Question

cot (4π-θ) formula of this
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Answer 1

Answer:

I am assuming that you want value of cot(4π-theta) in terms of theta.

For this, I would like to say a few facts to make this clear to you.

Fact 1:

If an angle is rotated whether clockwise or anticlockwise by an angle which is multiple of π, then the ratios don't change means sin remains sin, cos remains cos, tan remains tan and so on.

thus as soon as we come across something like sin(6π - A), it should come to our mind that it will remain sin, whatever be the polarity.

Fact 2.

when an angle is rotated by odd multiple of (π/2), sin becomes cos, cos becomes sin, tan becomes cot, cot becomes tan, sec becomes cosec, cosec becomes sec.

thus as soon as we come across something like

cos(7π/2 + A), it should come to our mind that it will be sin....

Note: odd multiple of π/2, because even multiple of π/2 will be a multiple of π.

Fact 3:

adding or subtracting an angle by a multiple of 2π does not make any difference.

thus sinA = sin(2π+A) = sin(4π+A) and so on..

If the given angle is too large to decide in which quadrant it falls, we can subtract multiple of 2π so that the resultant angle is less that 2π. and we can decide in which quadrant it falls

e.g. sin 1650° = sin(1650 - 4*360°)

= sin(1650-1440) = sin 210° which is in third quadrant

Fact 4

an angle measured anticlockwise is positive angle whereas an angle measured clockwise is taken as negative angle...

what I mean is

sin (π+A)

both π and A are positive. so π and A are both measured anticlockwise. this will bring the angle in third quadrant.

sin(π - A) means first measure π anticlockwise then measure A clockwise. this will bring the angle in second quadrant

now our problem

cot(4π - A)

since the angle is a multiple of π, cot will remain cot. now, (4π - A) = (4π - A- 2π) = (2π - A).

the angle will fall in fourth quadrant (2π anticlockwise then A clockwise).

In fourth quadrant only cos and sec are positive.

so our solution will be

cot (4π - A ) = -cotA