Download Trigonometric functions of the oriented angle

Trigonometry
( Lenka Šturmankinová, Erika Veselová )
Oriented angle
- an ordered quadruple: [ initial side, terminal side, orientation, size ]
Orientation: + ...counterclockwise
- ...clockwise

 
XVY = [VX, VY, +, 60]
Arch measure
Definition: The angle  has measure 1 rad, if the length of the arch defined by  on the unit
circle is 1 ( = radius of the unit circle ).
Observation: || = 1 rad  length of the arch = radius of the circle.
Measure of  [rad] = length of the arch on the unit circle.
Oriented angle in the coordinate system
M
M


Mapping of the set R of all real numbers onto the unit circle ( assignment of points M of the
unit circle to real numbers  ):
- initial side of the oriented angle  R ... x+

M  unit circle  terminal side of the oriented angle 
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Trigonometric functions
Definition: Let   R, let M [xM, yM] be the point on the unit circle assigned to the
oriented angle φ. Then: sin  = yM
cos  = xM
sin 
tg  =
cos 
cos 
cotg  =
sin 
M
yM

xM
Observations :
1. For 0 < φ < π/2 ( acute angles φ ) values: sinφ , cosφ, tgφ , cotgφ according to this
definition coincide with the values in right triangle ( sin  = a/c, cos  = b/c, tg  = a/b,
cotg  = b/a ).
y
sin φ = M
1
xM
cos φ =
1
sin  y M
tg φ =

cos  xM
cos  xM
cotg φ =

sin  y M
2. D (sin) = R = D (cos)
D (tg) = R – { φ  R ; cos φ = 0 }
D (cotg) = R – { φ  R ; sin φ = 0 }

sin

2  1 , tg   , cotg 0, cotg  ...analogically  undefined )
 0
2
2
cos
2
3. Geometrical interpretation of functions sin, cos: by the definition – the coordinates of Mφ.
Geometrical interpretation of functions tg, cotg:
yM
M A
tg φ =

 M A
xM
0A
( Thus: tg
cotg φ =
xM
yM


M B
0B
 MB
2
Geometrical interpretation: tg φ1 = the coordinate of the intercept M‘ of the terminal side
of φ1 with the real axis R‘ for tangent.
M’
M1
yM
M2
2
1
tg 1
A
xM
R’
tg 2
cotg φ = the coordinate of the intercept of the terminal side of φ
with the real axis R‘‘ for cotangent
R’’
-1
1
0
xM
4. H (sin) = -1,1 = H (cos), H (tg) = R = H (cotg).
5. Periodicity of sin, cos:
 φ є R : sin (φ+2  ) = sin φ = sin (φ-2  )
cos (φ+2  ) = cos φ = cos (φ-2  )

 
sin
= 1, sin    = -1
2
 2
cos 0 = 1, cos  = -1
sin(/2) = 1, but there exists no   (/2; /2+2) such that sin =1 ( for cos
analogically ).
That proves that 2  is the minimal length of the interval of repetition  sin, cos are
periodic with the period 2  .
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M = M+2

 + 2
6. Periodicity of tg, cotg :
 φ  R : tg   2  
sin   2  sin 

 tg , but a = 2 is not the least length of the
cos  2  cos 
repetition interval:
sin      sin  sin 
tg ( + ) =
= tg ,


cos     cos  cos 
cos   
cotg ( + ) =
= cotg 
sin    
tg, cotg are periodic with the period  ( follows immediately from the geom. interpretation ).
„Basic period“ of tg is the interval (-/2, /2).
„Basic period“ of cotg is the interval (0, ).
M
+

7. sin (-φ) = - sin φ
... ODD FUNCTION
cos (-φ) = cos φ
... EVEN FUNCTION
sin     sin 
...ODD FUNCTION
tg    

 tg
cos   cos 
cos   cos 
cotg (-φ) =

  cotg φ ...ODD FUNCTION
sin     sin 
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8. tg φ . cotg φ=
sin  cos 

1
cos  sin 
1 = |sin |2+ |cos |2  PYTHAGOREAN FORMULA ( RULE ): 1 = sin2  + cos2 
( 1= (sin )2 + (cos )2 )
1

sin 
cos 
Functions inverse to trigonometric functions
Observation: Trigonometric functions are periodic  trigonometric functions are not 1-1 
 trigonometric functions have not inverse functions.
Notation: Let us denote by f1, f2, f3, f4 the following functions:
f1: y = sin x for x  -/2;/2
f2: y = cos x for x   0; 
f3: y = tg x for x  (-/2;/2)
f4: y = cotg x for x  ( 0; )
Observation: f1, f2, f3, f4 are 1-1  f1, f2, f3, f4 have their inverse functions f1-1, f2-1, f3-1, f4-1and
D(f1) = -/2;/2 = H(f1-1), H(f1) = -1;1 = D(f1-1),
D(f2) =  0;  = H(f2-1), H(f2) = -1;1 = D(f2-1),
D(f3) = (-/2;/2) = H(f3-1), H(f3) = R = D(f3-1),
D(f4) = ( 0; ) = H(f4-1), H(f4) = R = D(f4-1).
Definition: Let f1, f2, f3, f4 be the functions according to the former notation. The functions
arcsin, arccos, arctg, arccotg are defined by the formulas
arcsin = f1-1: y = arcsin x = f1-1(x)
arccos = f2-1: y = arccos x = f2-1(x)
arctg = f3-1: y = arctg x = f3-1(x)
arccotg = f4-1: y = arccotg x = f4-1(x).
Consequence: y = arcsin x  ( x = sin y  y -/2;/2 )
y = arccos x  ( x = cos y  y  0;  )
y = arctg x  ( x = tg y  y (-/2;/2) )
y = arccotg x  ( x = cotg y  y ( 0; ) )
D(arcsin) = -1;1 = H(sin), H(arcsin) = -/2;/2
D(arccos) = -1;1 = H(cos), H(arccos) =  0; 
D(arctg) = R = H(tg), H(arctg) = (-/2;/2)
D(arccotg) = R = H(cotg), H(arccotg) = ( 0; ).
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Sine, cosine rules
Theorem: For each triangle ABC with the sides a, b, c, corresponding angles , ,  and the
circumradius r the following formulas hold:
a/sin = b/sin = c/sin =2r ……………………………………………………..…... sine rule
c2 = a2 + b2 – 2abcos, a2 = b2 + c2 – 2bccos, b2 = a2 + c2 – 2accos ……...… cosine rule
Area of a triangle
Theorem: For the area S of each triangle ABC with the sides a, b, c, corresponding angles
, , , the circumradius r, the radius of the circle inscribed  and the perimeter a+b+c=2s
the following formulas hold:
S = (absin) / 2 = (c2sinsin) / (2sin) = abc / 4r = s,
S2 = s(s-a)(s-b)(s-c) ( Heron’s formula ).
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[ cos  ; sin  ]
REMEMBER FOREVER !
2
= the side of the ( smaller ) square with the diagonal of length 1
2
1
= one half of the side of the isosceles triangle with the side of length 1
2
3
= the altitude of the isosceles triangle with the side of length 1
2
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Trigonometric identities:


sin   cos   
2



cos   sin    
2

sin  x  y   sin x  cos y  cos x  sin y
sin x  y   sin x  cos y  cos x  sin y 
cos x  y   cos x  cos y  sin x  sin y
cosx  y   cos x  cos y  sin x  sin y 
tan x  tan y
1  tan x  tan y
1  tan x  tan y
cot  x  y  
tan x  tan y
tan  x  y  
sin 2 x  2  sin x  cos x
cos 2 x  cos 2 x  sin 2 x
2  tan x
1  tan 2 x
1  tan 2 x
cot 2 x 
2  tan x
tan 2 x 
sin
x
1  cos x

2
2
cos
x
1  cos x

2
2
tan
x
1  cos x

2
1  cos x
cot
x
1  cos x

2
1  cos x
x y
x y
 cos
2
2
x y
x y
sin x  sin y  2  cos
 sin
2
2
x y
x y
cos x  cos y  2  cos
 cos
2
2
x y
x y
cos x  cos y  2  sin
 sin
2
2
sin x  sin y  2  sin
For technical reasons ( properties of the editor of equations ) the notation “tan” and “cot” is used for tg and cotg.
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