan x\ dx = \sec x + C}
8. ∫
csc
⁡
x
cot
⁡
x

d
# x
−
csc
⁡
x
C

{\displaystyle \int \csc x \cot x\ dx = -\csc x + C}
# 9. s
∫
v

d
t
 and
# v
∫
a

d
t

{\displaystyle s=\int v\ dt\text{ and }v=\int a\ dt}

, where:
	* s

	{\displaystyle s}

	is displacement
	* t

	{\displaystyle t}

	is time elapsed
	* v

	{\displaystyle v}

	is velocity
	* a

	{\displaystyle a}

	is acceleration
20. ∫
u

d
# v
u
v
−
∫
v

d
u

{\displaystyle \int u\ dv = uv-\int v\ du}

a
2

−

x
2

{\displaystyle \sqrt{a^2-x^2}}

x
2

−

a
2

{\displaystyle \sqrt{x^2-a^2}}

a
2

x
2

{\displaystyle \sqrt{a^2+x^2}}

subsitution

{\displaystyle \text{subsitution}}

x

=
a
sin
⁡
θ

d
x

d
θ

=
a
cos
⁡
θ

d
x

=
a
cos
⁡
θ

d
θ

{\displaystyle \begin{alignat}{2} x & = a \sin \theta \\ \frac{dx}{d\theta} & = a \cos \theta \\ dx & = a \cos \theta \ d\theta\\ \end{alignat}}

x

=
a
sec
⁡
θ

d
x

d
θ

=
a
sec
⁡
θ
tan
⁡
θ

d
x

=
a
sec
⁡
θ
tan
⁡
θ

d
θ

{\displaystyle \begin{alignat}{2} x & = a \sec \theta \\ \frac{dx}{d\theta} & = a \sec \theta \tan \theta \\ dx & = a \sec \theta \tan \theta \ d\theta\\ \end{alignat}}

x

=
a
tan
⁡
θ

d
x

d
θ

=
a

sec
2

⁡
θ

d
x

=
a

sec
2

⁡
θ

d
θ

{\displaystyle \begin{alignat}{2} x & = a \tan \theta \\ \frac{dx}{d\theta} & = a \sec^2 \theta \\ dx & = a \sec^2 \theta \ d\theta\\ \end{alignat}}

θ

{\displaystyle \theta}

# θ
arcsin
⁡

x
a

{\displaystyle \theta = \arcsin \frac{x}{a}}

# θ
arctan
⁡

x
a

{\displaystyle \theta = \arctan \frac{x}{a}}

a
2

−

a
2

s
i

n
2

θ

=
a

1
−

sin
2

⁡
θ

=
a
cos
⁡
θ

{\displaystyle \begin{alignat}{2} & \ \ \ \ \sqrt{a^2-a^2 sin^2 \theta} \\ & = a\sqrt{1-\sin^2\theta} \\ & = a \cos \theta \end{alignat}}

a
2

sec
2

⁡
θ
−

a
2

=
a

sec
2

⁡
θ
−
1

=
a
tan
⁡
θ

{\displaystyle \begin{alignat}{2} & \ \ \ \ \sqrt{a^2\sec^2\theta-a^2} \\ & = a\sqrt{\sec^2\theta-1} \\ & = a \tan \theta \end{alignat}}

a
2

a
2

t
a

n
2

θ

=
a

1

tan
2

⁡
θ

=
a
sec
⁡
θ

{\displaystyle \begin{alignat}{2} & \ \ \ \ \sqrt{a^2+a^2 tan^2 \theta} \\ & = a\sqrt{1+\tan^2\theta} \\ & = a \sec \theta \end{alignat}}

cos
⁡
# θ

a
2

−

x
2

x

{\displaystyle \cos \theta = \frac{\sqrt{a^2-x^2}}{x}}

tan
⁡
# θ

x
2

−

a
2

a

{\displaystyle \tan \theta = \frac{\sqrt{x^2-a^2}}{a}}

tan
⁡
# θ

x
a

{\displaystyle \tan \theta = \frac{x}{a}}

sec
⁡
# θ

a

a
2

x
2

{\displaystyle \sec \theta = \frac{a}{\sqrt{a^2+x^2}}}