24 ELEMENTARY FUNCTIONS
2.2. Trigonometric Functions
2.2.1. Trigonometric Circle. Definition of Trigonometric Functions
2.2.1-1. Trigonometric circle. Degrees and radians.
Trigonometric circle is the circle of unit radius with center at the origin of an orthogonal
coordinate system Oxy. The coordinate axes divide the circle into four quarters (quadrants);
see Fig. 2.5. Consider rotation of the polar radius issuing from the origin O and ending
at a point M of the trigonometric circle. Let α be the angle between the x-axis and the
polar radius OM measured from the positive direction of the x-axis. This angle is assumed
positive in the case of counterclockwise rotation and negative in the case of clockwise
rotation.
O
M
1
1
x
α
y
1
1
Figure 2.5. Trigonometric circle.
Angles are measured either in radians or in degrees. One radian is the angle at the vertex
of the sector of the trigonometric circle supported by its arc of unit length. One degree is
the angle at the vertex of the sector of the trigonometric circle supported by its arc of length
π/180. The radians are related to the degrees by the formulas
1 radian =
180
◦
π
; 1
◦

argument).
2.2. TRIGONOMETRIC FUNCTIONS 25
TABLE 2.1
Signs of trigonometric functions in different quarters
Quarter
Angle in radians sin α cos α
tan α cot α
sec α cosec α
I
0 <α<
π
2
+ + + + + +
II
π
2
<α < π
+
– – – –
+
III
π <α <
3π
2
– –
+ +
– –
IV
3π
2

2
2
√
3
2
1
√
3
2
√
2
2
1
2
0
cos α
1
√
3
2
√
2
2
1
2
0
–
1
2
–

√
3
3
0 –
√
3
3
–1
–
√
3
∞
2.2.2. Graphs of Trigonometric Functions
2.2.2-1. Sine: y =sinx.
This function is defined for all x and its range is y [–1, 1]. The sine is an odd, bounded,
periodic function (with period 2π). It crosses the axis Oy at the point y = 0 and crosses
the axis Ox at the points x = πn, n = 0,
1, 2, The sine is an increasing function
on every segment [–
π
2
+ 2πn,
π
2
+ 2πn] and is a decreasing function on every segment
[
π
2
+ 2πn,
3

+ πn. The cosine is an increasing function on every
segment [–π + 2πn, 2πn] and is a decreasing function on every segment [2πn, π + 2πn],
n = 0,
1, 2, For x = 2πn it attains its maximal value (y = 1), and for x = π + 2πn
it attains its minimal value (y =–1). The graph of the function y =cosx is a sinusoid
obtained by shifting the graph of the function y =sinx by
π
2
to the left along the axis Ox
(see Fig. 2.7).
O
1
π
x
y
π
1
yx= cos
π
2
π
2
Figure 2.7. The graph of the function y =cosx.
2.2.2-3. Tangent: y =tanx.
This function is defined for all x ≠
π
2
+ πn, n = 0, 1, 2, , and its range is the entire
y-axis. The tangent is an unbounded, odd, periodic function (with period π). It crosses the
axis Oy at the point y = 0 and crosses the axis Ox at the points x = πn.Thisisanincreasing

π
2
π
2
Figure 2.8. The graph of the function y =tanx.
O
1
x
y
π
π
1
yx= cot
π
2
π
2
3π
2
Figure 2.9. The graph of the function y =cotx.
2.2. TRIGONOMETRIC FUNCTIONS 27
2.2.3. Properties of Trigonometric Functions
2.2.3-1. Simplest relations.
sin
2
x +cos
2
x = 1,tanx cot x = 1,
sin(–x)=–sinx,cos(–x)=cosx,
tan x =

x
2n + 1
2
π

= (–1)
n
cos x,
sin

x
π
4

=
√
2
2
(sin x cos x),
tan(x
nπ)=tanx,
tan

x
2n + 1
2
π

=–cotx,
tan

4

=
√
2
2
(cos x sin x),
cot(x
nπ)=cotx,
cot

x
2n + 1
2
π

=–tanx,
cot

x
π
4

=
cot x
1
1 cot x
,
where n = 1, 2,
2.2.3-3. Relations between trigonometric functions of single argument.

1 +cot
2
x
,
tan x =
sin x
√
1 –sin
2
x
=
√
1 –cos
2
x
cos x
=
1
cot x
,
cot x =
√
1 –sin
2
x
sin x
=
cos x
√
1 –cos


,
cos x +cosy = 2 cos

x + y
2

cos

x – y
2

,
cos x –cosy =–2 sin

x + y
2

sin

x – y
2

,
sin
2
x –sin
2
y =cos
2

1
2
[cos(x – y)+cos(x + y)],
sin x cos y =
1
2
[sin(x – y)+sin(x + y)].
2.2.3-6. Powers of trigonometric functions.
cos
2
x =
1
2
cos 2x +
1
2
,
cos
3
x =
1
4
cos 3x +
3
4
cos x,
cos
4
x =
1

3
x =–
1
4
sin 3x +
3
4
sin x,
sin
4
x =
1
8
cos 4x –
1
2
cos 2x +
3
8
,
sin
5
x =
1
16
sin 5x –
5
16
sin 3x +
5

k=0
C
k
2n+1
cos[(2n – 2k + 1)x],
sin
2n
x =
1
2
2n–1
n–1

k=0
(–1)
n–k
C
k
2n
cos[2(n – k)x]+
1
2
2n
C
n
2n
,
sin
2n+1
x =

y)=
1 tan x tan y
tan x tan y
.
2.2.3-8. Trigonometric functions of multiple arguments.
cos 2x = 2 cos
2
x – 1 = 1 – 2 sin
2
x,
cos 3x =–3 cos x + 4 cos
3
x,
cos 4x = 1 – 8 cos
2
x + 8 cos
4
x,
cos 5x = 5 cosx – 20 cos
3
x + 16 cos
5
x,
sin 2x = 2 sin x cosx,
sin 3x = 3 sin x – 4 sin
3
x,
sin 4x = 4 cos x (sin x – 2 sin
3
x),


k=1
(–1)
k
[(2n+1)
2
–1][(2n+1)
2
–3
2
] [(2n+1)
2
–(2k–1)
2
]
(2k)!
sin
2k
x

,
sin(2nx)=2n cos x

sin x +
n

k=1
(–4)
k
(n

] [(2n+1)
2
–(2k–1)
2
]
(2k+1)!
sin
2k+1
x

,
tan 2x =
2 tan x
1 –tan
2
x
,tan3x =
3 tan x –tan
3
x
1 – 3 tan
2
x
,tan4x =
4 tan x – 4 tan
3
x
1 – 6 tan
2
x +tan

x
2
=
sin x
1 –cosx
=
1 +cosx
sin x
,
sin x =
2 tan
x
2
1 +tan
2
x
2
,cosx =
1 –tan
2
x
2
1 +tan
2
x
2
,tanx =
2 tan
x
2


sin xdx=–cosx + C,

cos xdx=sinx + C,

tan xdx=–ln| cos x| + C,

cot xdx =ln| sin x| + C,
where C is an arbitrary constant.
2.2.3-12. Expansion in power series.
cos x = 1 –
x
2
2!
+
x
4
4!
–
x
6
6!
+ ···+(–1)
n
x
2n
(2n)!
+ ··· (|x| < ∞),
sin x = x –
x

2n
(2
2n
– 1)|B
2n
|
(2n)!
x
2n–1
+ ··· (|x| < π/2),
cot x =
1
x
–

x
3
+
x
3
45
+
2x
5
945
+ ···+
2
2n
|B
2n




1 –
x
2
n
2
π
2


cos x =

1 –
4x
2
π
2

1 –
4x
2
9π
2

1 –
4x
2
25π