How to find the solution of this trigonometric equation cot ⁡ ( x + 110...

dream13rxs

dream13rxs

Answered

2022-07-08

How to find the solution of this trigonometric equation
cot ( x + 110 ) = cot ( x + 60 ) cot x cot ( x 60 )
I have used the formulae
cos ( x + 60 ) cos ( x 60 ) = cos 2 60 sin 2 x
sin ( x + 60 ) sin ( x 60 ) = sin 2 x sin 2 60
How to move further? What is the least positive value of x?

Answer & Explanation

Brendan Bush

Brendan Bush

Expert

2022-07-09Added 14 answers

cot ( x + 110 ) = cot ( x + 60 ) cot x cot ( x 60 )
cot ( x + 110 ) cot x = cot ( x + 60 ) cot ( x 60 )
cos ( x + 110 ) sin x sin ( x + 110 ) cos x = cos ( x + 60 ) cos ( x 60 ) sin ( x + 60 ) sin ( x 60 )
Now Using Componendo and Dividendo, We get
cos ( x + 110 ) sin x + sin ( x + 110 ) cos x cos ( x + 110 ) sin x sin ( x + 110 ) cos x = cos ( x + 60 ) cos ( x 60 ) + sin ( x + 60 ) sin ( x 60 ) cos ( x + 60 ) cos ( x 60 ) sin ( x + 60 ) sin ( x 60 )
sin [ ( x + 110 ) + x ] sin [ ( x + 110 ) x ] = cos [ ( x + 60 ) ( x 60 ) ] cos [ ( x + 60 ) + ( x 60 ) ]
So we get
sin ( 2 x + 110 ) sin ( 110 ) = cos ( 120 ) cos ( 2 x )
sin ( 2 x + 110 ) cos ( 2 x ) = 1 2 sin ( 110 0 )
So we get
2 sin ( 2 x + 110 ) cos ( 2 x ) = sin ( 110 0 )
sin ( 4 x + 110 ) + sin ( 110 ) = sin ( 110 )
So we get
sin ( 4 x + 110 ) = 0 4 x + 110 0 = n × 180 ,
Where n Z

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