Appendix A
Proof of Basic Rules of
Differentiation
Sum rule:
du dv
d
(u + v) =
+
dx
dx dx
u(x + δx) + v(x + δx) − u(x) − v(x)
δx→0
δx
u(x + δx) − u(x)
v(x + δx) − v(x)
= lim
+ lim
= RHS.
δx→0
δx→0
δx
δx
LHS =
lim
d
dv
du
(uv) = u + v
dx
dx
dx
Product rule:
Note that
∙
¸
∙
¸
u(x + δx)v(x + δx) − u(x)v(x)
v(x + δx) − v(x)
u(x + δx) − u(x)
= u(x+δx)
+v(x)
.
δx
δx
δx
Now just let δx → 0, observing that
∙
u(x + δx) − u(x)
u(x + δx) = u(x) + δx
δx
¸
tends to u(x) as δx → 0.
− u dv
d ³ u ´ v du
= dx 2 dx
dx v
v
Quotient rule:
Note that
u(x + δx) u(x)
v(x)[u(x + δx) − u(x)] − u(x)[v(x + δx) − v(x)]
−
=
.
v(x + δx) v(x)
v(x)v(x + δx)
Now divide through by δx and let δx → 0, observing that v(x + δx) → v(x) as δx → 0.
1
Chain rule
df du
d
[f {u(x)}] =
.
dx
du dx
The left hand side is the limit as δx → 0 of
∙
¸∙
¸
f {u(x + δx)} − f {u(x)}
f {u(x + δx)} − f {u(x)} u(x + δx) − u(x)
=
,
δx
u(x + δx) − u(x)
δx
¸∙
¸
∙
f {u + δu} − f {u} g(x + δx) − g(x)
,
=
δu
δx
where the x-dependent quantity δu = u(x + δx) − u(x) tends to 0 as δx → 0.
2
Appendix B
Basic Properties of the
Trigonometric Functions
B.1
Fundamental Identities for the Sine and Cosine
We recall the special values:
sin 0
π
cos
2
cos 0
π
sin
2
= 0,
(B.1)
= 0,
(B.2)
= 1,
(B.3)
= 1.
(B.4)
The trigonometric functions possess a high degree of symmetry and accordingly obey
an almost bewildering variety of identities. [NB Recall that an identity is an equation
which holds for all permitted values of its variables.] Of fundamental importance are the
addition and subtraction formulae:sin(x + y)
sin(x − y)
cos(x + y)
cos(x − y)
=
=
=
=
sin x cos y + cos x sin y,
sin x cos y − cos x sin y,
cos x cos y − sin x sin y,
cos x cos y + sin x sin y.
(B.5)
(B.6)
(B.7)
(B.8)
Take special note here of the signs in equations (B.7) and (B.8). The truth of the addition
formulae (B.5) and (B.7) for 0 < x < x + y < π/2 and of the subtraction formulae (B.6)
and (B.8) for 0 < y < x < π/2 is immediately apparent geometrically from the two
diagrams which follow, but note that (B.5)—(B.8) hold for all real values of the variables
x and y.
3
Diagram for proof of the addition formulae for 0 < x < x + y < π/2
Diagram for proof of the subtraction formulae for 0 < y < x < π/2
From (B.1)—(B.8) we can deduce many more formulae for the sine and cosine. Taking
y = x in (B.8) and using (B.3) gives the fundamental trigonometric identity
cos2 x + sin2 x = 1,
(B.9)
while taking y = x in (B.5) and (B.7) gives the double angle formulae
sin 2x = 2 sin x cos x,
cos 2x = cos2 x − sin2 x.
4
(B.10)
(B.11)
Adding and subtracting (B.9) and (B.11) back to front (and subsequently dividing
through by 2) gives
cos2 x =
sin2 x =
1
2
1
2
(1 + cos 2x),
(1 − cos 2x).
(B.12)
(B.13)
Adding (B.5) and (B.6) and adding and subtracting (B.7) and (B.8) yields
1
[sin(x + y) + sin(x − y)],
2
1
cos x cos y =
[cos(x − y) + cos(x + y)],
2
1
sin x sin y =
[cos(x − y) − cos(x + y)].
2
sin x cos y =
(B.14)
(B.15)
(B.16)
[Note the sign in (B.16). By (B.3), it is clear that (B.12) and (B.13) are the special cases
of (B.15) and (B.16) respectively with y = x. Formulae (B.12)-(B.16) can prove very
useful in integration.]
Putting x = 0 in (B.6) and (B.8), using (B.1) and (B.3) and then replacing y by x
(which is legitimate, since x and y are just labels standing for arbitrary real numbers)
gives
sin(−x) = − sin x,
cos(−x) = cos x,
(B.17)
(B.18)
or sin is an odd function (i.e. reverses sign when its variable does), while cos is an even
function (i.e. remains unchanged in value when its variable reverses sign).
Putting x = π/2 in (B.10) and (B.11) and using (B.2) and (B.4) gives
sin π = 0,
cos π = −1.
(B.19)
(B.20)
Putting y = π in (B.5) and (B.7) then gives
sin(x + π) = − sin x,
cos(x + π) = − cos x,
(B.21)
(B.22)
i.e., whenever we add π to x, we reverse the signs of both sin x and cos x (as is evident
from the graphs of these two functions). For any integer n (positive, negative or zero),
this allows us to generalize (B.1)—(B.4) to
µ
¶
1
sin nπ = cos n +
π = 0,
(B.23)
2
¶
½
¾
µ
1
1 if n is even
π =
= (−1)n .
cos nπ = sin n +
(B.24)
−1 if n is odd
2
Putting x = π/2 in (B.6) and (B.8), using (B.2) and (B.4) and then replacing y by x
gives the simple (and equivalent!) relations
´
³π
− x = cos x,
(B.25)
sin
2
´
³π
− x = sin x
(B.26)
cos
2
5
between the sine and cosine functions. Replacing x by −x in (B.21) and (B.22) and using
(B.17) and (B.18) gives
sin(π − x) = sin x,
cos(π − x) = − cos x.
B.2
(B.27)
(B.28)
Some Identities involving Other Trigonometric
Functions
Further identities may now be obtained involving the four extra trigonometric functions
³π
´
sin x
cos x
1
tan x =
, cot x =
=
= tan
− x [by (B.25) to (B.26)],
cos x
sin x
tan x
2
1
1
, cosec x =
,
cos x
sin x
of which tan x and sec x are undefined when x is an odd integer multiple of π/2 [since
this makes cos x = 0, by (B.23)], and cot x and cosec x are undefined when x is an integer
multiple of π [since this makes sin x = 0, by (B.23)]. We note here some of the more
important identities involving these functions. From (B.17) and (B.18) we deduce
sec x =
tan(−x)
cot(−x)
sec(−x)
cosec x
=
=
=
=
− tan x,
− cot x,
sec x,
−cosec x,
(B.29)
(B.30)
(B.31)
(B.32)
i.e. tan, cot and cosec x are odd functions, while sec is even. Dividing (B.9) by cos2 x or
sin2 x gives the useful formulae
1 + tan2 x = sec2 x,
1 + cot2 x = cosec2 x.
(B.33)
(B.34)
tan(x + π)= tan x,
(B.35)
Dividing (B.21) by (B.22) gives
or, as we say, tan is a periodic function with period π. For this reason, its graph, depicted
below, consists of a pattern which continually repeats itself after a horizontal distance π.
Note the vertical asymptotes occurring wherever x is equal to an odd integer multiple of
π/2.
6
The graph of tan x in the range −3π < x < 3π
Writing x + 2π as (x + π) + π and using (B.21) and (B.22), we get
sin(x + 2π) = sin x,
cos(x + 2π) = cos x,
(B.36)
(B.37)
i.e. sin and cos are periodic with period 2π. Geometrically, this reflects the fact that
2π radians = 360◦ is a complete revolution, so that x and x + 2π are essentially the
same angle. This periodicity of sin and cos is immediately evident from the form of their
graphs.
B.3
Trigonometric Functions of some Special Angles
Putting x = π/4 in (B.25) and (B.10) and using (B.4) gives sin(π/4) = cos(π/4), 1 =
2 sin2 (π/4), so
π
1
π
π
(B.38)
sin = cos = √ , tan = 1.
4
4
4
2
[NB Since π/4 is an acute angle, i.e. 0 < π/4 < π/2, we know that its sine and
cosine must both be positive.] Putting x = π/3 in (B.25) and (B.26) gives sin(π/6) =
cos(π/3),
cos(π/6) = sin(π/3). Putting x = π/6 in (B.10) then gives sin(π/3) =
2 sin(π/6) cos(π/6) = 2 sin(π/6) sin(π/3), whence use of (B.9) shows that
√
π
3
π
π
π
π
1
π √
1
sin = cos = 2 , cos = sin =
(B.39)
, tan = √ , tan = 3.
6
3
6
3
2
6
3
3
7
B.4
Evaluation of a Certain Limit
In the above diagram, the areas of triangles OAB and OAC are (sin x)/2 and (tan x)/2
respectively (each being equal to half the base times the vertical height). On the other
hand, the area of the sector OAB of the circle centre O radius 1 is a fraction x/(2π) of
the total area π of that circle, i.e. it is equal to x/2. Comparing these areas and doubling
through, we see immediately that
sin x < x < tan x
for 0 < x <
π
.
2
Dividing through by the positive quantity sin x gives
1<
x
< sec x
sin x
for 0 < x <
π
,
2
and “turning this upside down” yields
cos x <
sin x
<1
x
for 0 < x <
π
π
, and hence also for − < x < 0,
2
2
since cos x and sin x are respectively even and odd functions of x. Letting x tend to zero
(from either side), since cos x → cos 0 = 1, it follows that
sin x
= 1.
x→0 x
lim
i.e. sin x ≈ x for small x.
8