how is tan(pi/7) equal to cot(5pi/14)?

Question

i couldn't understand this, please explain, thanks

Answer 1

using tan(x) = cot(pi/2-x)

putting x= pi/7 we get tan (pi/7) = cot(pi/2-pi/7) = cot(5pi/14)

Answer 2

If you have a right triangle, and one of the angles is pi/7, then the other angles must be pi/2 and 5pi/14. This way you get pi/7 + 5pi/14 = 2pi/14 + 5pi/14 = 7pi/14 = pi/2.

tan(pi/7) will be the length of the side opposite of the pi/7 angle, divided by the length of the adjacent side. cot(5pi/14) will be the length of the side adjacent to the 5pi/14 angle, divided by the length of the side opposite of this. But these are actually the same two sides used to calculate tan(pi/7). If you draw a right triangle with the angles labeled and with some given names for the legs, it's easy to see why.

mil406's calculations are wrong. If you try comparing results using a calculator, make sure you're calculing in radians.

Answer 3

Because (pi/7) = (pi/2) - (5pi/14).

Draw the picture.

The circular arc from the X axis to pi/7 is the same length as the circular arc from (5pi/14) to the Y axis.

Answer 4

It is two different things!
It won't be equal!!

pi / 7 = 3.142 / 7 = 0.4489
5pi / 14 = (5)(3.142) / 14 = 1.121

tan (pi/7)
= tan 0.4489
= 0.007835

cot (5pi / 14)
= 1 / tan (5pi / 14)
= 1/ tan 1.121
= 1/ 0.01957
= 51.104

Answer 5

Draw a right angled triangle with angles x and y.
Let b be side opposite x
Let a be side opposite y
tan x = b / a
tan y = a / b
cot y = b / a (where y = π/2 - x)
tan x = cot y

Thus tan x = cot (π/2 - x)
tan (π/7) = cot (π/2 - π/7)
tan (π / 7) = cot (7π/14 - 2π/14)
tan (π / 7) = cot (5π / 14)

Answer 6

cot(5pi/14)=cot(pi/2 - 2pi/14)=tg(2pi/14)=tg(pi/7)