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 trigonome
Trang 12.2 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
1radian = 180◦
π ; 1◦ = π
180. 2.2.1-2 Definition of trigonometric functions
The sine of α is the ordinate (the projection to the axis Oy) of the point on the trigonometric circle corresponding to the angle of α radians The cosine of α is the abscissa (projection
to the axis Ox) of that point (see Fig 2.5) The sine and the cosine are basic trigonometric functions and are denoted, respectively, by sin α and cos α.
Other trigonometric functions are tangent, cotangent, secant, and cosecant These are
derived from the basic trigonometric functions, sine and cosine, as follows:
tan α = sin α
cos α, cot α =
cos α sin α, sec α =
1
cos α, cosec α =
1
sin α.
Table 2.1 gives the signs of the trigonometric functions in different quadrants The
signs and the values of sin α and cos α do not change if the argument α is incremented by
2πn , where n =1,2, The signs and the values of tan α and cot α do not change if the argument α is incremented by πn, where n =1, 2,
Table 2.2 gives the values of trigonometric functions for some values of their argument (the symbol ∞ means that the function is undefined for the corresponding value of its
argument)
Trang 2TABLE 2.1 Signs of trigonometric functions in different quarters
TABLE 2.2
Numerical values of trigonometric functions for some angles α (in radians)
6 π4 π3 π2 23π 34π 56π π
2
√
2 2
√
3
2
√
2
2
√
2
2 12 0 –12 –√22 –√23 – 1
3 0 –√33 – 1 –√
2.2.2 Graphs of Trigonometric Functions
2.2.2-1 Sine: y = sin x.
This function is defined for all x and its range is y [–1,1] The sine is an odd, bounded, periodic function (with period2π ) 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 [–π2 +2πn,π2 +2πn] and is a decreasing function on every segment [π2 +2πn,32π+2πn ] For x = π2 +2πn , it attains its maximal value (y = 1), and for
x= –π2 +2πn it attains its minimal value (y = –1) The graph of the function y = sin x is called the sinusoid or sine curve and is shown in Fig 2.6.
O
1
π π
x y
π
2 1
y= sinx
π
2
Figure 2.6 The graph of the function y = sin x.
2.2.2-2 Cosine: y = cos x.
This function is defined for all x and its range is y [–1,1] The cosine is a bounded, even, periodic function (with period 2π ) It crosses the axis Oy at the point y = 1, and
Trang 3crosses the axis Ox at the points x = π2 + 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 = cos x is a sinusoid obtained by shifting the graph of the function y = sin x by π2 to the left along the axis Ox
(see Fig 2.7)
O
1
π x y
π
1
y= cosx
π
2
π
2
Figure 2.7 The graph of the function y = cos x.
2.2.2-3 Tangent: y = tan x.
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 =0and crosses the axis Ox at the points x = πn This is an increasing
function on every interval (–π2 + πn, π2 + πn) This function has no points of extremum and has vertical asymptotes at x = π2 + πn, n =0, 1, 2, The graph of the function y = tan x
is given in Fig 2.8
2.2.2-4 Cotangent: y = cot x.
This function is defined for all x≠πn , n =0, 1, 2, , and its range is the entire y-axis The cotangent is an unbounded, odd, periodic function (with period π) It crosses the axis
Ox at the points x = π2+πn, and does not cross the axis Oy This is a decreasing function on every interval (πn, π +πn) This function has no extremal points and has vertical asymptotes
at x = πn, n =0, 1, 2, The graph of the function y = cot x is given in Fig 2.9.
O
1
x y
1
y=tanx
π
2
π
2
Figure 2.8 The graph of the function y = tan x.
O
1
x y
1
y=cotx
π
2
π
2 3π2
Figure 2.9 The graph of the function y = cot x.
Trang 42.2.3 Properties of Trigonometric Functions
2.2.3-1 Simplest relations
sin2x+ cos2x=1, tan x cot x =1,
sin(–x) = – sin x, cos(–x) = cos x, tan x = sin x
cos x sin x, tan(–x) = – tan x, cot(–x) = – cot x,
1+ tan2x= 1
cos2x, 1+ cot2x= 1
sin2x
2.2.3-2 Reduction formulas
sin(x 2nπ ) = sin x,
sin(x nπ) = (–1)n sin x,
sin
x 2n+1
= (–1)n cos x,
sin
x π
4
=
√
2
2 (sin x cos x),
tan(x nπ ) = tan x,
tan
x 2n+1
= – cot x,
tan
x π
4
= tan x 1
1tan x,
cos(x 2nπ ) = cos x, cos(x nπ) = (–1)n cos x,
cos
x 2n+1
=(–1)n sin x,
cos
x π
4
=
√
2
2 (cos xsin x), cot(x nπ ) = cot x,
cot
x 2n+1
= – tan x,
cot
x π
4
= cot x1
1 cot x, where n =1, 2,
2.2.3-3 Relations between trigonometric functions of single argument
sin x = √
1– cos2x= √ tan x
1+ tan2x = 1
√
1+ cot2x,
cos x = √
1– sin2x= √ 1
1+ tan2x = √ cot x
1+ cot2x,
tan x = √ sin x
1– sin2x =
√
1– cos2x
1
cot x, cot x =
√
1– sin2x
cos x
√
1– cos2x = 1
tan x.
The sign before the radical is determined by the quarter in which the argument takes its values
Trang 52.2.3-4 Addition and subtraction of trigonometric functions.
sin x + sin y =2sin
x + y 2
cos
x – y 2
,
sin x – sin y =2sinx – y
2
cosx + y 2
,
cos x + cos y =2cosx + y
2
cosx – y 2
,
cos x – cos y = –2sin
x + y 2
sin
x – y 2
, sin2x– sin2y= cos2y– cos2x = sin(x + y) sin(x – y),
sin2x– cos2y = – cos(x + y) cos(x – y), tan x tan y = sin(x y)
cos x cos y, cot x cot y =
sin(y x)
sin x sin y,
a cos x + b sin x = r sin(x + ϕ) = r cos(x – ψ).
Here, r = √
a2+ b2, sin ϕ = a/r, cos ϕ = b/r, sin ψ = b/r, and cos ψ = a/r.
2.2.3-5 Products of trigonometric functions
sin x sin y = 12[cos(x – y) – cos(x + y)], cos x cos y = 12[cos(x – y) + cos(x + y)], sin x cos y = 12[sin(x – y) + sin(x + y)].
2.2.3-6 Powers of trigonometric functions
cos2x= 12cos2x+ 12,
cos3x= 14cos3x+ 34cos x,
cos4x= 18cos4x+ 12cos2x+ 38,
cos5x= 161 cos5x+ 165 cos3x+ 58 cos x,
sin2x= –12 cos2x+ 12, sin3x= –14 sin3x+ 34sin x,
sin4x= 18cos4x– 12 cos2x+ 38, sin5x= 161 sin5x– 165 sin3x+ 58sin x,
cos2n x= 1
22n–1
n–1
k=0
C k
2ncos[2(n – k)x] + 1
22n C2n n,
cos2n+1x= 1
22n
n
k=0
C k
2n+1cos[(2n–2k+1)x],
sin2n x= 1
22n–1
n–1
k=0
(–1)n–k C k
2ncos[2(n – k)x] + 1
22n C2n n,
sin2n+1x= 1
22n
n
k=0
(–1)n–k C k
2n+1sin[(2n–2k+1)x].
Here, n =1, 2, and C m k = m!
k ! (m – k)! are binomial coefficients (0! =1)
Trang 62.2.3-7 Addition formulas.
sin(x y ) = sin x cos y cos x sin y,
tan(x y) = tan x tan y
1tan x tan y,
cos(x y ) = cos x cos ysin x sin y, cot(x y) = 1tan x tan y
tan x tan y .
2.2.3-8 Trigonometric functions of multiple arguments
cos 2x= 2 cos2x– 1 = 1 – 2 sin2x,
cos 3x= – 3cos x +4 cos3x,
cos 4x= 1 – 8 cos2x+ 8 cos4x,
cos 5x = 5cos x –20 cos3x+ 16 cos5x,
sin 2x = 2sin x cos x,
sin 3x = 3sin x –4 sin3x, sin 4x = 4cos x (sin x –2 sin3x), sin 5x = 5sin x –20 sin3x+ 16 sin5x, cos( 2nx) = 1 +
n
k=1
(– 1 )k n
2(n2– 1) [n2– (k –1 )2]
( 2k )! 4ksin2k x, cos[( 2n + 1)x] = cos x
1 +
n
k=1
(– 1 )k[(2n+ 1 )2– 1 ][( 2n+ 1 )2– 3 2] [(2n+ 1 )2–( 2k– 1 )2]
2k x ,
sin( 2nx ) = 2ncos x
sin x +
n
k=1 (– 4 )k (n
2 – 1)(n2– 2 2) (n2– k2) ( 2k – 1 )! sin
2k–1x ,
sin[( 2n + 1)x] = (2n + 1 )
sin x+
n
k=1
(– 1 )k[(2n + 1 )2– 1 ][( 2n + 1 )2– 3 2] [(2n + 1 )2–( 2k – 1 )2]
2k+1x ,
tan 2x = 2tan x
1 – tan2x, tan 3x = 3tan x – tan3x
1 – 3 tan2x , tan 4x = 4tan x –4 tan3x
1 – 6 tan2x+ tan4x,
where n =1, 2,
2.2.3-9 Trigonometric functions of half argument
sin2 x
2 =
1– cos x
2 , cos2
x
2 =
1+ cos x
tanx
2 =
sin x
1+ cos x =
1– cos x sin x , cot
x
2 =
sin x
1– cos x =
1+ cos x sin x , sin x = 2tanx2
1+ tan2 x2 , cos x =
1– tan2 x2
1+ tan2 x2 , tan x =
2tanx2
1– tan2x2 . 2.2.3-10 Differentiation formulas
d sin x
dx = cos x, d cos x
dx = – sin x, d tan x
dx = 1
cos2x, d cot x
dx = – 1
sin2x
Trang 72.2.3-11 Integration formulas.
sin x dx = – cos x + C,
cos x dx = sin x + C,
tan x dx = – ln|cos x|+ C,
cot x dx = ln|sin x|+ C, where C is an arbitrary constant.
2.2.3-12 Expansion in power series
cos x =1– x
2
2! +
x4
4! –
x6
6! +· · · + (–1)n x
2n
(2n)! +· · · (|x|<∞),
sin x = x – x
3
3! +
x5
5! –
x7
7! +· · · + (–1)n x
2n+1
(2n+1)! +· · · (|x|<∞),
tan x = x + x3
3 +
2x5
15 +
17x7
315 +· · · +
22n(22n–1)|B2n| (2n)! x
2n–1+· · · (|x|< π/2),
cot x = 1
x –
x
3 +
x3
45 +
2x5
945 +· · · +
22n|B2n| (2n)! x
2n–1+· · ·
(0<|x|< π), where B nare Bernoulli numbers (see Subsection 18.1.3)
2.2.3-13 Representation in the form of infinite products
sin x = x
1– x
2
π2
1– x
2
4π2
1– x
2
9π2
.
1– x
2
n2π2
.
cos x =
1– 4x2
π2
1– 4x2
9π2
1– 4x2
25π2
.
1– 4x2
(2n+1)2π2
.
2.2.3-14 Euler and de Moivre formulas Relationship with hyperbolic functions
e y+ix = e y (cos x + i sin x), (cos x + i sin x) n = cos(nx) + i sin(nx), i2= –1,
sin(ix) = i sinh x, cos(ix) = cosh x, tan(ix) = i tanh x, cot(ix) = –i coth x.
2.3 Inverse Trigonometric Functions
2.3.1 Definitions Graphs of Inverse Trigonometric Functions
2.3.1-1 Definitions of inverse trigonometric functions
Inverse trigonometric functions (arc functions) are the functions that are inverse to the trigonometric functions Since the trigonometric functions sin x, cos x, tan x, cot x are periodic, the corresponding inverse functions, denoted by Arcsin x, Arccos x, Arctan x,