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Basic Mathematics – I

Notes To ﬁnd the general solution of the equation sin = 0

It is given that sin = 0

But we know that sin 0, sin , sin 2 , ...., sin n are equal to 0
= n , n N
But we know that sin( ) = sin = 0
sin( ) , sin( 2 ), sin( 3 ) ,...., sin( n ) = 0

= n , n I.
Thus, the general solution of equations of the type sin = 0 is given by = n where is an
integer.

To ﬁnd the general solution of the equation cos = 0

It is given that cos = 0
But in practice we know that cos /2 = 0. Therefore, the ﬁrst value of is

= /2 …(1)
We know that cos ( + ) = cos or cos( + /2) = cos /2 = 0.
or cos 3 /2 = 0

In the same way, it can be found that 9
cos 5 /2, cos 7 /2,cos 9 /2,....., cos( 2n + 1) /2 are all zero
( 2n + 1) /2, n N
But we know that cos( ) = cos

cos( /2) = cos( 3 /2) = cos( 5 /2) = cos{ (2n 1) /2} = 0
= (2n + 1) /2, n I
Therefore, = (2n + 1) /2 is the solution of equations cos = 0 for all numbers whose cosine is
0.

To ﬁnd a general solution of the equation tan = 0
It is given that tan =0

or sin /cos = 0 or sin = 0
i.e. = n , n I.
We have consider above the general solution of trigonometric equations, where the right hand is
zero. In the following equation, we take up some cases where right hand side is non–zero.

To ﬁnd the general solution of the equation sin = sin

It is given that sin = sin

sin sin = 0
or 2cos ( + /2) sin( /2) = 0
Either cos ( + /2) = 0 or sin( /2) = 0

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