LOGARITHM UNEQUATIONS WITH TRIGONOMETRY

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Summary: The aim of this project is defining the way of solving and the field of

definition of the logarithm unequations combined with trigonometry. The methods of the

research are collecting assignments and literature as well as their interpretation and carrying out.

It is established that tha assignments of the logarithm unequations with trigonometry can be in

the form of comparison and proving or calculation of value of the unknown. They can be

combined with inequalities which are valid for middle points. With these unequations the field of

definition is achieved as bisection of definition of the logarithm and trigonometric function. The

solving of the unequation can be continued in several ways, some of which are shown in this

project.

Key words: inequality, unequation, function, logarithms, trigonometry

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(c).

a) f(x) f(x)=f(-x), f(x)= -f(-x).

b) x-.

c) A f(x+p)=f(x), f

. [3.]

b

(>0) . [3.]

log a x

log a x.

x=y.

() :

y=log a x ay=x [3.]

:

a>1

00 1

2. f(x) f(-x)

3. y=log a x=0 => x=a0

x=1

4.

5. : : x 0 y -, : x 0 y +

6. : - 0

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1 .

(1,0)

( 3). .

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(xo, yo) , :

sin = yo

cos = xo [2.]

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tg=sin /cos ; ctg=cos/sin =1/tg

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. [1.]

sin x cos x 2:

cos x= cos (x + 2)= cos (x + 2k) k-

tg x

cos x

sin x

ctg x

M

O A

3-

Graphic 3-Oriented angle and its sinus and

cosinus

yo

xo

sin x = sin (x + 2)= sin (x + 2k)

tg x ctg x ,

( + ) . tg ctg ( ):

tg x= tg (x + ) = tg (x + k) k-

ctg x= ctg (x + ) = ctg (x + k)

:

:

- !

1:

y- - .

2: x-

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:

- x.

sin (-x)= -sin x

cos (-x)= cos x

tg (-x)= -tg x

ctg (-x)= -ctg x

I II III IV

sin x + + - -

cos x + - - +

tg x + - + -

ctg x + - + -

-

+

-

sin x

cos x

+

+

- -

I II

III

IV

3-

Graphic 3-

1-

Table 1-Signs of the trigonometric functions in quadrants

4-

Graphic 4-Positive and negative angle

= .

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sin x cos x tg x ctg x

x x cos x 0 ; x /2+k sin x 0 ; x k

y=[-1,1] y=[-1,1]

a x=

/2+

a x=

k

x= 0+k x=/2+k x= 0+k x=/2+k

y>0, 0+2k 0,-/2+2k 0,0+k 0,0+k

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1. a>1

log a f(x)>log a g(x) (1)

f(x)>g(x)

2. 0

2)

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1)

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(1) (2), .

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sin 1< log 371/2

37< 7

4 ( )

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=c*sin

b=c*cos

1/2 log 1/2 (ab) log 1/2 (a+b) log 1/2 2

log1/2 (ab)1/2

log 1/2 (a+b)/2

(ab)1/2

(a+b)/2

A o ,

b,

=b.

- .

log tgx sinx-log ctgx cosx 3

:

I :

log tgx sinx +log tgx cosx 3

log tgx sinx*cosx 3

1o tgx>1 /4+2k

: sinx2 + sinx -1 0

:

arcsin +2k x< /4 +2k (*)

II :

1/log sinx tgx- 1/log cosx ctg x 3

1/(log sinx sinx-log sinx cosx)-1/(log cosx cosx-log cosx sinx) 3

t=log cosx sinx

1/(1-1/t)-1/(1-t) 3 , t/(t-1)+1/(t-1) 3 , (t+1)/(t-1) 3

(2-t)/(t-1) 0

- 1 2 +

2-t + + -

t-1 - + +

- + -

1sinx cosx2 sinx

cosx-cos(/2-x)>0 sinx2 + sinx -1 0

-2sin/2*sin(x-/4)>0

x- /4< 0+k

x

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[1.] , LAROUSSE, , , 1973.

[2.]. , ., . , . ,

, , , 1993.

[3.]. , ,

Ke, , 2004.