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Question Number 48851 by vajpaithegrate@gmail.com last updated on 29/Nov/18

the total no of solution of \sin {x} = \cos {x}

where {x} denotes fractional part of x in

[o 2 π] is equal to

ans : 6

Commented by tanmay.chaudhury50@gmail.com last updated on 29/Nov/18

s i n {x} = c o s {x}

s i n (x - [x]) = c o s (x - [x])

x = {x} + [x]

{x} = f r a c t i o n a l p a r t f u n c t i o n

[x] = g r e a t e s t i n t e g e r f u n c t i o n

f r o m g r a p h i t i s s e e n t h a t i n [0, 2 π]

s i x (06) s o l u t i o n

Commented by tanmay.chaudhury50@gmail.com last updated on 29/Nov/18

Commented by tanmay.chaudhury50@gmail.com last updated on 29/Nov/18

g r a p h o f s i n {x}

s i n (x - [x]

Commented by tanmay.chaudhury50@gmail.com last updated on 29/Nov/18

Commented by tanmay.chaudhury50@gmail.com last updated on 29/Nov/18

g r a p h o f c o s {x}

g r a p h o f c o s (x - [x])

Commented by tanmay.chaudhury50@gmail.com last updated on 29/Nov/18

Commented by tanmay.chaudhury50@gmail.com last updated on 29/Nov/18

2 π = 2 \times 3.14 = 6.28

i n g r a p h i t i s s e e n t h a t f r o m x = 0 t o x = 6

t h e r e a r e s i x p o i n t w h e r e s i n {x} = c o s {x}

i n t e r s e c t s . s o s i x s o l u t i o n i n x (0, 2 π)

Commented by vajpaithegrate@gmail.com last updated on 29/Nov/18

tnq sir

Commented by vajpaithegrate@gmail.com last updated on 29/Nov/18

any alternate method,

Answered by mr W last updated on 29/Nov/18

l e t t = {x}

0 ⩽ t < 1

\sin t = \cos t

\Rightarrow \tan t = 1

\Rightarrow t = π / 4 = 0.785

\Rightarrow s o l u t i o n i s x = n + t w i t h n = 0, 1, 2, \dots .

i . e . x = 0.785, 1.785, 2.785, \dots, 5.785, 6.785, \dots \dots

2 π = 6.283 < 6.785

\Rightarrow 6 s o l u t i o n s w i t h i n [0, 2 π] :

x = 0.785, 1.785, 2.785, \dots, 5.785

Commented by vajpaithegrate@gmail.com last updated on 29/Nov/18

tnq sir

Commented by tanmay.chaudhury50@gmail.com last updated on 29/Nov/18

e x c e l l e n t s i r \dots